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https://feeds.library.caltech.edu/people/Schröder-P/combined.rss
A Caltech Library Repository Feedhttp://www.rssboard.org/rss-specificationpython-feedgenenTue, 16 Apr 2024 15:57:50 +0000The virtual erector set: dynamic simulation with linear recursive constraint propagation
https://resolver.caltech.edu/CaltechAUTHORS:20230210-663927000.2
Authors: {'items': [{'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}, {'id': 'Zeltzer-David', 'name': {'family': 'Zeltzer', 'given': 'David'}}]}
Year: 1990
DOI: 10.1145/91385.91403
We have implemented an algorithm for rigid body dynamics which unifies the advantages of linear recursive algorithms with the advantages of earlier linear algebra based constraint force approaches. No restriction is placed on the joints between links. The algorithm is numerically robust and can deal with arbitrary trees of bodies, including kinematic loops. Motion as well as force constraints on the dynamic behavior of any member of the linkage can be added easily. Through the use of spatial algebra notation---including our extension to account for spatial position---the mathematical expressions are simplified and more efficient to execute. The algorithm has been implemented on workstation class machines and performs at interactive speeds.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/r0h7j-q5w70Interpolating Subdivision for meshes with arbitrary topology
https://resolver.caltech.edu/CaltechAUTHORS:20170110-142159612
Authors: {'items': [{'id': 'Zorin-D', 'name': {'family': 'Zorin', 'given': 'Denis'}}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}, {'id': 'Sweldens-W', 'name': {'family': 'Sweldens', 'given': 'Wim'}}]}
Year: 1996
DOI: 10.1145/237170.237254
Subdivision is a powerful paradigm for the generation of surfaces of arbitrary topology. Given an initial triangular mesh the goal is to produce a smooth and visually pleasing surface whose shape is controlled by the initial mesh. Of particular interest are interpolating
schemes since they match the original data exactly, and play an important role in fast multiresolution and wavelet techniques. Dyn, Gregory, and Levin introduced the Butterfly scheme, which yields C^1 surfaces in the topologically regular setting. Unfortunately it
exhibits undesirable artifacts in the case of an irregular topology. We examine these failures and derive an improved scheme, which retains the simplicity of the Butterfly scheme, is interpolating, and results in smoother surfaces.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/f611x-ncq34Interactive multiresolution mesh editing
https://resolver.caltech.edu/CaltechAUTHORS:20170110-142949739
Authors: {'items': [{'id': 'Zorin-D', 'name': {'family': 'Zorin', 'given': 'Denis'}}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}, {'id': 'Sweldens-W', 'name': {'family': 'Sweldens', 'given': 'Wim'}}]}
Year: 1997
DOI: 10.1145/258734.258863
We describe a multiresolution representation for meshes based on subdivision, which is a natural extension of the existing patch-based surface representations. Combining subdivision and the smoothing algorithms of Taubin [26] allows us to construct a set of algorithms for interactive multiresolution editing of complex hierarchical meshes of arbitrary topology. The simplicity of the underlying algorithms for refinement and coarsification enables us to make them local and adaptive, thereby considerably improving their efficiency. We have built a scalable interactive multiresolution editing system based on such algorithms.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/8pgy8-v9306Multiresolution signal processing for meshes
https://resolver.caltech.edu/CaltechAUTHORS:20161101-170259438
Authors: {'items': [{'id': 'Guskov-I', 'name': {'family': 'Guskov', 'given': 'Igor'}}, {'id': 'Sweldens-W', 'name': {'family': 'Sweldens', 'given': 'Wim'}}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}]}
Year: 1999
DOI: 10.1145/311535.311577
We generalize basic signal processing tools such as downsampling, upsampling, and filters to irregular connectivity triangle meshes. This is accomplished through the design of a non-uniform relaxation procedure whose weights depend on the geometry and we
show its superiority over existing schemes whose weights depend only on connectivity. This is combined with known mesh simplification methods to build subdivision and pyramid algorithms. We demonstrate the power of these algorithms through a number of application examples including smoothing, enhancement, editing, and texture mapping.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/pan4n-v6f04Multiresolution mesh morphing
https://resolver.caltech.edu/CaltechAUTHORS:20161121-171932398
Authors: {'items': [{'id': 'Lee-A-W-F', 'name': {'family': 'Lee', 'given': 'Aaron W. F.'}}, {'id': 'Dobkin-D', 'name': {'family': 'Dobkin', 'given': 'David'}}, {'id': 'Sweldens-W', 'name': {'family': 'Sweldens', 'given': 'Wim'}}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}]}
Year: 1999
DOI: 10.1145/311535.311586
We present a new method for user controlled morphing of two
homeomorphic triangle meshes of arbitrary topology. In particular we focus on the problem of establishing a correspondence map between source and target meshes. Our method employs the MAPS algorithm to parameterize both meshes over simple base domains and an additional harmonic map bringing the latter into correspondence.
To control the mapping the user specifies any number of
feature pairs, which control the parameterizations produced by the MAPS algorithm. Additional controls are provided through a direct manipulation interface allowing the user to tune the mapping between the base domains. We give several examples of æsthetically pleasing morphs which can be created in this manner with little user input. Additionally we demonstrate examples of temporal
and spatial control over the morph.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/nfb5k-mpm37Implicit Fairing of Irregular Meshes using Diffusion and Curvature Flow
https://resolver.caltech.edu/CaltechAUTHORS:20161006-145432340
Authors: {'items': [{'id': 'Desbrun-M', 'name': {'family': 'Desbrun', 'given': 'Mathieu'}, 'orcid': '0000-0003-3424-6079'}, {'id': 'Meyer-M', 'name': {'family': 'Meyer', 'given': 'Mark'}}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}, {'id': 'Barr-A-H', 'name': {'family': 'Barr', 'given': 'Alan H.'}}]}
Year: 1999
DOI: 10.1145/311535.311576
In this paper, we develop methods to rapidly remove rough features from irregularly triangulated data intended to portray a smooth surface. The main task is to remove undesirable noise and uneven edges while retaining desirable geometric features. The problem
arises mainly when creating high-fidelity computer graphics objects using imperfectly-measured data from the real world.
Our approach contains three novel features: an implicit integration method to achieve efficiency, stability, and large time-steps; a scale-dependent Laplacian operator to improve the diffusion process; and finally, a robust curvature flow operator that achieves a smoothing of the shape itself, distinct from any parameterization.
Additional features of the algorithm include automatic exact volume preservation, and hard and soft constraints on the positions of the points in the mesh.
We compare our method to previous operators and related algorithms, and prove that our curvature and Laplacian operators have several mathematically-desirable qualities that improve the appearance of the resulting surface. In consequence, the user can easily select the appropriate operator according to the desired type of fairing.
Finally, we provide a series of examples to graphically and
numerically demonstrate the quality of our results.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/8jp32-h3186Wavelets on irregular point sets
https://resolver.caltech.edu/CaltechAUTHORS:20200929-143508083
Authors: {'items': [{'id': 'Daubechies-I', 'name': {'family': 'Daubechies', 'given': 'Ingrid'}}, {'id': 'Guskov-I', 'name': {'family': 'Guskov', 'given': 'Igor'}}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}, {'id': 'Sweldens-W', 'name': {'family': 'Sweldens', 'given': 'Wim'}}]}
Year: 1999
DOI: 10.1098/rsta.1999.0439
In this article we review techniques for building and analysing wavelets on irregular point sets in one and two dimensions. We discuss current results both on the practical and theoretical side. In particular, we focus on subdivision schemes and commutation rules. Several examples are included.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/0c6gj-h3f35Building your own wavelets at home
https://resolver.caltech.edu/CaltechAUTHORS:20200212-145736556
Authors: {'items': [{'id': 'Sweldens-W', 'name': {'family': 'Sweldens', 'given': 'Wim'}}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}]}
Year: 2000
DOI: 10.1007/bfb0011093
Wavelets have been making an appearance in many pure and applied areas of science and engineering. Computer graphics with its many and varied computational problems has been no exception to this rule. In these notes we will attempt to motivate and explain the basic ideas behind wavelets and what makes them so successful in application areas.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/gsa51-bm390Subdivision surfaces: a new paradigm for thin-shell finite-element analysis
https://resolver.caltech.edu/CaltechAUTHORS:20171213-090454075
Authors: {'items': [{'id': 'Cirak-F', 'name': {'family': 'Cirak', 'given': 'Fehmi'}}, {'id': 'Ortiz-M', 'name': {'family': 'Ortiz', 'given': 'Michael'}, 'orcid': '0000-0001-5877-4824'}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}]}
Year: 2000
DOI: 10.1002/(SICI)1097-0207(20000430)47:12<2039::AID-NME872>3.0.CO;2-1
We develop a new paradigm for thin-shell finite-element analysis based on the use of subdivision surfaces for (i) describing the geometry of the shell in its undeformed configuration, and (ii) generating smooth interpolated displacement fields possessing bounded energy within the strict framework of the Kirchhoff–Love theory of thin shells. The particular subdivision strategy adopted here is Loop's scheme, with extensions such as required to account for creases and displacement boundary conditions. The displacement fields obtained by subdivision are H2 and, consequently, have a finite Kirchhoff–Love energy. The resulting finite elements contain three nodes and element integrals are computed by a one-point quadrature. The displacement field of the shell is interpolated from nodal displacements only. In particular, no nodal rotations are used in the interpolation. The interpolation scheme induced by subdivision is non-local, i.e. the displacement field over one element depend on the nodal displacements of the element nodes and all nodes of immediately neighbouring elements. However, the use of subdivision surfaces ensures that all the local displacement fields thus constructed combine conformingly to define one single limit surface. Numerical tests, including the Belytschko et al. [10] obstacle course of benchmark problems, demonstrate the high accuracy and optimal convergence of the method.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/kw8e6-wxc29Progressive geometry compression
https://resolver.caltech.edu/CaltechAUTHORS:20161116-151004261
Authors: {'items': [{'id': 'Khodakovsky-A', 'name': {'family': 'Khodakovsky', 'given': 'Andrei'}}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}, {'id': 'Sweldens-W', 'name': {'family': 'Sweldens', 'given': 'Wim'}}]}
Year: 2000
DOI: 10.1145/344779.344922
We propose a new progressive compression scheme for arbitrary topology, highly detailed and densely sampled meshes arising from geometry scanning. We observe that meshes consist of three distinct components: geometry, parameter, and connectivity information. The latter two do not contribute to the reduction of error in a compression setting. Using semi-regular meshes, parameter and connectivity information can be virtually eliminated. Coupled with semi-regular wavelet transforms, zerotree coding, and subdivision based reconstruction we see improvements in error by a factor four (12dB) compared to other progressive coding schemes.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/nh6mt-v8v09Normal meshes
https://resolver.caltech.edu/CaltechAUTHORS:20161101-171342645
Authors: {'items': [{'id': 'Guskov-I', 'name': {'family': 'Guskov', 'given': 'Igor'}}, {'id': 'Vidimče-K', 'name': {'family': 'Vidimče', 'given': 'Kiril'}}, {'id': 'Sweldens-W', 'name': {'family': 'Sweldens', 'given': 'Wim'}}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}]}
Year: 2000
DOI: 10.1145/344779.344831
Normal meshes are new fundamental surface descriptions inspired by differential geometry. A normal mesh is a multiresolution mesh where each level can be written as a normal offset from a coarser version. Hence the mesh can be stored with a single float per vertex. We present an algorithm to approximate any surface arbitrarily closely with a normal semi-regular mesh. Normal meshes can be useful in numerous applications such as compression, filtering, rendering, texturing, and modeling.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/adw4q-q6e15Semi-regular mesh extraction from volumes
https://resolver.caltech.edu/CaltechAUTHORS:20170110-140538420
Authors: {'items': [{'id': 'Wood-Z-J', 'name': {'family': 'Wood', 'given': 'Zoë J.'}}, {'id': 'Desbrun-M', 'name': {'family': 'Desbrun', 'given': 'Mathieu'}, 'orcid': '0000-0003-3424-6079'}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}, {'id': 'Breen-D', 'name': {'family': 'Breen', 'given': 'David'}}]}
Year: 2000
DOI: 10.1109/VISUAL.2000.885705
We present a novel method to extract iso-surfaces from distance volumes. It generates high quality semi-regular multiresolution meshes of arbitrary topology. Our technique proceeds in two stages. First, a very coarse mesh with guaranteed topology is extracted. Subsequently an iterative multi-scale force-based solver refines the initial mesh into a semi-regular mesh with geometrically adaptive sampling rate and good aspect ratio triangles. The coarse mesh extraction is performed using a new approach we call surface wavefront propagation. A set of discrete iso-distance ribbons are rapidly built and connected while respecting the topology of the iso-surface implied by the data. Subsequent multi-scale refinement is driven by a simple force-based solver designed to combine good iso-surface fit and high quality sampling through reparameterization. In contrast to the Marching Cubes technique our output meshes adapt gracefully to the iso-surface geometry, have a natural multiresolution structure and good aspect ratio triangles, as demonstrated with a number of examples.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/zkjvh-vp666Curvature integrability of subdivision surfaces
https://resolver.caltech.edu/CaltechAUTHORS:20190829-131533096
Authors: {'items': [{'id': 'Reif-U', 'name': {'family': 'Reif', 'given': 'Ulrich'}, 'orcid': '0000-0001-6015-1722'}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}]}
Year: 2001
DOI: 10.1023/a:1016685104156
We examine the smoothness properties of the principal curvatures of subdivision surfaces near irregular points. In particular we give an estimate of their L_p class based on the eigenstructure of the subdivision matrix. As a result we can show that the popular Loop and Catmull–Clark schemes (among many others) have square integrable principal curvatures enabling their use as shape functions in FEM treatments of the thin shell equations.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/rehnc-60844Surface drawing: creating organic 3D shapes with the hand and tangible tools
https://resolver.caltech.edu/CaltechAUTHORS:20161215-155712856
Authors: {'items': [{'id': 'Schkolne-S', 'name': {'family': 'Schkolne', 'given': 'Steven'}}, {'id': 'Pruett-M', 'name': {'family': 'Pruett', 'given': 'Michael'}}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}]}
Year: 2001
DOI: 10.1145/365024.365114
Surface Drawing is a system for creating organic 3D shapes in a manner which supports the needs and interests of artists. This medium facilitates the early stages of creative design which many 3D modeling programs neglect. Much like traditional media such as line drawing and painting, Surface Drawing lets users construct shapes through repeated marking. In our case, the hand is used to mark 3D space in a semi-immersive virtual environment. The interface is completed with tangible tools to edit and manipulate models. We introduce the use of tongs to move and scale 3D shapes and demonstrate a magnet tool which is comfortably held without restricting hand motion. We evaluated our system through collaboration with artists and designers, exhibition before hundreds of users, our own extensive exploration of the medium, and an informal user study. Response was especially positive from users with an artistic background.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/12r6y-23d65Surface Drawing
https://resolver.caltech.edu/CaltechCSTR:1999.cs-tr-99-03
Authors: {'items': [{'id': 'Schkolne-S', 'name': {'family': 'Schkolne', 'given': 'Steven'}}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}]}
Year: 2001
DOI: 10.7907/Z97H1GKB
We present Surface Drawing, a medium which provides direct control over the creation of a wide range of intricate shapes. Surface Drawing addresses several key issues in creative expression and perceptual thinking by providing a direct link between the motions of the hand and the forging of shapes. Surfaces are created by moving a hand, instrumented with a special glove, through space in a semi-immersive 3D display and interaction environment (the Responsive Workbench). This technique allows both novices and experts to create intricate forms without the perceptual constraints of a rigid mathematical structure, large toolset, or a reduction of modeling to editing. In Surface Drawing the design space can be freely explored during the modeling process without the need to plan the construction of the final shape. In particular it supports unconstrained erasing and buildup of new geometry. This is achieved through the use of a novel incremental construction method for triangulated meshes, the Cookie Cutter algorithm. It allows the user to freely grow, join, and erase surfaces based on hand motions. We report on our experiences with the system and present results created by artists and designers exploring problems in industrial design, character design, and fine art.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/h1dxp-kfv03Interpolating Subdivision for Meshes of Arbitary Topology
https://resolver.caltech.edu/CaltechCSTR:1996.cs-tr-96-06
Authors: {'items': [{'id': 'Zorin-D', 'name': {'family': 'Zorin', 'given': 'Denis'}}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}, {'id': 'Sweldens-W', 'name': {'family': 'Sweldens', 'given': 'Wim'}}]}
Year: 2001
DOI: 10.7907/Z93B5X5W
Subdivision is a powerful paradigm for the generation of surfaces of arbitrary topology. Given an initial triangular mesh the goal is to produce a smooth and visually pleasing surface whose shape is controlled by the initial mesh. Of particular interest are interpolating schemes since they match the original data exactly, and are crucial for fast mutliresolution and wavelet techniques. Dyn, Gregory, and Levin introduced the Butterfly scheme [17], which yields C1 surfaces in the topologically regular setting. Unfortunately it exhibits undesirable artifacts in the case of an irregular topology. We examine these failures and derive an improved scheme, which retains the simplicity of the Butterfly scheme, is interpolating, and results in smoother surfaces.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/3bk4f-8wy57Subdivision, multiresolution and the construction of scalable algorithms in computer graphics
https://resolver.caltech.edu/CaltechAUTHORS:20230210-140523000.1
Authors: {'items': [{'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'P.'}, 'orcid': '0000-0002-0323-7674'}]}
Year: 2001
DOI: 10.1017/cbo9780511569616.009
Multiresolution representations are a critical tool in addressing complexity issues (time and memory) for the large scenes typically found in computer graphics applications. Many of these techniques are based on classical subdivision techniques and their generalizations. In this chapter we review two exemplary applications from this area, multiresolution surface editing and semi-regular remeshing. The former is directed towards building algorithms which are fast enough for interactive manipulation of complex surfaces of arbitrary topology. The latter is concerned with constructing smooth parameterizations for arbitrary topology surfaces as they typically arise from 3D scanning techniques. Remeshing such surfaces then allows the use of classical subdivision ideas. We focus in particular on the practical aspects of making the well-understood mathematical machinery applicable and accessible to the very general settings encountered in practice.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/qxzrv-j6944Normal bounds for subdivision-surface interference detection
https://resolver.caltech.edu/CaltechAUTHORS:20161031-164830677
Authors: {'items': [{'id': 'Grinspun-Eitan', 'name': {'family': 'Grinspun', 'given': 'Eitan'}}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}]}
Year: 2001
DOI: 10.1109/VISUAL.2001.964529
Subdivision surfaces are an attractive representation when modeling arbitrary-topology free-form surfaces and show great promise for applications in engineering design and computer animation. Interference detection is a critical tool in many of these applications. In this paper, we derive normal bounds for subdivision surfaces and use these to develop an efficient algorithm for (self-) interference detection.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/29wyc-py283Rapid Evaluation of Catmull-Clark Subdivision Surfaces
https://resolver.caltech.edu/CaltechAUTHORS:20160809-163838344
Authors: {'items': [{'id': 'Bolz-J', 'name': {'family': 'Bolz', 'given': 'Jeffrey'}}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}]}
Year: 2002
DOI: 10.1145/504502.504505
Using subdivision as a basic primitive for the construction of arbitrary topology, smooth, free-form surfaces is attractive for content destined for display on devices with greatly varying rendering performance. Subdivision naturally supports level of detail rendering and powerful compression algorithms. While the underlying algorithms are conceptually simple it is difficult to implement player engines which achieve optimal performance on modern CPUs such as the Intel Pentium family.
