as-killing-as-possible vector fields for planar deformation
TRANSCRIPT
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Justin Solomon Geometric Computing Group Stanford University
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Compute shape descriptorshttp://graphics.stanford.edu/projects/lgl/papers/sog-hks-09/sog-hks-09.pdf
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Extract important featureshttp://people.cs.umass.edu/~kalo/papers/LabelMeshes/LabelMeshes.pdf
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Map shapes to one another
Á
http://www.stanford.edu/~justso1/assets/fmaps.pdf
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Relate new scans to known modelshttp://www.hao-li.com/publications/papers/siggraph2011RPBFA.pdf
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Understand collections of shapeshttp://graphics.stanford.edu/projects/lgl/papers/nbwyg-oaicsm-11/nbwyg-oaicsm-11.pdf
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Graphics
http://people.csail.mit.edu/tmertens/papers/textransfer_electronic.pdf http://graphics.stanford.edu/~mdfisher/Data/Context.pdf
http://graphics.stanford.edu/~niloy/research/symmetrization/paper_docs/symmetrization_sig_07.pdf http://www.mpi-inf.mpg.de/~mbokeloh/project_dockingSites.html
Editing
Retrieval
Exploiting patterns
Transfer
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Vision
http://eijournal.com/newsite/wp-content/uploads/2012/01/VELODYNE-IMAGE.jpg http://www.stanford.edu/~jinhae/iccv09/
http://www.stanford.edu/~justso1/assets/intrinsic_part_discovery.pdf http://www.cs.technion.ac.il/~ron/PAPERS/BroBroKimIJCV05.pdf
Recognition
Segmentation
Navigation
Reconstruction
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Medical Imaging
http://dmfr.birjournals.org/content/33/4/226/F3.large.jpg http://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
http://www.conduitprojects.com/php/images/scan.jpg http://www.emeraldinsight.com/content_images/fig/0330290204005.png
Scanning Defect detection
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Analysis and correspondence form a
large and diverse field.
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Summarize approaches to
Local descriptors Shape understanding Correspondence Shape collections
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vision.in.tum.de/_media/spezial/bib/aubry-et-al-4dmod11.pdf
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http://liris.cnrs.fr/meshbenchmark/images/fig_attacks.jpg
§
Rn
Pointwise quantity
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http://isg.cs.tcd.ie/spheretree/pics/bunny.gif
§
RjV j£n
Pointwise quantity
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Distinguishing Provides useful information about a point
Stable Numerically and geometrically
Intrinsic No dependence on embedding
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Distinguishing Provides useful information about a point
Stable Numerically and geometrically
Intrinsic No dependence on embedding
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Isometry [ahy-som-i-tree]: Bending without stretching.
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Isometry invarianthttp://www.revedreams.com/crochet/yarncrochet/nonorientable-crochet/
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http://www.flickr.com/photos/melvinvoskuijl/galleries/72157624236168459
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Few shapes can deform isometricallyhttp://www.4tnz.com/content/got-toilet-paper
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Few shapes can deform isometricallyhttp://www.4tnz.com/content/got-toilet-paper
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http://www.sciencedirect.com/science/article/pii/S0010448510001983
K = ·1·2 H = 1=2(·1 +·2)
Gaussian and mean curvature
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http://www.sciencedirect.com/science/article/pii/S0010448510001983
K = ·1·2
Gaussian and mean curvature
Theorema Egregium (“Remarkable Theorem”):
Gaussian curvature is intrinsic.
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Localized differential descriptors
K = ·1·2
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Nonuniquehttp://www.integrityware.com/images/MerceedesGaussianCurvature.jpg
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Principal curvatures
Shape index
Curvedness
·1; ·2
2
¼arctan
µ·1 + ·2
·1 ¡ ·2
¶
r1
2(·21 + ·22)
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Incorporate neighborhood information
in an intrinsic fashion.
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Incorporate neighborhood information
in an intrinsic fashion.
http://ddg.cs.columbia.edu/SIGGRAPH06/DDGCourse2006.pdf
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¢= d ? d ?+ ? d ? d
An intrinsic operator
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¢= d ? d ?+ ? d ? d
¢ ¢ ¢
An intrinsic operator
¢Á1 = ¸1Á1 ¢Á2 = ¸2Á2 ¢Á3 = ¸3Á3 ¢Á4 = ¸4Á4 ¢Á5 = ¸5Á5
(¢Á0 = 0)
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GPS(p) =
µ1p¸1Á1(p);
1p¸2Á2(p);
1p¸3Á3(p); ¢ ¢ ¢
¶
Good properties:
Isometry-invariant Unique to each point Complete description of intrinsic
geometry Dot products, distances meaningful
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GPS(p) =
µ1p¸1Á1(p);
1p¸2Á2(p);
1p¸3Á3(p); ¢ ¢ ¢
¶
Bad properties:
Assumes unique λ’s Potential for eigenfunction
“switching” upon deformation Nonlocal feature
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Heat equationhttp://graphics.stanford.edu/courses/cs468-10-fall/LectureSlides/11_shape_matching.pdf
@u
@t= ¡¢u
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Wave equation
@2u
@t2= ¡i¢u
http://www.ceremade.dauphine.fr/~peyre/numerical-tour/tours/meshproc_5_pde/index_06.png
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@2u
@t2= ¡i¢u
Wave equationhttp://www.ceremade.dauphine.fr/~peyre/numerical-tour/tours/meshproc_5_pde/index_06.png
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@u
@t= ¡¢u
Heat equation
u =
1X
n=0
ane¡¸ntÁn(x)
µan =
Z
§
u0 ¢ Án dA¶
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kt(x; x) =
1X
n=0
e¡¸itÁn(x)2
Continuous function on [0,∞)
How much heat diffuses from x to
itself in time t?
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kt(x; x) =
1X
n=0
e¡¸itÁn(x)2
http://graphics.stanford.edu/projects/lgl/papers/sog-hks-09/sog-hks-09.pdf
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Good properties:
Isometry-invariant Multiscale Not subject to switching Easy to compute Related to curvature at small scales
kt(x; x) =
1X
n=0
e¡¸itÁn(x)2
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Bad properties:
Issues remain with repeated eigenvalues
Theoretical guarantees require (near-)isometry
kt(x; x) =
1X
n=0
e¡¸itÁn(x)2
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Scale-Invariant HKS (SI-HKS)http://www.cs.technion.ac.il/~mbron/publications_conference.html
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Affine-Invariant HKShttp://www.cs.technion.ac.il/~darav/RavBroBroKimAffine10TR.pdf
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Photometric HKShttp://www.cs.technion.ac.il/~mbron/publications_conference.html
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Volumetric HKShttp://www.cs.technion.ac.il/~mbron/publications_conference.html
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WKS(E;x) = limT!1
1
T
Z T
0
jÃE(x; t)j2 dt =1X
n=0
Án(x)2fE(¸k)
2
Average probability over time that particle is at x.
Initial energy distribution
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WKS(E;x) = limT!1
1
T
Z T
0
jÃE(x; t)j2 dt =1X
n=0
Án(x)2fE(¸k)
2
HKS WKS vision.in.tum.de/_media/spezial/bib/aubry-et-al-4dmod11.pdf
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WKS(E;x) = limT!1
1
T
Z T
0
jÃE(x; t)j2 dt =1X
n=0
Án(x)2fE(¸k)
2
Good properties:
[Similar to HKS] