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6.838:
Shape AnalysisJustin Solomon
Spring 2021
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Instructor: Justin SolomonEmail: [email protected]
Geometric Data Processing Group:http://gdp.csail.mit.edu
Will cover administrative details over Zoom.
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CodingJulia, Python, or Matlab preferred
MathFluency in linear algebra and multivariable calculus
Not required (won’t hurt):Graphics, differential geometry, numerics, ML
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We want you to take this course!Assignments intended to be interesting
(may be unintentionally easy/hard!).
Image from postermywall.com
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I. Theoretical toolbox
II. Computational toolbox
III. Application areas
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I. Theoretical toolbox
II. Computational toolbox
III. Application areas
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http://www.turbosquid.com/3d-models/medically-human-brain-3d-model/437885
?
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http://www.turbosquid.com/3d-models/medically-human-brain-3d-model/437885
? ?
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Spivak: A Comprehensive Introduction to Differential Geometry
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http://en.wikipedia.org/wiki/Differential_geometry
Study of smooth manifolds
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http://upload.wikimedia.org/wikipedia/commons/b/bc/Double_torus_illustration.png
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http://www.sciencedirect.com/science/article/pii/S0010448510001983
Curvature and shape properties
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Distances
Crane, Weischedel, Wardetzky. Geodesics in heat. TOG 2013.Wittman. Manifold learning techniques.
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Flows and vector fields
Vaxman et al.Directional field synthesis, design, and processing.
EG STAR 2016.
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Differential operators
Vallet and Lévy. Spectral Geometry Processing with Manifold Harmonics. EG 2008
“Enneahedra”
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Only need angles and distanceshttp://upload.wikimedia.org/wikipedia/commons/2/2c/Hobo%E2%80%93Dyer_projection_SW.jpg
Same distance?
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Only need angles and distanceshttp://www.phy.syr.edu/courses/modules/LIGHTCONE/pics/curved.jpg
Ant’s view
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Heeren et al.Splines in the Space of Shells.
SGP 2016.
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http://en.wikipedia.org/wiki/Double_pendulumhttp://www.ualberta.ca/dept/math/gauss/fcm/BscIdeas/SpcDmnsn/pndlm2.htm
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Peyré, Cuturi, and Solomon.Gromov-Wasserstein Averaging of Kernel and Distance Matrices.
ICML 2016.
Tenenbaum et al.A Global Geometric Framework for Nonlinear Dimensionality Reduction.Science 2000.
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https://www.researchgate.net/publication/274071948_Space_charge_and_resistive_wall_impedance_computation_in_the_frequency_domain_using_the_finite_element_methodhttp://blog.forty.to/wp-content/uploads/2013/05/Bars1.jpg
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I. Theoretical toolbox
II. Computational toolbox
III. Application areas
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Triangle mesh Triangle soup
Graph Point cloud
Pairwise distance matrix Dataset Network
Nearly anything with a notion of proximity/distance/curvature/…
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• Collection of flat triangles
• Approximates a smooth surface
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Can a triangle mesh have curvature?
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Combine smooth and discrete
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http://www.geomtop.org/teaching/conformal_2012Summer/title_image.jpg
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Discretevs.
Discretized
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Discrete theory parallelingdifferential geometry.
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Structure preservation[struhk-cher pre-zur-vey-shuh n]:Keeping properties from the continuous abstraction exactly true in a discretization.
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Images from: Grinspun and Secord, “The Geometry of Plane Curves” (SIGGRAPH 2006)
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Convergence[kuh n-vur-juh ns]:Increasing approximation quality as a discretization is refined.
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Can you have it all?
Fey et al. 30 Rock (2006-2013).
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Pick and choosewhich properties you need.
But there is a huge toolbox of algorithms to draw from!
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Chuang and Kazhdan. Fast Mean-Curvature Flow via Finite-Elements Tracking. CGF 2011.Coifman & Lafon. Diffusion Maps. ACHA 2006.
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Smith and Schaefer. Bijective parameterization with free boundaries. SIGGRAPH 2015.
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Bommes, Zimmer, Kobbelt. Mixed-integer quadrangulation. SIGGRAPH 2009.
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Huang, Guibas. Consistent shape maps via semidefinite
programming. SGP 2013.
Krishnan, Fattal, Szeliski. Efficient preconditioning of
Laplacian matrices for computer graphics. SIGGRAPH 2013.
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Palmer, Bommes, & Solomon.Algebraic Representations for Volumetric Frame Fields.TOG 2020.
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I. Theoretical toolbox
II. Computational toolbox
III. Application areas
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Graphics
Mertens et al. “Texture Transfer Using Geometry Correlation.”Fisher et al. “Context-Based Search for 3D Models.”
Mitra et al. “Symmetrization.”Bokeloh et al. “A connection between partial symmetry and inverse procedural modeling.”
Editing
Retrieval
Exploiting patterns
Transfer
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Vision
http://eijournal.com/newsite/wp-content/uploads/2012/01/VELODYNE-IMAGE.jpgKim et al. “Multi-view image and tof sensor fusion for dense 3d reconstruction.”
Solomon et al. “Discovery of Intrinsic Primitives on Triangle Meshes.”Bronstein et al. “Three-Dimensional Face Recognition.”
Recognition
Segmentation
Navigation
Reconstruction
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Medical Imaging
http://dmfr.birjournals.org/content/33/4/226/F3.large.jpghttp://www-sop.inria.fr/asclepios/software/inriaviz4d/SphericalImTransp.png
http://www.creatis.insa-lyon.fr/site/sites/default/files/segm2.png
Analysis
Registration
Segmentation
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Manufacturing and Fabrication
http://www.conduitprojects.com/php/images/scan.jpghttp://www.emeraldinsight.com/content_images/fig/0330290204005.png
Scanning Defect detection
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Architecture
Design and analysisVouga et al. “Design of self-supporting surfaces.”
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Shape collectionsHuang et al. “Consistent shape maps via semidefinite programming.”
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CorrespondenceOvsjanikov et al. “Functional maps.”
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Deformation transferWeise et al. “Realtime performance-based facial animation.”
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Simulation
Bergou et al. “Discrete viscous threads.”Wardetzky et al. “Discrete quadratic curvature energies.”
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Scientific visualizationCrane et al. “Spin Transformations of Discrete Surfaces.”
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Segmentation
http://www.bioinformaticslaboratory.nl/twiki/pub/EBioScience/News/freesurfer-3d.jpghttp://hal.inria.fr/docs/00/40/21/30/IMG/vivodtzev_et_al-Dagstuhl03.jpg
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Machine learningZhu et al. Semi-Supervised Learning Using Gaussian Fields and Harmonic Functions. ICML 2003.
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StatisticsHou et al. Novel semisupervised high-dimensional correspondences learning method. Opt. Eng. 2008.
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6.838:
Shape AnalysisJustin Solomon
Spring 2021