In this paper we describe a novel table driven evaluation strategy for subdivision surfaces using as an example the scheme of Catmull and Clark. Cache conscious design and exploitation of SIMD instructions allows us to achieve nearly 100% FPU utilization in the inner loop and achieve a composite performance of 1.2 flop/cycle on the Intel PIII and 1.8 flop/cycle on the Intel P4 including all memory transfers. The algorithm supports tradeoffs between cache size and memory bus usage which we examine. A library which implements this engine is freely available from the authors.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/avvac-b2z04Integrated modeling, finite-element analysis, and engineering design for thin-shell structures using subdivision
https://resolver.caltech.edu/CaltechAUTHORS:20171208-164357957
Authors: {'items': [{'id': 'Cirak-F', 'name': {'family': 'Cirak', 'given': 'Fehmi'}}, {'id': 'Scott-M-J', 'name': {'family': 'Scott', 'given': 'Michael J.'}}, {'id': 'Antonsson-E-K', 'name': {'family': 'Antonsson', 'given': 'Erik K.'}}, {'id': 'Ortiz-M', 'name': {'family': 'Ortiz', 'given': 'Michael'}, 'orcid': '0000-0001-5877-4824'}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}]}
Year: 2002
DOI: 10.1016/S0010-4485(01)00061-6
Many engineering design applications require geometric modeling and mechanical simulation of thin flexible structures, such as those found in the automotive and aerospace industries. Traditionally, geometric modeling, mechanical simulation, and engineering design are treated as separate modules requiring different methods and representations. Due to the incompatibility of the involved representations the transition from geometric modeling to mechanical simulation, as well as in the opposite direction, requires substantial effort. However, for engineering design purposes efficient transition between geometric modeling and mechanical simulation is essential. We propose the use of subdivision surfaces as a common foundation for modeling, simulation, and design in a unified framework. Subdivision surfaces provide a flexible and efficient tool for arbitrary topology free-form surface modeling, avoiding many of the problems inherent in traditional spline patch based approaches. The underlying basis functions are also ideally suited for a finite-element treatment of the so-called thin-shell equations, which describe the mechanical behavior of the modeled structures. The resulting solvers are highly scalable, providing an efficient computational foundation for design exploration and optimization. We demonstrate our claims with several design examples, showing the versatility and high accuracy of the proposed method.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/drcyt-3y668Near-Optimal Connectivity Encoding of 2-Manifold Polygon Meshes
https://resolver.caltech.edu/CaltechAUTHORS:20230210-546014000.1
Authors: {'items': [{'id': 'Khodakovsky-Andrei', 'name': {'family': 'Khodakovsky', 'given': 'Andrei'}}, {'id': 'Alliez-Pierre', 'name': {'family': 'Alliez', 'given': 'Pierre'}}, {'id': 'Desbrun-M', 'name': {'family': 'Desbrun', 'given': 'Mathieu'}, 'orcid': '0000-0003-3424-6079'}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}]}
Year: 2002
DOI: 10.1006/gmod.2002.0575
Encoders for triangle mesh connectivity based on enumeration of vertex valences are among the best reported to date. They are both simple to implement and report the best compressed file sizes for a large corpus of test models. Additionally they have recently been shown to be near-optimal since they realize the Tutte entropy bound for all planar triangulations. In this paper we introduce a connectivity encoding method which extends these ideas to 2-manifold meshes consisting of faces with arbitrary degree. The encoding algorithm exploits duality by applying valence enumeration to both the primal and the dual mesh in a symmetric fashion. It generates two sequences of symbols, vertex valences, and face degrees, and encodes them separately using two context-based arithmetic coders. This allows us to exploit vertex or face regularity if present. When the mesh exhibits perfect face regularity (e.g., a pure triangle or quad mesh) or perfect vertex regularity (valence six or four respectively) the corresponding bit rate vanishes to zero asymptotically. For triangle meshes, our technique is equivalent to earlier valence-driven approaches. We report compression results for a corpus of standard meshes. In all cases we are able to show coding gains over earlier coders, sometimes as large as 50%. Remarkably, we even slightly gain over coders specialized to triangle or quad meshes. A theoretical analysis reveals that our approach is near-optimal as we achieve the Tutte entropy bound for arbitrary planar graphs of two bits per edge in the worst case.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/87x5k-hv089Assembling and Rearranging Digital Objects in Physical Space with Tongs, a Gluegun, and a Lightsaber
https://resolver.caltech.edu/CaltechCSTR:2002.005
Authors: {'items': [{'id': 'Schkolne-S', 'name': {'family': 'Schkolne', 'given': 'Steven'}}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}]}
Year: 2002
DOI: 10.7907/Z9PC30C8
We present an interface for the arrangement of objects in three-dimensional space. Physical motions of the user are mapped to interface commands through tangible props. Tongs move objects freely, a gluegun binds objects together, and a lightsaber breaks these bonds. The experimental interface is implemented on the Responsive Workbench, a semi-immersive 3D computer. We conducted a small user study comparing our approach with the 2D interface of Maya. The results suggest that our system is much faster than Maya for object assembly. Users qualitatively found the system to be far more intuitive than the monitor-based alternative.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/bgr95-tr082Hybrid Meshes: Multiresolution using regular and irregular refinement
https://resolver.caltech.edu/CaltechAUTHORS:20161031-172646085
Authors: {'items': [{'id': 'Guskov-I', 'name': {'family': 'Guskov', 'given': 'Igor'}}, {'id': 'Khodakovsky-A', 'name': {'family': 'Khodakovsky', 'given': 'Andrei'}}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}, {'id': 'Sweldens-W', 'name': {'family': 'Sweldens', 'given': 'Wim'}}]}
Year: 2002
DOI: 10.1145/513400.513443
A hybrid mesh is a multiresolution surface representation that combines advantages from regular and irregular meshes. Irregular operations allow a hybrid mesh to change topology throughout the hierarchy and approximate detailed features at multiple scales. A preponderance of regular refinements allows for efficient data-structures and processing algorithms. We provide a user driven procedure for creating a hybrid mesh from scanned geometry and present a progressive hybrid mesh compression algorithm.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/p3qje-mxq48CHARMS: a simple framework for adaptive simulation
https://resolver.caltech.edu/CaltechAUTHORS:20161031-164831941
Authors: {'items': [{'id': 'Grinspun-Eitan', 'name': {'family': 'Grinspun', 'given': 'Eitan'}}, {'id': 'Krysl-Petr', 'name': {'family': 'Krysl', 'given': 'Petr'}}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}]}
Year: 2002
DOI: 10.1145/566570.566578
Finite element solvers are a basic component of simulation applications; they are common in computer graphics, engineering, and medical simulations. Although adaptive solvers can be of great value in reducing the often high computational cost of simulations they are not employed broadly. Indeed, building adaptive solvers can be a daunting task especially for 3D finite elements. In this paper we are introducing a new approach to produce conforming, hierarchical, adaptive refinement methods (CHARMS). The basic principle of our approach is to refine basis functions, not elements. This removes a number of implementation headaches associated with other approaches and is a general technique independent of domain dimension (here 2D and 3D), element type (e.g., triangle, quad, tetrahedron, hexahedron), and basis function order (piece-wise linear, higher order B-splines, Loop subdivision, etc.). The (un-)refinement algorithms are simple and require little in terms of data structure support. We demonstrate the versatility of our new approach through 2D and 3D examples, including medical applications and thin-shell animations.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/n3p6m-3wf28CHARMS: A Simple Framework for Adaptive Simulation
https://resolver.caltech.edu/CaltechAUTHORS:20111102-153618063
Authors: {'items': [{'id': 'Grinspun-Eitan', 'name': {'family': 'Grinspun', 'given': 'Eitan'}}, {'id': 'Krysl-Petr', 'name': {'family': 'Krysl', 'given': 'Petr'}}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}]}
Year: 2002
DOI: 10.1145/566654.566578
Finite element solvers are a basic component of simulation applications; they are common in computer graphics, engineering, and medical simulations. Although adaptive solvers can be of great value in reducing the often high computational cost of simulations they are not employed broadly. Indeed, building adaptive solvers can be a daunting task especially for 3D finite elements. In this paper we are introducing a new approach to produce conforming, hierarchical, adaptive refinement methods (CHARMS). The basic principle of our approach is to refine basis functions, not elements. This removes a number of implementation headaches associated with other approaches and is a general technique independent of domain dimension (here 2D and 3D), element type (e.g., triangle, quad, tetrahedron, hexahedron), and basis function order (piece-wise linear, higher order B-splines, Loop subdivision, etc.). The (un-)refinement algorithms are simple and require little in terms of data structure support. We demonstrate the versatility of our new approach through 2D and 3D examples, including medical applications and thin-shell animations.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/vb3f5-bkp29Subdivision as a fundamental building block of digital geometry processing algorithms
https://resolver.caltech.edu/CaltechAUTHORS:20230210-376243000.1
Authors: {'items': [{'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}]}
Year: 2002
DOI: 10.1016/s0377-0427(02)00530-7
Digital Geometry Processing (DGP) is a newly emerging discipline which aims to build the mathematical and algorithmic foundations for the next generation of digital multimedia. Digital signal processing was highly successful for earlier generations of digital multimedia such as sound, image, and video. Now that 3D scanning technologies make sampled surfaces broadly available a similar apparatus is needed to deal with this new multimedia datatype. Due to the inherent curvature of such surfaces though traditional tools are not immediately applicable. In this paper we give a brief overview of some recent developments in subdivision surfaces and how these help to build a foundation for DGP algorithms.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/16y10-bw273Tangible + Virtual = A Flexible 3D Interface for Spatial Construction Applied to DNA
https://resolver.caltech.edu/CaltechAUTHORS:20230210-153275000.2
Authors: {'items': [{'id': 'Schkolne-Steven', 'name': {'family': 'Schkolne', 'given': 'Steven'}}, {'id': 'Ishii-Hiroshi', 'name': {'family': 'Ishii', 'given': 'Hiroshi'}}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}]}
Year: 2003
Ideas from tangible interfaces and VR ease a difficult spatial design task: construct a DNA molecule with desired characteristics. Our hybrid interface has both the physical intimacy of tangible media and the versatility of 3D digital display. Two new physical affordances: a raygun and a grip tool, enable kinesthetic control of the addition and removal of structure. We introduce 3D local menus which select multiple functions for each tool. New interactions for sensed tongs enable the sophisticated multi-object arrangement that the delicate, intricate DNA construction task demands. These flexible tools allow UI designers to create multiple interfaces upon the same physical substrate. In a user study, practicing research scientists expressed a strong preference for Silkworm, our 3D interface, when compared to mouse/monitor UI. We show that 3D tangible interfaces, heretofore only applied to freeform artistic creation, also facilitate intuition in the highly structured task that is our focus.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/j2sjq-vxf82Discrete Differential-Geometry Operators for Triangulated 2-Manifolds
https://resolver.caltech.edu/CaltechAUTHORS:20191009-101951616
Authors: {'items': [{'id': 'Meyer-Mark', 'name': {'family': 'Meyer', 'given': 'Mark'}}, {'id': 'Desbrun-M', 'name': {'family': 'Desbrun', 'given': 'Mathieu'}, 'orcid': '0000-0003-3424-6079'}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}, {'id': 'Barr-A-H', 'name': {'family': 'Barr', 'given': 'Alan H.'}}]}
Year: 2003
DOI: 10.1007/978-3-662-05105-4_2
This paper proposes a unified and consistent set of flexible tools to approximate important geometric attributes, including normal vectors and curvatures on arbitrary triangle meshes. We present a consistent derivation of these first and second order differential properties using averaging Voronoi cells and the mixed Finite-Element/Finite-Volume method, and compare them to existing formulations. Building upon previous work in discrete geometry, these operators are closely related to the continuous case, guaranteeing an appropriate extension from the continuous to the discrete setting: they respect most intrinsic properties of the continuous differential operators. We show that these estimates are optimal in accuracy under mild smoothness conditions, and demonstrate their numerical quality. We also present applications of these operators, such as mesh smoothing, enhancement, and quality checking, and show results of denoising in higher dimensions, such as for tensor images.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/z23e7-31x10Data-dependent fairing of subdivision surfaces
https://resolver.caltech.edu/CaltechAUTHORS:20161024-173523014
Authors: {'items': [{'id': 'Friedel-I', 'name': {'family': 'Friedel', 'given': 'Ilja'}}, {'id': 'Mullen-P', 'name': {'family': 'Mullen', 'given': 'Patrick'}}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}]}
Year: 2003
DOI: 10.1145/781606.781635
In this paper we present a new algorithm for solving the data dependent fairing problem for subdivision surfaces, using Catmull-Clark surfaces as an example. Earlier approaches to subdivision surface fairing encountered problems with singularities in the parametrization of the surface. We address these issues through the use of the characteristic map parametrization, leading to well defined membrane and bending energies even at irregular vertices. Combining this approach with ideas from data-dependent energy operators we are able to express the associated nonlinear stiffness matrices for Catmull-Clark surfaces as linear combinations of precomputed energy matrices. This machinery also provides exact, inexpensive gradients and Hessians of the new energy operators. With these the nonlinear minimization problem can be solved in a stable and efficient way using Steihaug's Newton/CG trust-region method. We compare properties of linear and nonlinear methods through a number of examples and report on the performance of the algorithm.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/61t5m-4f904Composite primal/dual √3-subdivision schemes
https://resolver.caltech.edu/CaltechAUTHORS:20170408-163038224
Authors: {'items': [{'id': 'Oswald-Peter', 'name': {'family': 'Oswald', 'given': 'Peter'}}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}]}
Year: 2003
DOI: 10.1016/S0167-8396(03)00026-8
We present new families of primal and dual subdivision schemes for triangle meshes and 3-refinement. The proposed schemes use two simple local rules which cycle between primal and dual meshes a number of times. The resulting surfaces become very smooth at regular vertices if the number of cycles is ⩾2. The C^1-property is violated only at low-valence irregular vertices, and can be restored by slight modifications of the local rules used.
As a generalization, we introduce a wide class of composite subdivision schemes suitable for arbitrary topologies and refinement rules. A composite scheme is defined by a simple upsampling from the coarse to a refined topology, embedded into a cascade of geometric averaging operators acting on coarse and/or refined topologies. We propose a small set of such averaging rules (and some of their parametric extensions) which allow for the switching between control nets associated with the same or different topologic elements (vertices, edges, faces), and show a number of examples, based on triangles, that the resulting class of composite subdivision schemes contains new and old, primal and dual schemes for 3-refinement as well as for quadrisection. As a common observation from the examples considered, we found that irregular vertex treatment is necessary only at vertices of low valence, and can easily be implemented by using generic modifications of some elementary averaging rules.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/tgc6t-z8091Sparse Matrix Solvers on the GPU: Conjugate Gradients and Multigrid
https://resolver.caltech.edu/CaltechAUTHORS:20160809-162351341
Authors: {'items': [{'id': 'Bolz-Jeff', 'name': {'family': 'Bolz', 'given': 'Jeff'}}, {'id': 'Farmer-Ian', 'name': {'family': 'Farmer', 'given': 'Ian'}}, {'id': 'Grinspun-Eitan', 'name': {'family': 'Grinspun', 'given': 'Eitan'}}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}]}
Year: 2003
DOI: 10.1145/1201775.882364
Many computer graphics applications require high-intensity numerical simulation. We show that such computations can be performed efficiently on the GPU, which we regard as a full function streaming processor with high floating-point performance. We implemented two basic, broadly useful, computational kernels: a sparse matrix conjugate gradient solver and a regular-grid multigrid solver. Real-time applications ranging from mesh smoothing and parameterization to fluid solvers and solid mechanics can greatly benefit from these, evidence our example applications of geometric flow and fluid simulation running on NVIDIA's GeForce FX.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/j76nk-kg240Sparse Matrix Solvers on the GPU: Conjugate Gradients and Multigrid
https://resolver.caltech.edu/CaltechAUTHORS:20160809-162211967
Authors: {'items': [{'id': 'Bolz-Jeff', 'name': {'family': 'Bolz', 'given': 'Jeff'}}, {'id': 'Farmer-Ian', 'name': {'family': 'Farmer', 'given': 'Ian'}}, {'id': 'Grinspun-Eitan', 'name': {'family': 'Grinspun', 'given': 'Eitan'}}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}]}
Year: 2003
DOI: 10.1145/882262.882364
Many computer graphics applications require high-intensity numerical simulation. We show that such computations can be performed efficiently on the GPU, which we regard as a full function streaming processor with high floating-point performance. We implemented two basic, broadly useful, computational kernels: a sparse matrix conjugate gradient solver and a regular-grid multigrid solver. Real-time applications ranging from mesh smoothing and parameterization to fluid solvers and solid mechanics can greatly benefit from these, evidence our example applications of geometric flow and fluid simulation running on NVIDIA's GeForce FX.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/5whfr-hf924Progressive encoding of complex isosurfaces
https://resolver.caltech.edu/CaltechAUTHORS:20161121-173322261
Authors: {'items': [{'id': 'Lee-Haeyoung', 'name': {'family': 'Lee', 'given': 'Haeyoung'}}, {'id': 'Desbrun-M', 'name': {'family': 'Desbrun', 'given': 'Mathieu'}, 'orcid': '0000-0003-3424-6079'}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}]}
Year: 2003
DOI: 10.1145/1201775.882294
We present a progressive encoding technique specifically designed for complex isosurfaces. It achieves better rate distortion performance than all standard mesh coders, and even improves on all previous single rate isosurface coders. Our novel algorithm handles isosurfaces with or without sharp features, and deals gracefully with high topologic and geometric complexity. The inside/outside function of the volume data is progressively transmitted through the use of an adaptive octree, while a local frame based encoding is used for the fine level placement of surface samples. Local patterns in topology and local smoothness in geometry are exploited by context-based arithmetic encoding, allowing us to achieve an average of 6.10 bits per vertex (b/v) at very low distortion. Of this rate only 0.65 b/v are dedicated to connectivity data: this improves by 24% over the best previous single rate isosurface encoder.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/7j7gt-q9k05Progressive encoding of complex isosurfaces
https://resolver.caltech.edu/CaltechAUTHORS:20161121-173902493
Authors: {'items': [{'id': 'Lee-Haeyoung', 'name': {'family': 'Lee', 'given': 'Haeyoung'}}, {'id': 'Desbrun-M', 'name': {'family': 'Desbrun', 'given': 'Mathieu'}, 'orcid': '0000-0003-3424-6079'}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}]}
Year: 2003
DOI: 10.1145/882262.882294
We present a progressive encoding technique specifically designed for complex isosurfaces. It achieves better rate distortion performance than all standard mesh coders, and even improves on all previous single rate isosurface coders. Our novel algorithm handles isosurfaces with or without sharp features, and deals gracefully with high topologic and geometric complexity. The inside/outside function of the volume data is progressively transmitted through the use of an adaptive octree, while a local frame based encoding is used for the fine level placement of surface samples. Local patterns in topology and local smoothness in geometry are exploited by context-based arithmetic encoding, allowing us to achieve an average of 6.10 bits per vertex (b/v) at very low distortion. Of this rate only 0.65 b/v are dedicated to connectivity data: this improves by 24% over the best previous single rate isosurface encoder.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/66g1z-srz51Globally smooth parameterizations with low distortion
https://resolver.caltech.edu/CaltechAUTHORS:20161116-142436970
Authors: {'items': [{'id': 'Khodakovsky-A', 'name': {'family': 'Khodakovsky', 'given': 'Andrei'}}, {'id': 'Litke-N', 'name': {'family': 'Litke', 'given': 'Nathan'}}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}]}
Year: 2003
DOI: 10.1145/1201775.882275
Good parameterizations are of central importance in many digital geometry processing tasks. Typically the behavior of such processing algorithms is related to the smoothness of the parameterization and how much distortion it contains. Since a parameterization maps a bounded region of the plane to the surface, a parameterization for a surface which is not homeomorphic to a disc must be made up of multiple pieces. We present a novel parameterization algorithm for arbitrary topology surface meshes which computes a globally smooth parameterization with low distortion. We optimize the patch layout subject to criteria such as shape quality and metric distortion, which are used to steer a mesh simplification approach for base complex construction. Global smoothness is achieved through simultaneous relaxation over all patches, with suitable transition functions between patches incorporated into the relaxation procedure. We demonstrate the quality of our parameterizations through numerical evaluation of distortion measures and the excellent rate distortion performance of semi-regular remeshes produced with these parameterizations. The numerical algorithms required to compute the parameterizations are robust and run on the order of minutes even for large meshes.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/y3x49-v7a86Globally smooth parameterizations with low distortion
https://resolver.caltech.edu/CaltechAUTHORS:20161116-144000826
Authors: {'items': [{'id': 'Khodakovsky-A', 'name': {'family': 'Khodakovsky', 'given': 'Andrei'}}, {'id': 'Litke-N', 'name': {'family': 'Litke', 'given': 'Nathan'}}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}]}
Year: 2003
DOI: 10.1145/882262.882275
Good parameterizations are of central importance in many digital geometry processing tasks. Typically the behavior of such processing algorithms is related to the smoothness of the parameterization and how much distortion it contains. Since a parameterization maps a bounded region of the plane to the surface, a parameterization for a surface which is not homeomorphic to a disc must be made up of multiple pieces. We present a novel parameterization algorithm for arbitrary topology surface meshes which computes a globally smooth parameterization with low distortion. We optimize the patch layout subject to criteria such as shape quality and metric distortion, which are used to steer a mesh simplification approach for base complex construction. Global smoothness is achieved through simultaneous relaxation over all patches, with suitable transition functions between patches incorporated into the relaxation procedure. We demonstrate the quality of our parameterizations through numerical evaluation of distortion measures and the excellent rate distortion performance of semi-regular remeshes produced with these parameterizations. The numerical algorithms required to compute the parameterizations are robust and run on the order of minutes even for large meshes.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/fvnnm-whq77Discrete shells
https://resolver.caltech.edu/CaltechAUTHORS:20161031-163604983
Authors: {'items': [{'id': 'Grinspun-E', 'name': {'family': 'Grinspun', 'given': 'Eitan'}}, {'id': 'Hirani-A-N', 'name': {'family': 'Hirani', 'given': 'Anil N.'}}, {'id': 'Desbrun-M', 'name': {'family': 'Desbrun', 'given': 'Mathieu'}, 'orcid': '0000-0003-3424-6079'}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}]}
Year: 2003
DOI: 10.2312/SCA03/062-067
In this paper we introduce a discrete shell model describing the behavior of thin flexible structures, such as hats, leaves, and aluminum cans, which are characterized by a curved undeformed configuration. Previously such models required complex continuum mechanics formulations and correspondingly complex algorithms. We show that a simple shell model can be derived geometrically for triangle meshes and implemented quickly by modifying a standard cloth simulator. Our technique convincingly simulates a variety of curved objects with materials ranging from paper to metal, as we demonstrate with several examples including a comparison of a real and simulated falling hat.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/nagsj-q9097Removing excess topology from isosurfaces
https://resolver.caltech.edu/CaltechAUTHORS:20170110-135149738
Authors: {'items': [{'id': 'Wood-Z', 'name': {'family': 'Wood', 'given': 'Zoë'}}, {'id': 'Hoppe-H', 'name': {'family': 'Hoppe', 'given': 'Hugues'}}, {'id': 'Desbrun-M', 'name': {'family': 'Desbrun', 'given': 'Mathieu'}, 'orcid': '0000-0003-3424-6079'}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}]}
Year: 2004
DOI: 10.1145/990002.990007
Many high-resolution surfaces are created through isosurface extraction from volumetric representations, obtained by 3D photography, CT, or MRI. Noise inherent in the acquisition process can lead to geometrical and topological errors. Reducing geometrical errors during reconstruction is well studied. However, isosurfaces often contain many topological errors in the form of tiny handles. These nearly invisible artifacts hinder subsequent operations like mesh simplification, remeshing, and parametrization. In this article we present a practical method for removing handles in an isosurface. Our algorithm makes an axis-aligned sweep through the volume to locate handles, compute their sizes, and selectively remove them. The algorithm is designed to facilitate out-of-core execution. It finds the handles by incrementally constructing and analyzing a Reeb graph. The size of a handle is measured by a short nonseparating cycle. Handles are removed robustly by modifying the volume rather than attempting "mesh surgery." Finally, the volumetric modifications are spatially localized to preserve geometrical detail. We demonstrate topology simplification on several complex models, and show its benefits for subsequent surface processing.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/w5j4z-mk277Variational normal meshes
https://resolver.caltech.edu/CaltechAUTHORS:20161024-174513528
Authors: {'items': [{'id': 'Friedel-I', 'name': {'family': 'Friedel', 'given': 'Ilja'}}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}, {'id': 'Khodakovsky-A', 'name': {'family': 'Khodakovsky', 'given': 'Andrei'}}]}
Year: 2004
DOI: 10.1145/1027411.1027418
Hierarchical representations of surfaces have many advantages for digital geometry processing applications. Normal meshes are particularly attractive since their level-to-level displacements are in the local normal direction only. Consequently, they only require scalar coefficients to specify. In this article, we propose a novel method to approximate a given mesh with a normal mesh. Instead of building an associated parameterization on the fly, we assume a globally smooth parameterization at the beginning and cast the problem as one of perturbing this parameterization. Controlling the magnitude of this perturbation gives us explicit control over the range between fully constrained (only scalar coefficients) and unconstrained (3-vector coefficients) approximations. With the unconstrained problem giving the lowest approximation error, we can thus characterize the error cost of normal meshes as a function of the number of nonnormal offsets---we find a significant gain for little (error) cost. Because the normal mesh construction creates a geometry driven approximation, we can replace the difficult geometric distance minimization problem with a much simpler least squares problem. This variational approach reduces magnitude and structure (aliasing) of the error further. Our method separates the parameterization construction into an initial setup followed only by subsequent perturbations, giving us an algorithm which is far simpler to implement, more robust, and significantly faster.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/78cpf-b3k30Immersive design of DNA molecules with a tangible interface
https://resolver.caltech.edu/CaltechAUTHORS:20230210-663941000.4
Authors: {'items': [{'id': 'Schkolne-Steven', 'name': {'family': 'Schkolne', 'given': 'Steven'}}, {'id': 'Ishii-Hiroshi', 'name': {'family': 'Ishii', 'given': 'Hiroshi'}}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}]}
Year: 2004
DOI: 10.1109/visual.2004.47
This paper presents an experimental immersive interface for designing DNA components for application in nanotechnology. While much research has been done on immersive visualization, this is one of the first systems to apply advanced interface techniques to a scientific design problem. This system uses tangible 3D input devices (tongs, a raygun, and a multipurpose handle tool) to create and edit a purely digital representation of DNS. The tangible controllers are associated with functions (not data) while a virtual display is used to render the model. This interface was built in collaboration with a research group investigating the design of DNA tiles. A user study shows that scientists find the immersive interface more satisfying than a 2D interface due to the enhanced understanding gained by directly interacting with molecules in 3D space.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/0x6d3-g4132Immersive design of DMA molecules with a tangible interface
https://resolver.caltech.edu/CaltechAUTHORS:20161215-153951824
Authors: {'items': [{'id': 'Schkolne-S', 'name': {'family': 'Schkolne', 'given': 'Steven'}}, {'id': 'Ishii-Hiroshi', 'name': {'family': 'Ishii', 'given': 'Hiroshi'}}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}]}
Year: 2004
DOI: 10.1109/VISUAL.2004.47
This work presents an experimental immersive interface for designing DNA components for application in nanotechnology. While much research has been done on immersive visualization, this is one of the first systems to apply advanced interface techniques to a scientific design problem. This system uses tangible 3D input devices (tongs, a raygun, and a multipurpose handle tool) to create and edit a purely digital representation of DNA. The tangible controllers are associated with functions (not data) while a virtual display is used to render the model. This interface was built in collaboration with a research group investigating the design of DNA tiles. A user study shows that scientists find the immersive interface more satisfying than a 2D interface due to the enhanced understanding gained by directly interacting with molecules in 3D space.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/azba5-9bf97Multilevel Solvers for Unstructured Surface Meshes
https://resolver.caltech.edu/CaltechAUTHORS:AKSsiamjsc05
Authors: {'items': [{'id': 'Aksoylu-B', 'name': {'family': 'Aksoylu', 'given': 'Burak'}}, {'id': 'Khodakovsky-A', 'name': {'family': 'Khodakovsky', 'given': 'Andrei'}}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}]}
Year: 2005
DOI: 10.1137/S1064827503430138
Parameterization of unstructured surface meshes is of fundamental importance in many applications of digital geometry processing. Such parameterization approaches give rise to large and exceedingly ill-conditioned systems which are difficult or impossible to solve without the use of sophisticated multilevel preconditioning strategies. Since the underlying meshes are very fine to begin with, such multilevel preconditioners require mesh coarsening to build an appropriate hierarchy. In this paper we consider several strategies for the construction of hierarchies using ideas from mesh simplification algorithms used in the computer graphics literature. We introduce two novel hierarchy construction schemes and demonstrate their superior performance when used in conjunction with a multigrid preconditioner.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/6161n-hjf23Discrete Willmore flow
https://resolver.caltech.edu/CaltechAUTHORS:20160725-115639350
Authors: {'items': [{'id': 'Bobenko-A-I', 'name': {'family': 'Bobenko', 'given': 'Alexander I.'}}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}]}
Year: 2005
The Willmore energy of a surface, ∫(H^2 -- K) dA, as a function of mean and Gaussian curvature, captures the deviation of a surface from (local) sphericity. As such this energy and its associated gradient flow play an important role in digital geometry processing, geometric modeling, and physical simulation. In this paper we consider a discrete Willmore energy and its flow. In contrast to traditional approaches it is not based on a finite element discretization, but rather on an ab initio discrete formulation which preserves the Möbius symmetries of the underlying continuous theory in the discrete setting. We derive the relevant gradient expressions including a linearization (approximation of the Hessian), which are required for non-linear numerical solvers. As examples we demonstrate the utility of our approach for surface restoration, n-sided hole filling, and non-shrinking surface smoothing.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/rjf0f-x9266Discrete Willmore Flow
https://resolver.caltech.edu/CaltechAUTHORS:20160725-114846575
Authors: {'items': [{'id': 'Bobenko-A-I', 'name': {'family': 'Bobenko', 'given': 'Alexander I.'}}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}]}
Year: 2005
DOI: 10.1145/1198555.1198664
The Willmore energy of a surface, ∫(H^2 - K) dA, as a function of mean and Gaussian curvature, captures the deviation of a surface from (local) sphericity. As such this energy and its associated gradient flow play an important role in digital geometry processing, geometric modeling, and physical simulation. In this paper we consider a discrete Willmore energy and its flow. In contrast to traditional approaches it is not based on a finite element discretization, but rather on an ab initio discrete formulation which preserves the Möbius symmetries of the underlying continuous theory in the discrete setting. We derive the relevant gradient expressions including a linearization (approximation of the Hessian), which are required for non-linear numerical solvers. As examples we demonstrate the utility of our approach for surface restoration, n-sided hole filling, and non-shrinking surface smoothing.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/0j26w-82a57What can we measure?
https://resolver.caltech.edu/CaltechAUTHORS:20161215-161533538
Authors: {'items': [{'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}]}
Year: 2005
DOI: 10.1145/1198555.1198661
When characterizing a shape or changes in shape we must first ask, what can we measure about a shape? For example, for a region in ℝ^3 we may ask for its volume or its surface area. If the object at hand undergoes deformation due to forces acting on it we may need to formulate the laws governing the change in shape in terms of measurable quantities and their change over time. Usually such measurable quantities for a shape are defined with the help of integral calculus and often require some amount of smoothness on the object to be well defined. In this chapter we will take a more abstract approach to the question of measurable quantities which will allow us to define notions such as mean curvature integrals and the curvature tensor for piecewise linear meshes without having to worry about the meaning of second derivatives in settings in which they do not exist. In fact in this chapter we will give an account of a classical result due to Hadwiger, which shows that for a convex, compact set in ℝ^n there are only n + 1 unique measurements if we require that the measurements be invariant under Euclidian motions (and satisfy certain "sanity" conditions). We will see how these measurements are constructed in a very straightforward and elementary manner and that they can be read off from a characteristic polynomial due to Steiner. This polynomial describes the volume of a family of shapes which arise when we "grow" a given shape. As a practical tool arising from these consideration we will see that there is a well defined notion of the curvature tensor for piece-wise linear meshes and we will see very simple formulas for quantities needed in physical simulation with piecewise linear meshes. Much of the treatment here will initially be limited to convex bodies to keep things simple. This limitation that will be removed at the very end.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/hsbr1-mq960Unconstrained spherical parameterization
https://resolver.caltech.edu/CaltechAUTHORS:20161024-174100275
Authors: {'items': [{'id': 'Friedel-I', 'name': {'family': 'Friedel', 'given': 'Ilja'}}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}, {'id': 'Desbrun-M', 'name': {'family': 'Desbrun', 'given': 'Mathieu'}, 'orcid': '0000-0003-3424-6079'}]}
Year: 2005
DOI: 10.1145/1187112.1187274
[no abstract]https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/c019p-pq895Discrete, Vorticity-Preserving, and Stable Simplicial Fluids
https://resolver.caltech.edu/CaltechAUTHORS:20161011-165249647
Authors: {'items': [{'id': 'Elcott-S', 'name': {'family': 'Elcott', 'given': 'Sharif'}}, {'id': 'Tong-Yiying', 'name': {'family': 'Tong', 'given': 'Yiying'}}, {'id': 'Kanso-E', 'name': {'family': 'Kanso', 'given': 'Eva'}}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}, {'id': 'Desbrun-M', 'name': {'family': 'Desbrun', 'given': 'Mathieu'}, 'orcid': '0000-0003-3424-6079'}]}
Year: 2005
DOI: 10.1145/1198555.1198668
Visual accuracy, low computational cost, and numerical stability are foremost goals in computer animation. An important ingredient in achieving these goals is the conservation of fundamental motion invariants. For example, rigid or deformable body simulation have benefited greatly from conservation of linear and angular momenta. In the case of fluids, however, none of the current techniques focuses on conserving invariants, and consequently, they often introduce a visually disturbing numerical diffusion of
vorticity. Visually just as important is the resolution of complex simulation domains. Doing so with regular (even if adaptive) grid techniques can be computationally delicate.
In this chapter, we propose a novel technique for the simulation of fluid flows. It is designed to respect the defining differential properties, i.e., the
conservation of circulation along arbitrary loops as
they are transported by the flow. Consequently, our method offers several new and desirable properties: (1) arbitrary simplicial meshes (triangles in 2D, tetrahedra in 3D) can be used to define the fluid domain; (2) the computations are efficient due to discrete operators with small support; (3) the method is stable for arbitrarily large time steps; and (4) it preserves a discrete circulation
avoiding numerical diffusion of vorticity. The underlying ideas are easy to incorporate in current approaches to fluid simulation and should thus prove valuable in many applications.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/66jk7-ms511Discrete conformal mappings via circle patterns
https://resolver.caltech.edu/CaltechAUTHORS:20161116-140825377
Authors: {'items': [{'id': 'Kharevych-L', 'name': {'family': 'Kharevych', 'given': 'Liliya'}}, {'id': 'Springborn-B', 'name': {'family': 'Springborn', 'given': 'Boris'}}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}]}
Year: 2005
DOI: 10.1145/1198555.1198665
We introduce a novel method for the construction of discrete conformal mappings from (regions of) embedded meshes to the plane. Our approach is based on circle patterns, i.e., arrangements of circles---one for each face---with prescribed intersection angles. Given these angles the circle radii follow as the unique minimizer of a convex energy. The method has two principal advantages over earlier approaches based on discrete harmonic mappings: (1) it supports very flexible boundary conditions ranging from natural boundaries to control of the boundary shape via prescribed curvatures; (2) the solution is based on a convex energy as a function of logarithmic radius variables with simple explicit expressions for gradients and Hessians, greatly facilitating robust and efficient numerical treatment. We demonstrate the versatility and performance of our algorithm with a variety of examples.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/fhw0y-xf962Building your own DEC at home
https://resolver.caltech.edu/CaltechAUTHORS:20161011-163903630
Authors: {'items': [{'id': 'Elcott-S', 'name': {'family': 'Elcott', 'given': 'Sharif'}}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}]}
Year: 2005
DOI: 10.1145/1198555.1198667
The methods of Discrete Exterior Calculus (DEC) have given birth to many new algorithms applicable to areas such as fluid simulation, deformable body simulation, and others. Despite the (possibly intimidating) mathematical theory that went into deriving these algorithms, in the end they lead to simple, elegant, and straightforward implementations. However, readers interested in implementing them should note that the algorithms presume the existence of a suitable simplicial complex data structure. Such a data structure needs to support local traversal of elements, adjacency information for all dimensions of simplices, a notion of a dual mesh, and all simplices must be oriented. Unfortunately, most publicly available tetrahedral mesh libraries provide only unoriented representations with little more than vertex-tet adjacency information (while we need vertex-edge, edge-triangle, edge-tet, etc.). For those eager to implement and build on the algorithms presented in this course without having to worry about these details, we provide an implementation of a DEC-friendly tetrahedral mesh data structure in C++. This chapter documents the ideas behind the implementation.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/b0vp9-dvf44An Image Processing Approach to Surface Matching
https://resolver.caltech.edu/CaltechAUTHORS:20161122-170753654
Authors: {'items': [{'id': 'Litke-N', 'name': {'family': 'Litke', 'given': 'Nathan'}}, {'id': 'Droske-M', 'name': {'family': 'Droske', 'given': 'Marc'}}, {'id': 'Rumpf-M', 'name': {'family': 'Rumpf', 'given': 'Martin'}}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}]}
Year: 2005
DOI: 10.2312/SGP/SGP05/207-216
Establishing a correspondence between two surfaces is a basic ingredient in many geometry processing applications. Existing approaches, which attempt to match two meshes directly in 3D, can be cumbersome to implement and it is often hard to produce accurate results in a reasonable amount of time. In this paper, we present a new variational method for matching surfaces that addresses these issues. Instead of matching two surfaces directly in 3D, we apply well-established matching methods from image processing in the parameter domains of the surfaces. A matching energy is introduced that can depend on curvature, feature demarcations or surface textures, and a regularization energy controls length and area changes in the induced non-rigid deformation between the two surfaces. The metric on both surfaces is properly incorporated into the formulation of the energy. This approach reduces all computations to the 2D setting while accounting for the original geometries. Consequently a fast multiresolution numerical algorithm for regular image grids can be used to solve the global optimization problem. The final algorithm is robust, generically much simpler than direct matching methods, and very fast for highly resolved triangle meshes.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/0tj9s-zwe89Sparse Matrix Solvers on the GPU: Conjugate Gradients and Multigrid
https://resolver.caltech.edu/CaltechAUTHORS:20160725-161316334
Authors: {'items': [{'id': 'Bolz-J', 'name': {'family': 'Bolz', 'given': 'Jeff'}}, {'id': 'Farmer-I', 'name': {'family': 'Farmer', 'given': 'Ian'}}, {'id': 'Grinspun-E', 'name': {'family': 'Grinspun', 'given': 'Eitan'}}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}]}
Year: 2005
DOI: 10.1145/1198555.1198781
Many computer graphics applications require high-intensity numerical simulation. We show that such computations can be performed efficiently on the GPU, which we regard as a full function streaming processor with high floating-point performance. We implemented two basic, broadly useful, computational kernels: a sparse matrix conjugate gradient solver and a regular-grid multigrid solver. Real-time applications ranging from mesh smoothing and parameterization to fluid solvers and solid mechanics can greatly benefit from these, evidence our example applications of geometric flow and fluid simulation running on NVIDIA's GeForce FX.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/eq00s-hq297Multiscale Representations for Manifold-Valued Data
https://resolver.caltech.edu/CaltechAUTHORS:RAHmms05
Authors: {'items': [{'id': 'Ur-Rahman-I', 'name': {'family': 'Ur Rahman', 'given': 'Inam'}}, {'id': 'Drori-I', 'name': {'family': 'Drori', 'given': 'Iddo'}}, {'id': 'Stodden-V-C', 'name': {'family': 'Stodden', 'given': 'Victoria C.'}}, {'id': 'Donoho-D-L', 'name': {'family': 'Donoho', 'given': 'David L.'}, 'orcid': '0000-0003-1830-710X'}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}]}
Year: 2005
DOI: 10.1137/050622729
We describe multiscale representations for data observed on equispaced grids and taking values in manifolds such as the sphere $S^2$, the special orthogonal group $SO(3)$, the positive definite matrices $SPD(n)$, and the Grassmann manifolds $G(n,k)$. The representations are based on the deployment of Deslauriers--Dubuc and average-interpolating pyramids "in the tangent plane" of such manifolds, using the $Exp$ and $Log$ maps of those manifolds. The representations provide "wavelet coefficients" which can be thresholded, quantized, and scaled in much the same way as traditional wavelet coefficients. Tasks such as compression, noise removal, contrast enhancement, and stochastic simulation are facilitated by this representation. The approach applies to general manifolds but is particularly suited to the manifolds we consider, i.e., Riemannian symmetric spaces, such as $S^{n-1}$, $SO(n)$, $G(n,k)$, where the $Exp$ and $Log$ maps are effectively computable. Applications to manifold-valued data sources of a geometric nature (motion, orientation, diffusion) seem particularly immediate. A software toolbox, SymmLab, can reproduce the results discussed in this paper.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/55a65-bfq94Geometric, Variational Integrators for Computer Animation
https://resolver.caltech.edu/CaltechAUTHORS:20101005-093706360
Authors: {'items': [{'id': 'Kharevych-L', 'name': {'family': 'Kharevych', 'given': 'L.'}}, {'id': 'Wei-W', 'name': {'family': 'Wei', 'given': 'W.'}}, {'id': 'Tong-Y', 'name': {'family': 'Tong', 'given': 'Y.'}}, {'id': 'Kanso-E', 'name': {'family': 'Kanso', 'given': 'E.'}}, {'id': 'Marsden-J-E', 'name': {'family': 'Marsden', 'given': 'J. E.'}}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'P.'}, 'orcid': '0000-0002-0323-7674'}, {'id': 'Desbrun-M', 'name': {'family': 'Desbrun', 'given': 'M.'}, 'orcid': '0000-0003-3424-6079'}]}
Year: 2006
We present a general-purpose numerical scheme for time integration of Lagrangian dynamical systems—an important
computational tool at the core of most physics-based animation techniques. Several features make this
particular time integrator highly desirable for computer animation: it numerically preserves important invariants,
such as linear and angular momenta; the symplectic nature of the integrator also guarantees a correct energy
behavior, even when dissipation and external forces are added; holonomic constraints can also be enforced quite
simply; finally, our simple methodology allows for the design of high-order accurate schemes if needed. Two key
properties set the method apart from earlier approaches. First, the nonlinear equations that must be solved during
an update step are replaced by a minimization of a novel functional, speeding up time stepping by more than a
factor of two in practice. Second, the formulation introduces additional variables that provide key flexibility in the
implementation of the method. These properties are achieved using a discrete form of a general variational principle
called the Pontryagin-Hamilton principle, expressing time integration in a geometric manner. We demonstrate
the applicability of our integrators to the simulation of non-linear elasticity with implementation details.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/nsh6p-s5d70Discrete Conformal Mappings via Circle Patterns
https://resolver.caltech.edu/CaltechAUTHORS:20110630-141124886
Authors: {'items': [{'id': 'Kharevych-L', 'name': {'family': 'Kharevych', 'given': 'Liliya'}}, {'id': 'Springborn-B', 'name': {'family': 'Springborn', 'given': 'Boris'}}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}]}
Year: 2006
DOI: 10.1145/1138450.1138461
We introduce a novel method for the construction of discrete conformal mappings from surface meshes of arbitrary topology to the plane. Our approach is based on circle patterns, that is, arrangements of circles---one for each face---with prescribed intersection angles. Given these angles, the circle radii follow as the unique minimizer of a convex energy. The method supports very flexible boundary conditions ranging from free boundaries to control of the boundary shape via prescribed curvatures. Closed meshes of genus zero can be parameterized over the sphere. To parameterize higher genus meshes, we introduce cone singularities at designated vertices. The parameter domain is then a piecewise Euclidean surface. Cone singularities can also help to reduce the often very large area distortion of global conformal maps to moderate levels. Our method involves two optimization problems: a quadratic program and the unconstrained minimization of the circle pattern energy. The latter is a convex function of logarithmic radius variables with simple explicit expressions for gradient and Hessian. We demonstrate the versatility and performance of our algorithm with a variety of examples.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/hye6q-pw318What can we measure?
https://resolver.caltech.edu/CaltechAUTHORS:20161215-162318208
Authors: {'items': [{'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}]}
Year: 2006
DOI: 10.1145/1185657.1185660
When characterizing a shape or changes in shape we must first ask, what can we measure about a shape? For example, for a region in ℝ^3 we may ask for its volume or its surface area. If the object at hand undergoes deformation due to forces acting on it we may need to formulate the laws governing the change in shape in terms of measurable quantities and their change over time. Usually such measurable quantities for a shape are defined with the help of integral calculus and often require some amount of smoothness on the object to be well defined. In this chapter we will take a more abstract approach to the question of measurable quantities which will allow us to define notions such as mean curvature integrals and the curvature tensor for piecewise linear meshes without having to worry about the meaning of second derivatives in settings in which they do not exist. In fact in this chapter we will give an account of a classical result due to Hadwiger, which shows that for a convex, compact set in ℝ^n there are only n + 1 unique measurements if we require that the measurements be invariant under Euclidian motions (and satisfy certain "sanity" conditions). We will see how these measurements are constructed in a very straightforward and elementary manner and that they can be read off from a characteristic polynomial due to Steiner. This polynomial describes the volume of a family of shapes which arise when we "grow" a given shape. As a practical tool arising from these consideration we will see that there is a well defined notion of the curvature tensor for piecewise linear meshes and we will see very simple formulas for quantities needed in physical simulation with piecewise linear meshes. Much of the treatment here will be limited to convex
bodies to keep things simple.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/djww5-zfy81Stable, circulation-preserving, simplicial fluids
https://resolver.caltech.edu/CaltechAUTHORS:20161013-131839051
Authors: {'items': [{'id': 'Elcott-S', 'name': {'family': 'Elcott', 'given': 'Sharif'}}, {'id': 'Tong-Yiying', 'name': {'family': 'Tong', 'given': 'Yiying'}}, {'id': 'Kanso-E', 'name': {'family': 'Kanso', 'given': 'Eva'}}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}, {'id': 'Desbrun-M', 'name': {'family': 'Desbrun', 'given': 'Mathieu'}, 'orcid': '0000-0003-3424-6079'}]}
Year: 2006
DOI: 10.1145/1185657.1185667
Visual quality, low computational cost, and numerical stability are foremost goals in computer animation. An important ingredient in achieving these goals is the conservation of fundamental motion invariants. For example, rigid and deformable body simulation benefits greatly from conservation of linear and angular momenta. In the case of fluids, however, none of the current techniques focuses on conserving invariants, and consequently, they often introduce a visually disturbing numerical diffusion of vorticity. Visually just as important is the resolution of complex simulation domains. Doing so with regular (even if adaptive) grid techniques can be computationally delicate. In this chapter, we propose a novel technique for the simulation of fluid flows. It is designed to respect the defining differential properties, i.e., the conservation of circulation along arbitrary loops as they are transported by the flow. Consequently, our method offers several new and desirable properties: (1) arbitrary simplicial meshes (triangles in 2D, tetrahedra in 3D) can be used to define the fluid domain; (2) the computations are efficient due to discrete operators with small support; (3) the method is stable for arbitrarily large time steps; (4) it preserves discrete circulation avoiding numerical diffusion of vorticity; and (5) its implementation is straightforward.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/nad7m-5bz38Edge Subdivision Schemes and the Construction of Smooth Vector Fields
https://resolver.caltech.edu/CaltechAUTHORS:20110804-135119206
Authors: {'items': [{'id': 'Wang-Ke', 'name': {'family': 'Wang', 'given': 'Ke'}}, {'id': 'Wei-Wei', 'name': {'family': 'Wei', 'given': 'Wei'}, 'orcid': '0000-0002-1018-7708'}, {'id': 'Tong-Yiying', 'name': {'family': 'Tong', 'given': 'Yiying'}}, {'id': 'Desbrun-M', 'name': {'family': 'Desbrun', 'given': 'Mathieu'}, 'orcid': '0000-0003-3424-6079'}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}]}
Year: 2006
DOI: 10.1145/1179352.1141991
Vertex- and face-based subdivision schemes are now routinely used in geometric modeling and computational science, and their primal/dual relationships are well studied. In this paper, we interpret these schemes as defining bases for discrete differential 0- resp. 2-forms, and complete the picture by introducing edge-based subdivision schemes to construct the missing bases for discrete differential 1-forms. Such subdivision schemes map scalar coefficients on edges from the coarse to the refined mesh and are intrinsic to the surface. Our construction is based on treating vertex-, edge-, and face-based subdivision schemes as a joint triple and enforcing that subdivision commutes with the topological exterior derivative. We demonstrate our construction for the case of arbitrary topology triangle meshes. Using Loop's scheme for 0-forms and generalized half-box splines for 2-forms results in a unique generalized spline scheme for 1-forms, easily incorporated into standard subdivision surface codes. We also provide corresponding boundary stencils. Once a metric is supplied, the scalar 1-form coefficients define a smooth tangent vector field on the underlying subdivision surface. Design of tangent vector fields is made particularly easy with this machinery as we demonstrate.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/ez8ek-q1796Building your own DEC at home
https://resolver.caltech.edu/CaltechAUTHORS:20161011-164535652
Authors: {'items': [{'id': 'Elcott-S', 'name': {'family': 'Elcott', 'given': 'Sharif'}}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}]}
Year: 2006
DOI: 10.1145/1185657.1185666
The methods of Discrete Exterior Calculus (DEC) have given birth to many new algorithms applicable to areas such as fluid simulation, deformable body simulation, and others. Despite the (possibly intimidating) mathematical theory that went into deriving these algorithms, in the end they lead to simple, elegant, and straightforward implementations. However, readers interested in implementing
them should note that the algorithms presume the existence
of a suitable simplicial complex data structure. Such a data
structure needs to support local traversal of elements, adjacency information for all dimensions of simplices, a notion of a dual mesh, and all simplices must be oriented. Unfortunately, most publicly available tetrahedral mesh libraries provide only unoriented representations
with little more than vertex-tet adjacency information
(while we need vertex-edge, edge-triangle, edge-tet, etc.). For those eager to implement and build on the algorithms presented in this course without having to worry about these details, we provide an implementation of a DEC-friendly tetrahedral mesh data structure
in C++. This chapter documents the ideas behind the implementation.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/s5x80-k2a68An algorithm for the construction of intrinsic Delaunay triangulations with applications to digital geometry processing
https://resolver.caltech.edu/CaltechAUTHORS:20161019-160617640
Authors: {'items': [{'id': 'Fisher-Matthew-David-CompSci', 'name': {'family': 'Fisher', 'given': 'Matthew'}, 'orcid': '0000-0002-8908-3417'}, {'id': 'Springborn-B', 'name': {'family': 'Springborn', 'given': 'Boris'}}, {'id': 'Bobenko-A-I', 'name': {'family': 'Bobenko', 'given': 'Alexander I.'}}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}]}
Year: 2006
DOI: 10.1145/1185657.1185668
The discrete Laplace-Beltrami operator plays a prominent role in many Digital Geometry Processing applications ranging from denoising to parameterization, editing, and physical simulation. The standard discretization uses the cotangents of the angles in the immersed mesh which leads to a variety of numerical problems. We advocate use of the
intrinsic Laplace-Beltrami operator. It satisfies a local maximum principle, guaranteeing, e.g., that no flipped
triangles can occur in parameterizations. It also leads to better conditioned linear systems. The intrinsic Laplace-Beltrami operator is based on an intrinsic
Delaunay triangulation of the surface. We give
an incremental algorithm to construct such triangulations together with an overlay structure which captures the relationship between the extrinsic and intrinsic triangulations. Using a variety of example meshes we demonstrate the numerical benefits of the intrinsic
Laplace-Beltrami operator.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/1cg4x-t0880Edge subdivision schemes and the construction of smooth vector fields
https://resolver.caltech.edu/CaltechAUTHORS:20170104-161620229
Authors: {'items': [{'id': 'Wang-Ke', 'name': {'family': 'Wang', 'given': 'Ke'}}, {'id': 'Weiwei-', 'name': {'family': 'Weiwei'}}, {'id': 'Tong-Yiying', 'name': {'family': 'Tong', 'given': 'Yiying'}}, {'id': 'Desbrun-M', 'name': {'family': 'Desbrun', 'given': 'Mathieu'}, 'orcid': '0000-0003-3424-6079'}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}]}
Year: 2006
DOI: 10.1145/1179352.1141991
Vertex- and face-based subdivision schemes are now routinely used in geometric modeling and computational science, and their primal/dual relationships are well studied. In this paper, we interpret these schemes as defining bases for discrete differential 0- resp. 2-forms, and complete the picture by introducing edge-based subdivision schemes to construct the missing bases for discrete differential 1-forms. Such subdivision schemes map scalar coefficients on edges from the coarse to the refined mesh and are intrinsic to the surface. Our construction is based on treating vertex-, edge-, and face-based subdivision schemes as a joint triple and enforcing that subdivision commutes with the topological exterior derivative. We demonstrate our construction for the case of arbitrary topology triangle meshes. Using Loop's scheme for 0-forms and generalized half-box splines for 2-forms results in a unique generalized spline scheme for 1-forms, easily incorporated into standard subdivision surface codes. We also provide corresponding boundary stencils. Once a metric is supplied, the scalar 1-form coefficients define a smooth tangent vector field on the underlying subdivision surface. Design of tangent vector fields is made particularly easy with this machinery as we demonstrate.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/rw6cz-hh755Stable, circulation-preserving, simplicial fluids
https://resolver.caltech.edu/CaltechAUTHORS:20161013-130615464
Authors: {'items': [{'id': 'Elcott-S', 'name': {'family': 'Elcott', 'given': 'Sharif'}}, {'id': 'Tong-Yiying', 'name': {'family': 'Tong', 'given': 'Yiying'}}, {'id': 'Kanso-E', 'name': {'family': 'Kanso', 'given': 'Eva'}}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}, {'id': 'Desbrun-M', 'name': {'family': 'Desbrun', 'given': 'Mathieu'}, 'orcid': '0000-0003-3424-6079'}]}
Year: 2007
DOI: 10.1145/1189762.1189766
Visual quality, low computational cost, and numerical stability are foremost goals in computer animation. An important ingredient in achieving these goals is the conservation of fundamental motion invariants. For example, rigid and deformable body simulation benefits greatly from the conservation of linear and angular momenta. In the case of fluids, however, none of the current techniques focuses on conserving invariants, and consequently, often introduce a visually disturbing numerical diffusion of vorticity. Just as important visually is the resolution of complex simulation domains. Doing so with regular (even if adaptive) grid techniques can be computationally delicate. In this article, we propose a novel technique for the simulation of fluid flows. It is designed to respect the defining differential properties, that is, the conservation of circulation along arbitrary loops as they are transported by the flow. Consequently, our method offers several new and desirable properties: Arbitrary simplicial meshes (triangles in 2D, tetrahedra in 3D) can be used to define the fluid domain; the computations involved in the update procedure are efficient due to discrete operators with small support; and it preserves discrete circulation, avoiding numerical diffusion of vorticity.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/ask9g-sad25Design of tangent vector fields
https://resolver.caltech.edu/CaltechAUTHORS:20100507-094443477
Authors: {'items': [{'id': 'Fisher-Matthew-David-CompSci', 'name': {'family': 'Fisher', 'given': 'Matthew'}, 'orcid': '0000-0002-8908-3417'}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}, {'id': 'Desbrun-M', 'name': {'family': 'Desbrun', 'given': 'Mathieu'}, 'orcid': '0000-0003-3424-6079'}, {'id': 'Hoppe-Hugues', 'name': {'family': 'Hoppe', 'given': 'Hugues'}, 'orcid': '0000-0002-9699-2539'}]}
Year: 2007
DOI: 10.1145/1239451.1239507
Tangent vector fields are an essential ingredient in controlling surface appearance for applications ranging from anisotropic shading to texture synthesis and non-photorealistic rendering. To achieve a desired effect one is typically interested in smoothly varying fields that satisfy a sparse set of user-provided constraints. Using tools from Discrete Exterior Calculus, we present a simple and efficient algorithm for designing such fields over arbitrary triangle meshes. By representing the field as scalars over mesh edges (i.e., discrete 1-forms), we obtain an intrinsic, coordinate-free formulation in which field smoothness is enforced through discrete Laplace operators. Unlike previous methods, such a formulation leads to a linear system whose sparsity permits efficient pre-factorization. Constraints are incorporated through weighted least squares and can be updated rapidly enough to enable interactive design, as we demonstrate in the context of anisotropic texture synthesis.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/0vqc5-p7268Design of tangent vector fields
https://resolver.caltech.edu/CaltechAUTHORS:20161019-155301872
Authors: {'items': [{'id': 'Fisher-Matthew-David-CompSci', 'name': {'family': 'Fisher', 'given': 'Matthew'}, 'orcid': '0000-0002-8908-3417'}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}, {'id': 'Desbrun-M', 'name': {'family': 'Desbrun', 'given': 'Mathieu'}, 'orcid': '0000-0003-3424-6079'}, {'id': 'Hoppe-Hugues', 'name': {'family': 'Hoppe', 'given': 'Hugues'}, 'orcid': '0000-0002-9699-2539'}]}
Year: 2007
DOI: 10.1145/1275808.1276447
Tangent vector fields are an essential ingredient in controlling surface appearance for applications ranging from anisotropic shading to texture synthesis and non-photorealistic rendering. To achieve a desired effect one is typically interested in smoothly varying fields that satisfy a sparse set of user-provided constraints. Using tools from Discrete Exterior Calculus, we present a simple
and efficient algorithm for designing such fields over arbitrary triangle meshes. By representing the field as scalars over mesh edges (i.e., discrete 1-forms), we obtain an intrinsic, coordinate-free formulation in which field smoothness is enforced through discrete Laplace operators. Unlike previous methods, such a formulation leads to a linear system whose sparsity permits efficient
pre-factorization. Constraints are incorporated through weighted least squares and can be updated rapidly enough to enable interactive design, as we demonstrate in the context of anisotropic texture synthesis.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/rmh81-z3w27An algorithm for the construction of intrinsic Delaunay triangulations with applications to digital geometry processing
https://resolver.caltech.edu/CaltechAUTHORS:20100820-152952201
Authors: {'items': [{'id': 'Fisher-Matthew-David-CompSci', 'name': {'family': 'Fisher', 'given': 'M.'}, 'orcid': '0000-0002-8908-3417'}, {'id': 'Springborn-B', 'name': {'family': 'Springborn', 'given': 'B.'}}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'P.'}, 'orcid': '0000-0002-0323-7674'}, {'id': 'Bobenko-A-I', 'name': {'family': 'Bobenko', 'given': 'A. I.'}}]}
Year: 2007
DOI: 10.1007/s00607-007-0249-8
The discrete Laplace–Beltrami operator plays a prominent role in many digital geometry processing applications ranging from denoising to parameterization, editing, and physical simulation. The standard discretization uses the cotangents of the angles in the immersed mesh which leads to a variety of numerical problems. We advocate the use of the intrinsic Laplace–Beltrami operator. It satisfies a local maximum principle, guaranteeing, e.g., that no flipped triangles can occur in parameterizations. It also leads to better conditioned linear systems. The intrinsic Laplace–Beltrami operator is based on an intrinsic Delaunay triangulation of the surface. We detail an incremental algorithm to construct such triangulations together with an overlay structure which captures the relationship between the extrinsic and intrinsic triangulations. Using a variety of example meshes we demonstrate the numerical benefits of the intrinsic Laplace–Beltrami operator.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/1zzbj-x4f73Conformal equivalence of triangle meshes
https://resolver.caltech.edu/CaltechAUTHORS:20161220-135710541
Authors: {'items': [{'id': 'Springborn-B', 'name': {'family': 'Springborn', 'given': 'Boris'}}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}, {'id': 'Pinkall-U', 'name': {'family': 'Pinkall', 'given': 'Ulrich'}}]}
Year: 2008
DOI: 10.1145/1399504.1360676
We present a new algorithm for conformal mesh parameterization. It is based on a precise notion of discrete conformal equivalence for triangle meshes which mimics the notion of conformal equivalence for smooth surfaces. The problem of finding a flat mesh that is discretely conformally equivalent to a given mesh can be solved efficiently by minimizing a convex energy function, whose Hessian turns out to be the well known cot-Laplace operator. This method can also be used to map a surface mesh to a parameter domain which is flat except for isolated cone singularities, and we show how these can be placed automatically in order to reduce the distortion of the parameterization. We present the salient features of the theory and elaborate the algorithms with a number of examples.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/3jr66-h6j47Conformal Equivalence of Triangle Meshes
https://resolver.caltech.edu/CaltechAUTHORS:20100722-120116172
Authors: {'items': [{'id': 'Springborn-B', 'name': {'family': 'Springborn', 'given': 'Boris'}}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}, {'id': 'Pinkall-U', 'name': {'family': 'Pinkall', 'given': 'Ulrich'}}]}
Year: 2008
DOI: 10.1145/1360612.1360676
We present a new algorithm for conformal mesh parameterization. It is based on a precise notion of discrete conformal equivalence for triangle meshes which mimics the notion of conformal equivalence for smooth surfaces. The problem of finding a flat mesh that is discretely conformally equivalent to a given mesh can be solved efficiently by minimizing a convex energy function, whose Hessian turns out to be the well known cot-Laplace operator. This method can also be used to map a surface mesh to a parameter domain which is flat except for isolated cone singularities, and we show how these can be placed automatically in order to reduce the distortion of the parameterization. We present the salient features of the theory and elaborate the algorithms with a number of examples.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/q040e-9ya82Trivial Connections on Discrete Surfaces
https://resolver.caltech.edu/CaltechAUTHORS:20101011-145011916
Authors: {'items': [{'id': 'Crane-K', 'name': {'family': 'Crane', 'given': 'Keenan'}}, {'id': 'Desbrun-M', 'name': {'family': 'Desbrun', 'given': 'Mathieu'}, 'orcid': '0000-0003-3424-6079'}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}]}
Year: 2010
DOI: 10.1111/j.1467-8659.2010.01761.x
This paper presents a straightforward algorithm for constructing connections on discrete surfaces that are as
smooth as possible everywhere but on a set of isolated singularities with given index. We compute these connections
by solving a single linear system built from standard operators. The solution can be used to design rotationally
symmetric direction fields with user-specified singularities and directional constraints.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/bz991-9gc31A simple geometric model for elastic deformations
https://resolver.caltech.edu/CaltechAUTHORS:20100803-090608485
Authors: {'items': [{'id': 'Chao-Isaac', 'name': {'family': 'Chao', 'given': 'Isaac'}}, {'id': 'Pinkall-U', 'name': {'family': 'Pinkall', 'given': 'Ulrich'}}, {'id': 'Sanan-Patrick', 'name': {'family': 'Sanan', 'given': 'Patrick'}}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}]}
Year: 2010
DOI: 10.1145/1778765.1778775
We advocate a simple geometric model for elasticity: distance between the differential of a deformation and the rotation group. It comes with rigorous differential geometric underpinnings, both smooth and discrete, and is computationally almost as simple and efficient as linear elasticity. Owing to its geometric non-linearity, though, it does not suffer from the usual linearization artifacts. A material model with standard elastic moduli (Lamé parameters) falls out naturally, and a minimizer for static problems is easily augmented to construct a fully variational 2nd order time integrator. It has excellent conservation properties even for very coarse simulations, making it very robust.
Our analysis was motivated by a number of heuristic, physics-like algorithms from geometry processing (editing, morphing, parameterization, and simulation). Starting with a continuous energy formulation and taking the underlying geometry into account, we simplify and accelerate these algorithms while avoiding common pitfalls. Through the connection with the Biot strain of mechanics, the intuition of previous work that these ideas are "like" elasticity is shown to be spot on.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/snw3h-rfm17A Simple Geometric Model for Elastic Deformations
https://resolver.caltech.edu/CaltechAUTHORS:20160819-110910323
Authors: {'items': [{'id': 'Chao-Isaac', 'name': {'family': 'Chao', 'given': 'Isaac'}}, {'id': 'Pinkall-U', 'name': {'family': 'Pinkall', 'given': 'Ulrich'}}, {'id': 'Sanan-P', 'name': {'family': 'Sanan', 'given': 'Patrick'}}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}]}
Year: 2010
DOI: 10.1145/1833349.1778775
We advocate a simple geometric model for elasticity: distance between the differential of a deformation and the rotation group. It comes with rigorous differential geometric underpinnings, both smooth and discrete, and is computationally almost as simple and efficient as linear elasticity. Owing to its geometric non-linearity,
though, it does not suffer from the usual linearization artifacts. A material model with standard elastic moduli (Lamé parameters) falls out naturally, and a minimizer for static problems is easily augmented to construct a fully variational 2nd order time integrator. It has excellent conservation properties even for very coarse
simulations, making it very robust.
Our analysis was motivated by a number of heuristic, physics-like algorithms from geometry processing (editing, morphing, parameterization, and simulation). Starting with a continuous energy formulation and taking the underlying geometry into account, we simplify and accelerate these algorithms while avoiding common pitfalls. Through the connection with the Biot strain of mechanics, the intuition of previous work that these ideas are "like" elasticity is shown to be spot on.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/ybmfn-rqw41Spin Transformations of Discrete Surfaces
https://resolver.caltech.edu/CaltechAUTHORS:20120104-114455813
Authors: {'items': [{'id': 'Crane-K', 'name': {'family': 'Crane', 'given': 'Keenan'}}, {'id': 'Pinkall-U', 'name': {'family': 'Pinkall', 'given': 'Ulrich'}}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}]}
Year: 2011
DOI: 10.1145/1964921.1964999
We introduce a new method for computing conformal transformations of triangle meshes in R^3. Conformal maps are desirable in digital geometry processing because they do not exhibit shear, and therefore preserve texture fidelity as well as the quality of the mesh itself. Traditional discretizations consider maps into the complex plane, which are useful only for problems such as surface parameterization and planar shape deformation where the target surface is flat. We instead consider maps into the quaternions H, which allows us to work directly with surfaces sitting in R^3. In particular, we introduce a quaternionic Dirac operator and use it to develop a novel integrability condition on conformal deformations. Our discretization of this condition results in a sparse linear system that is simple to build and can be used to efficiently edit surfaces by manipulating curvature and boundary data, as demonstrated via several mesh processing applications.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/rtq0a-9ww02Spin transformations of discrete surfaces
https://resolver.caltech.edu/CaltechAUTHORS:20161005-155130201
Authors: {'items': [{'id': 'Crane-K', 'name': {'family': 'Crane', 'given': 'Keenan'}}, {'id': 'Pinkall-U', 'name': {'family': 'Pinkall', 'given': 'Ulrich'}}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}]}
Year: 2011
DOI: 10.1145/1964921.1964999
We introduce a new method for computing conformal transformations of triangle meshes in ℝ^3. Conformal maps are desirable in digital geometry processing because they do not exhibit shear, and therefore preserve texture fidelity as well as the quality of the mesh itself. Traditional discretizations consider maps into the
complex plane, which are useful only for problems such as surface parameterization and planar shape deformation where the target surface is flat. We instead consider maps into the quaternions H, which allows us to work directly with surfaces sitting in ℝ^3. In particular, we introduce a
quaternionic Dirac operator and use it to develop a novel integrability condition on conformal deformations. Our discretization of this condition results in a
sparse linear system that is simple to build and can be used to efficiently edit surfaces by manipulating curvature and boundary data, as demonstrated via several mesh processing applications.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/75r74-39v92√3-Based 1-Form Subdivision
https://resolver.caltech.edu/CaltechAUTHORS:20120725-131858923
Authors: {'items': [{'id': 'Huang-Jinghao', 'name': {'family': 'Huang', 'given': 'Jinghao'}}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}]}
Year: 2012
DOI: 10.1007/978-3-642-27413-8_22
In this paper we construct an edge based, or 1-form, subdivision scheme consistent with √3 subdivision. It produces smooth differential 1-forms in the limit. These can be identified with tangent vector fields, or viewed as edge elements in the sense of finite elements. In this construction, primal (0-form) and dual (2-form) subdivision schemes for surfaces are related through the exterior derivative with an edge (1-form) based subdivision scheme, amounting to a generalization of the well known formulé de commutation.
Starting with the classic √3 subdivision scheme as a 0-form subdivision scheme, we derive conditions for appropriate 1- and 2-form subdivision schemes without fixing the dual (2-form) subdivision scheme a priori. The resulting degrees of freedom are resolved through spectrum considerations and a conservation condition analogous to the usual moment
condition for primal subdivision schemes.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/vp6wy-fzh35Robust Fairing via Conformal Curvature Flow
https://resolver.caltech.edu/CaltechAUTHORS:20130829-152759082
Authors: {'items': [{'id': 'Crane-K', 'name': {'family': 'Crane', 'given': 'Keenan'}}, {'id': 'Pinkall-U', 'name': {'family': 'Pinkall', 'given': 'Ulrich'}}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}]}
Year: 2013
DOI: 10.1145/2461912.2461986
We present a formulation of Willmore flow for triangulated surfaces
that permits extraordinarily large time steps and naturally
preserves the quality of the input mesh. The main insight is
that Willmore flow becomes remarkably stable when expressed
in curvature space – we develop the precise conditions under
which curvature is allowed to evolve. The practical outcome is
a highly efficient algorithm that naturally preserves texture and
does not require remeshing during the flow. We apply this algorithm
to surface fairing, geometric modeling, and construction
of constant mean curvature (CMC) surfaces. We also present a
new algorithm for length-preserving flow on planar curves, which
provides a valuable analogy for the surface case.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/3c0bw-8fg61Globally Optimal Direction Fields
https://resolver.caltech.edu/CaltechAUTHORS:20130829-154018080
Authors: {'items': [{'id': 'Knöppel-F', 'name': {'family': 'Knöppel', 'given': 'Felix'}}, {'id': 'Crane-K', 'name': {'family': 'Crane', 'given': 'Keenan'}}, {'id': 'Pinkall-U', 'name': {'family': 'Pinkall', 'given': 'Ulrich'}}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}]}
Year: 2013
DOI: 10.1145/2461912.2462005
We present a method for constructing smooth n-direction fields
(line fields, cross fields, etc.) on surfaces that is an order of
magnitude faster than state-of-the-art methods, while still producing
fields of equal or better quality. Fields produced by the
method are globally optimal in the sense that they minimize a
simple, well-defined quadratic smoothness energy over all possible
configurations of singularities (number, location, and index).
The method is fully automatic and can optionally produce fields
aligned with a given guidance field such as principal curvature
directions. Computationally the smoothest field is found via a
sparse eigenvalue problem involving a matrix similar to the cotan-Laplacian. When a guidance field is present, finding the optimal
field amounts to solving a single linear system.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/14r9f-ckd18Digital geometry processing with discrete exterior calculus
https://resolver.caltech.edu/CaltechAUTHORS:20131008-161552884
Authors: {'items': [{'id': 'Crane-Keenan', 'name': {'family': 'Crane', 'given': 'Keenan'}}, {'id': 'de-Goes-Fernando', 'name': {'family': 'de Goes', 'given': 'Fernando'}}, {'id': 'Desbrun-M', 'name': {'family': 'Desbrun', 'given': 'Mathieu'}, 'orcid': '0000-0003-3424-6079'}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}]}
Year: 2013
DOI: 10.1145/2504435.2504442
These notes provide an introduction to working with real-world geometric data, expressed
in the language of discrete exterior calculus (DEC). DEC is a simple, flexible, and efficient framework
which provides a unified platform for geometry processing. The notes provide essential
mathematical background as well as a large array of real-world examples, with an emphasis on
applications and implementation. The material should be accessible to anyone with some exposure
to basic linear algebra and vector calculus, though most of the key concepts are reviewed as needed.
Coding exercises depend on a basic knowledge of C++, though knowledge of any programming
language is likely sufficient: we do not make heavy use of paradigms like inheritance, templates,
etc. The notes also provide guided written exercises that can be used to deepen understanding of
the material.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/em1fn-3b721Tree Shape Priors with Connectivity Constraints Using Convex Relaxation on General Graphs
https://resolver.caltech.edu/CaltechAUTHORS:20230210-663266000.1
Authors: {'items': [{'id': 'Stühmer-Jan', 'name': {'family': 'Stühmer', 'given': 'Jan'}}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}, {'id': 'Cremers-Daniel', 'name': {'family': 'Cremers', 'given': 'Daniel'}, 'orcid': '0000-0002-3079-7984'}]}
Year: 2013
DOI: 10.1109/iccv.2013.290
In this work we propose a novel method to include a connectivity prior into image segmentation that is based on a binary labeling of a directed graph, in this case a geodesic shortest path tree. Specifically we make two contributions: First, we construct a geodesic shortest path tree with a distance measure that is related to the image data and the bending energy of each path in the tree. Second, we include a connectivity prior in our segmentation model, that allows to segment not only a single elongated structure, but instead a whole connected branching tree. Because both our segmentation model and the connectivity constraint are convex a global optimal solution can be found. To this end, we generalize a recent primal-dual algorithm for continuous convex optimization to an arbitrary graph structure. To validate our method we present results on data from medical imaging in angiography and retinal blood vessel segmentation.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/mcpgt-ayd62Smoke Rings from Smoke
https://resolver.caltech.edu/CaltechAUTHORS:20140820-105147740
Authors: {'items': [{'id': 'Weißmann-S', 'name': {'family': 'Weißmann', 'given': 'Steffen'}}, {'id': 'Pinkall-U', 'name': {'family': 'Pinkall', 'given': 'Ulrich'}}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}]}
Year: 2014
DOI: 10.1145/2601097.2601171
We give an algorithm which extracts vortex filaments ("smoke rings") from a given 3D velocity field. Given a filament strength h > 0, an optimal number of vortex filaments, together with their extent and placement, is given by the zero set of a complex valued function over the domain. This function is the global minimizer of a quadratic energy based on a Schrödinger operator. Computationally this amounts to finding the eigenvector belonging to the smallest eigenvalue of a Laplacian type sparse matrix.
Turning traditional vector field representations of flows, for example, on a regular grid, into a corresponding set of vortex filaments is useful for visualization, analysis of measured flows, hybrid simulation methods, and sparse representations. To demonstrate our method we give examples from each of these.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/vd7e1-jnn80Exploring the Geometry of the Space of Shells
https://resolver.caltech.edu/CaltechAUTHORS:20140929-111525392
Authors: {'items': [{'id': 'Heeren-B', 'name': {'family': 'Heeren', 'given': 'B.'}}, {'id': 'Rumpf-M', 'name': {'family': 'Rumpf', 'given': 'M.'}}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'P.'}, 'orcid': '0000-0002-0323-7674'}, {'id': 'Wardetzky-M', 'name': {'family': 'Wardetzky', 'given': 'M.'}}, {'id': 'Wirth-B', 'name': {'family': 'Wirth', 'given': 'B.'}}]}
Year: 2014
DOI: 10.1111/cgf.12450
We prove both in the smooth and discrete setting that the Hessian of an elastic deformation energy results in a proper Riemannian metric on the space of shells (modulo rigid body motions). Based on this foundation we develop a time- and space-discrete geodesic calculus. In particular we show how to shoot geodesics with prescribed initial data, and we give a construction for parallel transport in shell space. This enables, for example, natural extrapolation of paths in shell space and transfer of large nonlinear deformations from one shell to another with applications in animation, geometric, and physical modeling. Finally, we examine some aspects of curvature on shell space.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/32ehc-0j734Stripe patterns on surfaces
https://resolver.caltech.edu/CaltechAUTHORS:20150814-144538364
Authors: {'items': [{'id': 'Knöppel-F', 'name': {'family': 'Knöppel', 'given': 'Felix'}}, {'id': 'Crane-K', 'name': {'family': 'Crane', 'given': 'Keenan'}}, {'id': 'Pinkall-U', 'name': {'family': 'Pinkall', 'given': 'Ulrich'}}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}]}
Year: 2015
DOI: 10.1145/2767000
Stripe patterns are ubiquitous in nature, describing macroscopic phenomena such as stripes on plants and animals, down to material impurities on the atomic scale. We propose a method for synthesizing stripe patterns on triangulated surfaces, where singularities are automatically inserted in order to achieve user-specified orientation and line spacing. Patterns are characterized as global minimizers of a convex-quadratic energy which is well-defined in the smooth setting. Computation amounts to finding the principal eigenvector of a symmetric positive-definite matrix with the same sparsity as the standard graph Laplacian. The resulting patterns are globally continuous, and can be applied to a variety of tasks in design and texture synthesis.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/fwrsr-ffd43Close-to-conformal deformations of volumes
https://resolver.caltech.edu/CaltechAUTHORS:20150814-145256783
Authors: {'items': [{'id': 'Chern-A', 'name': {'family': 'Chern', 'given': 'Albert'}}, {'id': 'Pinkall-U', 'name': {'family': 'Pinkall', 'given': 'Ulrich'}}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}]}
Year: 2015
DOI: 10.1145/2766916
Conformal deformations are infinitesimal scale-rotations, which can be parameterized by quaternions. The condition that such a quaternion field gives rise to a conformal deformation is nonlinear and in any case only admits Möbius transformations as solutions. We propose a particular decoupling of scaling and rotation which allows us to find near to conformal deformations as minimizers of a quadratic, convex Dirichlet energy. Applied to tetrahedral meshes we find deformations with low quasiconformal distortion as the principal eigenvector of a (quaternionic) Laplace matrix. The resulting algorithms can be implemented with highly optimized standard linear algebra libraries and yield deformations comparable in quality to far more expensive approaches.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/pjzqt-5aj43Splines in the Space of Shells
https://resolver.caltech.edu/CaltechAUTHORS:20160715-133020806
Authors: {'items': [{'id': 'Heeren-B', 'name': {'family': 'Heeren', 'given': 'Behrend'}}, {'id': 'Rumpf-M', 'name': {'family': 'Rumpf', 'given': 'Martin'}}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}, {'id': 'Wardetzky-M', 'name': {'family': 'Wardetzky', 'given': 'Max'}}, {'id': 'Wirth-B', 'name': {'family': 'Wirth', 'given': 'Benedikt'}}]}
Year: 2016
DOI: 10.1111/cgf.12968
Cubic splines in Euclidean space minimize the mean squared acceleration among all curves interpolating a given set of data points. We extend this observation to the Riemannian manifold of discrete shells in which the associated metric measures both bending and membrane distortion. Our generalization replaces the acceleration with the covariant derivative of the velocity. We introduce an effective time-discretization for this novel paradigm for navigating shell space. Further transferring this concept to the space of triangular surface descriptors-edge lengths, dihedral angles, and triangle areas-results in a simplified interpolation method with high computational efficiency.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/fwtaa-vz202Schrödinger's smoke
https://resolver.caltech.edu/CaltechAUTHORS:20160727-095740856
Authors: {'items': [{'id': 'Chern-A', 'name': {'family': 'Chern', 'given': 'Albert'}}, {'id': 'Knöppel-F', 'name': {'family': 'Knöppel', 'given': 'Felix'}}, {'id': 'Pinkall-U', 'name': {'family': 'Pinkall', 'given': 'Ulrich'}}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}, {'id': 'Weißmann-S', 'name': {'family': 'Weißmann', 'given': 'Steffen'}}]}
Year: 2016
DOI: 10.1145/2897824.2925868
We describe a new approach for the purely Eulerian simulation of incompressible fluids. In it, the fluid state is represented by a C^2-valued wave function evolving under the Schrödinger equation subject to incompressibility constraints. The underlying dynamical system is Hamiltonian and governed by the kinetic energy of the fluid together with an energy of Landau-Lifshitz type. The latter ensures that dynamics due to thin vortical structures, all
important for visual simulation, are faithfully reproduced. This enables robust simulation of intricate phenomena such as vortical wakes and interacting vortex filaments, even on modestly sized grids. Our implementation uses a simple splitting method for time integration, employing the FFT for Schrödinger evolution as well as constraint projection. Using a standard penalty method we also allow arbitrary obstacles. The resulting algorithm
is simple, unconditionally stable, and efficient. In particular it does not require any Lagrangian techniques for advection or to counteract the loss of vorticity. We demonstrate its use in a variety of scenarios, compare it with experiments, and evaluate it against benchmark tests. A full implementation is included in the ancillary materials.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/829zt-g0315Inside Fluids: Clebsch Maps for Visualization and Processing
https://resolver.caltech.edu/CaltechAUTHORS:20170814-144752570
Authors: {'items': [{'id': 'Chern-A', 'name': {'family': 'Chern', 'given': 'Albert'}}, {'id': 'Knöppel-F', 'name': {'family': 'Knöppel', 'given': 'Felix'}}, {'id': 'Pinkall-U', 'name': {'family': 'Pinkall', 'given': 'Ulrich'}}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}]}
Year: 2017
DOI: 10.1145/3072959.3073591
Clebsch maps encode velocity fields through functions. These functions contain valuable information about the velocity field. For example, closed integral curves of the associated vorticity field are level lines of the vorticity Clebsch map. This makes Clebsch maps useful for visualization and fluid dynamics analysis. Additionally they can be used in the context of simulations to enhance flows through the introduction of subgrid vorticity. In this paper we study spherical Clebsch maps, which are particularly attractive. Elucidating their geometric structure, we show that such maps can be found as minimizers of a non-linear Dirichlet energy. To illustrate our approach we use a number of benchmark problems and apply it to numerically given flow fields. Code and a video can be found in the ACM Digital Library.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/d7dk2-yza57Shape from metric
https://resolver.caltech.edu/CaltechAUTHORS:20180731-140714307
Authors: {'items': [{'id': 'Chern-A', 'name': {'family': 'Chern', 'given': 'Albert'}}, {'id': 'Knöppel-F', 'name': {'family': 'Knöppel', 'given': 'Felix'}}, {'id': 'Pinkall-U', 'name': {'family': 'Pinkall', 'given': 'Ulrich'}}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}]}
Year: 2018
DOI: 10.1145/3197517.3201276
We study the isometric immersion problem for orientable surface triangle meshes endowed with only a metric: given the combinatorics of the mesh together with edge lengths, approximate an isometric immersion into R^3. To address this challenge we develop a discrete theory for surface immersions into R^3. It precisely characterizes a discrete immersion, up to subdivision and small perturbations. In particular our discrete theory correctly represents the topology of the space of immersions, i.e., the regular homotopy classes which represent its connected components. Our approach relies on unit quaternions to represent triangle orientations and to encode, in their parallel transport, the topology of the immersion. In unison with this theory we develop a computational apparatus based on a variational principle. Minimizing a non-linear Dirichlet energy optimally finds extrinsic geometry for the given intrinsic geometry and ensures low metric approximation error.
We demonstrate our algorithm with a number of applications from mathematical visualization and art directed isometric shape deformation, which mimics the behavior of thin materials with high membrane stiffness.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/a4qmc-g7j50On bubble rings and ink chandeliers
https://resolver.caltech.edu/CaltechAUTHORS:20190712-133502830
Authors: {'items': [{'id': 'Padilla-M', 'name': {'family': 'Padilla', 'given': 'Marcel'}}, {'id': 'Chern-A', 'name': {'family': 'Chern', 'given': 'Albert'}}, {'id': 'Knöppel-F', 'name': {'family': 'Knöppel', 'given': 'Felix'}}, {'id': 'Pinkall-U', 'name': {'family': 'Pinkall', 'given': 'Ulrich'}}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}]}
Year: 2019
DOI: 10.1145/3306346.3322962
We introduce variable thickness, viscous vortex filaments. These can model such varied phenomena as underwater bubble rings or the intricate "chandeliers" formed by ink dropping into fluid. Treating the evolution of such filaments as an instance of Newtonian dynamics on a Riemannian configuration manifold we are able to extend classical work in the dynamics of vortex filaments through inclusion of viscous drag forces. The latter must be accounted for in low Reynolds number flows where they lead to significant variations in filament thickness and form an essential part of the observed dynamics. We develop and document both the underlying theory and associated practical numerical algorithms.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/gkk0f-0kw89Constrained Willmore surfaces
https://resolver.caltech.edu/CaltechAUTHORS:20210809-200325321
Authors: {'items': [{'id': 'Soliman-Yousuf', 'name': {'family': 'Soliman', 'given': 'Yousuf'}}, {'id': 'Chern-Albert', 'name': {'family': 'Chern', 'given': 'Albert'}}, {'id': 'Diamanti-Olga', 'name': {'family': 'Diamanti', 'given': 'Olga'}}, {'id': 'Knöppel-Felix', 'name': {'family': 'Knöppel', 'given': 'Felix'}}, {'id': 'Pinkall-Ulrich', 'name': {'family': 'Pinkall', 'given': 'Ulrich'}}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}]}
Year: 2021
DOI: 10.1145/3450626.3459759
Smooth curves and surfaces can be characterized as minimizers of squared curvature bending energies subject to constraints. In the univariate case with an isometry (length) constraint this leads to classic non-linear splines. For surfaces, isometry is too rigid a constraint and instead one asks for minimizers of the Willmore (squared mean curvature) energy subject to a conformality constraint. We present an efficient algorithm for (conformally) constrained Willmore surfaces using triangle meshes of arbitrary topology with or without boundary. Our conformal class constraint is based on the discrete notion of conformal equivalence of triangle meshes. The resulting non-linear constrained optimization problem can be solved efficiently using the competitive gradient descent method together with appropriate Sobolev metrics. The surfaces can be represented either through point positions or differential coordinates. The latter enable the realization of abstract metric surfaces without an initial immersion. A versatile toolkit for extrinsic conformal geometry processing, suitable for the construction and manipulation of smooth surfaces, results through the inclusion of additional point, area, and volume constraints.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/wjkgh-axt61Filament based plasma
https://resolver.caltech.edu/CaltechAUTHORS:20220726-998094000
Authors: {'items': [{'id': 'Padilla-Marcel', 'name': {'family': 'Padilla', 'given': 'Marcel'}}, {'id': 'Gross-Oliver', 'name': {'family': 'Gross', 'given': 'Oliver'}}, {'id': 'Knöppel-Felix', 'name': {'family': 'Knöppel', 'given': 'Felix'}}, {'id': 'Chern-Albert', 'name': {'family': 'Chern', 'given': 'Albert'}}, {'id': 'Pinkall-Ulrich', 'name': {'family': 'Pinkall', 'given': 'Ulrich'}}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}]}
Year: 2022
DOI: 10.1145/3528223.3530102
Simulation of stellar atmospheres, such as that of our own sun, is a common task in CGI for scientific visualization, movies and games. A fibrous volumetric texture is a visually dominant feature of the solar corona---the plasma that extends from the solar surface into space. These coronal fibers can be modeled as magnetic filaments whose shape is governed by the magnetohydrostatic equation. The magnetic filaments provide a Lagrangian curve representation and their initial configuration can be prescribed by an artist or generated from magnetic flux given as a scalar texture on the sun's surface. Subsequently, the shape of the filaments is determined based on a variational formulation. The output is a visual rendering of the whole sun. We demonstrate the fidelity of our method by comparing the resulting renderings with actual images of our sun's corona.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/4xa06-8t644Subdivision as a Fundamental Building Block of Digital Geometry Processing Algorithms
https://resolver.caltech.edu/CaltechAUTHORS:20230210-215324263
Authors: {'items': [{'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}]}
Year: 2023
Multi media data types such as digital sound, images, and video are now ubiquitous in all areas of computing and daily life. This wide impact was made possible by a number of factors. A key factor in the wide use of a given data type is the ease and economy of acquiring it. Using a rough time line one can observe that this was true for sound in the 70s, images in the 80s, and finally video in the 90s, roughly following the development of computing hardware with its ever increasing cpu and memory resources (Figure 1). Another key factor in the wide use of a given data type is the existence of efficient algorithms for creation, storage, transmission, editing and other manipulations of the data. The mathematical foundation for these algorithms has for a very long time rested on sampling and associated Fourier techniques. Even more recent developments, such as the use of wavelets for image and video compression still rest upon the foundation laid by Fourier analysis. As such, these methods now codified as "Digital Signal Processing" (DSP) have been extraordinarily successful impacting areas ranging from cheap consumer devices such as cell phones and MP3 players to high end scientific computing applications solving some of today's most demanding PDEs, for example.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/3ysyx-xw717Near-Optimal Connectivity Encoding of 2-Manifold Polygon Meshes
https://resolver.caltech.edu/CaltechAUTHORS:20230210-221859753
Authors: {'items': [{'id': 'Khodakovsky-Andrei', 'name': {'family': 'Khodakovsky', 'given': 'Andrei'}}, {'id': 'Alliez-Pierre', 'name': {'family': 'Alliez', 'given': 'Pierre'}}, {'id': 'Desbrun-M', 'name': {'family': 'Desbrun', 'given': 'Mathieu'}, 'orcid': '0000-0003-3424-6079'}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}]}
Year: 2023
Encoders for triangle mesh connectivity based on enumeration of vertex valences are among the best reported to date. They are both simple to implement and report the best compressed file sizes for a large corpus of test models. Additionally they have recently been shown to be near-optimal since they realize the Tutte entropy bound for all planar triangulations. In this paper we introduce a connectivity encoding method which extends these ideas to 2-manifold meshes consisting of faces with arbitrary degree. The encoding algorithm exploits duality by applying valence enumeration to both the primal and the dual mesh in a symmetric fashion. It generates two sequences of symbols, vertex valences, and face degrees, and encodes them separately using two context-based arithmetic coders. This allows us to exploit vertex or face regularity if present. When the mesh exhibits perfect face regularity (e.g., a pure triangle or quad mesh) or perfect vertex regularity (valence six or four respectively) the corresponding bit rate vanishes to zero asymptotically. For triangle meshes, our technique is equivalent to earlier valence-driven approaches. We report compression results for a corpus of standard meshes. In all cases we are able to show coding gains over earlier coders, sometimes as large as 50%. Remarkably, we even slightly gain over coders specialized to triangle or quad meshes. A theoretical analysis reveals that our approach is near-optimal as we achieve the Tutte entropy bound for arbitrary planar graphs of two bits per edge in the worst case.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/vvw3z-fex26Fitting Subdivision Surfaces
https://resolver.caltech.edu/CaltechAUTHORS:20230210-222759443
Authors: {'items': [{'id': 'Litke-Nathan', 'name': {'family': 'Litke', 'given': 'Nathan'}}, {'id': 'Levin-Adi', 'name': {'family': 'Levin', 'given': 'Adi'}}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}]}
Year: 2023
We introduce a new algorithm for fitting a Catmull-Clark subdivision surface to a given shape within a prescribed tolerance, based on the method of quasi-interpolation. The fitting algorithm is fast, local and scales well since it does not require the solution of linear systems. Its convergence rate is optimal for regular meshes and our experiments show that it behaves very well for irregular meshes. We demonstrate the power and versatility of our method with examples from interactive modeling, surface fitting, and scientific visualization.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/mz5yp-ms293Subdivision, multiresolution and the construction of
scalable algorithms in computer graphics
https://resolver.caltech.edu/CaltechAUTHORS:20230210-184408242
Authors: {'items': [{'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'P.'}, 'orcid': '0000-0002-0323-7674'}]}
Year: 2023
Multiresolution representations are a critical tool in addressing complexity
issues (time and memory) for the large scenes typically found in computer
graphics applications. Many of these techniques are based on classical subdivision
techniques and their generalizations. In this paper we review two
exemplary applications from this area, multiresolution surface editing and
semi-regular remeshing. The former is directed towards building algorithms
which are fast enough for interactive manipulation of complex surfaces of
arbitrary topology. The latter is concerned with constructing smooth parameterizations
for arbitrary topology surfaces as they typically arise from
3D scanning techniques. Remeshing such surfaces then allows the use of
classical subdivision ideas. We focus in particular on the practical aspects
of making the well understood mathematical machinery applicable and accessible
to the very general settings encountered in practice.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/x3seg-1xy36Sparse Matrix Solvers on the GPU: Conjugate Gradients and Multigrid
https://resolver.caltech.edu/CaltechAUTHORS:20230209-232912684
Authors: {'items': [{'id': 'Bolz-Jeff', 'name': {'family': 'Bolz', 'given': 'Jeff'}}, {'id': 'Farmer-Ian', 'name': {'family': 'Farmer', 'given': 'Ian'}}, {'id': 'Grinspun-Eitan', 'name': {'family': 'Grinspun', 'given': 'Eitan'}}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}]}
Year: 2023
Many computer graphics applications require high-intensity numerical simulation. We show that such computations can be performed efficiently on the GPU, which we regard as a full function streaming processor with high floating-point performance. We implemented two basic, broadly useful, computational kernels: a sparse matrix conjugate gradient solver and a regular-grid multigrid solver. Real-time applications ranging from mesh smoothing and parameterization to fluid solvers and solid mechanics can greatly benefit from these, evidence our example applications of geometric flow and fluid simulation running on NVIDIA's GeForce FX.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/e7fwb-j3238Normal Bounds for Subdivision-Surface Interference Detection
https://resolver.caltech.edu/CaltechAUTHORS:20230210-224940391
Authors: {'items': [{'id': 'Grinspun-Eitan', 'name': {'family': 'Grinspun', 'given': 'Eitan'}}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}]}
Year: 2023
Subdivision surfaces are an attractive representation when modeling arbitrary-topology free-form surfaces and show great promise for applications in engineering design and computer animation. Interference detection is a critical tool in many of these applications. In this paper, we derive normal bounds for subdivision surfaces and use these to develop an efficient algorithm for (self-) interference detection.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/8d6td-rax36Fitting subdivision surfaces
https://resolver.caltech.edu/CaltechAUTHORS:20230210-463137000.2
Authors: {'items': [{'id': 'Litke-Nathan', 'name': {'family': 'Litke', 'given': 'Nathan'}}, {'id': 'Levin-Adi', 'name': {'family': 'Levin', 'given': 'Adi'}}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}]}
Year: 2023
DOI: 10.1109/visual.2001.964527
We introduce a new algorithm for fitting a Catmull-Clark subdivision surface to a given shape within a prescribed tolerance, based on the method of quasi-interpolation. The fitting algorithm is fast, local and scales well since it does not require the solution of linear systems. Its convergence rate is optimal for regular meshes and our experiments show that it behaves very well for irregular meshes. We demonstrate the power and versatility of our method with examples from interactive modeling, surface fitting, and scientific visualization.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/tm0ke-g1c61Discrete Differential-Geometry Operators for Triangulated 2-Manifolds
https://resolver.caltech.edu/CaltechAUTHORS:20230210-221150053
Authors: {'items': [{'id': 'Meyer-Mark', 'name': {'family': 'Meyer', 'given': 'Mark'}}, {'id': 'Desbrun-M', 'name': {'family': 'Desbrun', 'given': 'Mathieu'}, 'orcid': '0000-0003-3424-6079'}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}, {'id': 'Barr-A-H', 'name': {'family': 'Barr', 'given': 'Alan H.'}}]}
Year: 2023
This paper proposes a unified and consistent set of flexible tools to approximate important geometric attributes, including normal vectors and curvatures on arbitrary triangle meshes. We present a consistent derivation of these first and second order differential properties using averaging Voronoi cells and the mixed Finite-Element/Finite-Volume method, and compare them to existing formulations. Building upon previous work in discrete geometry, these operators are closely related to the continuous case, guaranteeing an appropriate extension from the continuous to the discrete setting: they respect most intrinsic properties of the continuous differential operators. We show that these estimates are optimal in accuracy under mild smoothness conditions, and demonstrate their numerical quality. We also present applications of these operators, such as mesh smoothing, enhancement, and quality checking, and show results of denoising in higher dimensions, such as for tensor images.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/0rsjd-50h08C¹-continuous Terrain Reconstruction from Sparse Contours
https://resolver.caltech.edu/CaltechAUTHORS:20230209-232128443
Authors: {'items': [{'id': 'Hormann-Kai', 'name': {'family': 'Hormann', 'given': 'Kai'}}, {'id': 'Spinello-Salvatore', 'name': {'family': 'Spinello', 'given': 'Salvatore'}}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}]}
Year: 2023
Contour lines from topographic maps are still the most common form of elevation data for the Earth's surface and in the case of historical landscapes, they often are the only available source of information. In this paper we present a new contour interpolation method that solves this bivariate problem by considering univariate curve interpolation along the approximate gradient directions of the unknown surface. For a point between two contours the height value is computed with Hermite interpolation based on the shortest distances to the contours and height and derivative information at the contours. The surfaces generated are C¹ except at terrain characteristics such as ridges and valleys which are reconstructed as sharp features. The method also faithfully reconstructs summits, pits, and saddles and is especially well-suited for sparse sets of contours. The approach allows for an efficient numerical implementation as we demonstrate with a number of examples.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/yeth0-26786Composite primal/dual √3-subdivision schemes
https://resolver.caltech.edu/CaltechAUTHORS:20230210-191814502
Authors: {'items': [{'id': 'Oswald-Peter', 'name': {'family': 'Oswald', 'given': 'Peter'}}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}]}
Year: 2023
We present new families of primal and dual subdivision schemes for triangle meshes and 3-refinement. The proposed schemes use two simple local rules which cycle between primal and dual meshes a number of times. The resulting surfaces become very smooth at regular vertices if the number of cycles is ⩾2. The C¹-property is violated only at low-valence irregular vertices, and can be restored by slight modifications of the local rules used.
As a generalization, we introduce a wide class of composite subdivision schemes suitable for arbitrary topologies and refinement rules. A composite scheme is defined by a simple upsampling from the coarse to a refined topology, embedded into a cascade of geometric averaging operators acting on coarse and/or refined topologies. We propose a small set of such averaging rules (and some of their parametric extensions) which allow for the switching between control nets associated with the same or different topologic elements (vertices, edges, faces), and show a number of examples, based on triangles, that the resulting class of composite subdivision schemes contains new and old, primal and dual schemes for 3-refinement as well as for quadrisection. As a common observation from the examples considered, we found that irregular vertex treatment is necessary only at vertices of low valence, and can easily be implemented by using generic modifications of some elementary averaging rules.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/z01yk-srn46CHARMS: A Simple Framework for Adaptive Simulation
https://resolver.caltech.edu/CaltechAUTHORS:20230210-220110128
Authors: {'items': [{'id': 'Grinspun-Eitan', 'name': {'family': 'Grinspun', 'given': 'Eitan'}}, {'id': 'Krysl-Petr', 'name': {'family': 'Krysl', 'given': 'Petr'}}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}]}
Year: 2023
Finite element solvers are a basic component of simulation applications; they are common in computer graphics, engineering, and medical simulations. Although adaptive solvers can be of great value in reducing the often high computational cost of simulations they are not employed broadly. Indeed, building adaptive solvers can be a daunting task especially for 3D finite elements. In this paper we are introducing a new approach to produce conforming, hierarchical, adaptive refinement methods (CHARMS). The basic principle of our approach is to refine basis functions, not elements. This removes a number of implementation headaches associated with other approaches and is a general technique independent of domain dimension (here 2D and 3D), element type (e.g., triangle, quad, tetrahedron, hexahedron), and basis function order (piece-wise linear, higher order B-splines, Loop subdivision, etc.). The (un-)refinement algorithms are simple and require little in terms of data structure support. We demonstrate the versatility of our new approach through 2D and 3D examples, including medical applications and thin-shell animations.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/rv6d9-a9n32Motion from Shape Change
https://authors.library.caltech.edu/records/zgpj4-6p551
Authors: {'items': [{'id': 'Gross-Oliver', 'name': {'family': 'Gross', 'given': 'Oliver'}, 'orcid': '0000-0001-7970-5596'}, {'id': 'Soliman-Yousuf', 'name': {'family': 'Soliman', 'given': 'Yousuf'}, 'orcid': '0000-0003-4023-5026'}, {'id': 'Padilla-Marcel', 'name': {'family': 'Padilla', 'given': 'Marcel'}, 'orcid': '0000-0001-8962-9355'}, {'id': 'Knöppel-Felix', 'name': {'family': 'Knöppel', 'given': 'Felix'}, 'orcid': '0000-0002-0191-2859'}, {'id': 'Pinkall-Ulrich', 'name': {'family': 'Pinkall', 'given': 'Ulrich'}, 'orcid': '0000-0002-2087-6435'}, {'id': 'Schröder-P', 'name': {'family': 'Schröder', 'given': 'Peter'}, 'orcid': '0000-0002-0323-7674'}]}
Year: 2023
DOI: 10.1145/3592417
We consider motion effected by shape change. Such motions are ubiquitous in nature and the human made environment, ranging from single cells to platform divers and jellyfish. The shapes may be immersed in various media ranging from the very viscous to air and nearly inviscid fluids. In the absence of external forces these settings are characterized by constant momentum. We exploit this in an algorithm which takes a sequence of changing shapes, say, as modeled by an animator, as input and produces corresponding motion in world coordinates. Our method is based on the geometry of shape change and an appropriate variational principle. The corresponding Euler-Lagrange equations are first order ODEs in the unknown rotations and translations and the resulting time stepping algorithm applies to all these settings without modification as we demonstrate with a broad set of examples.https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/zgpj4-6p551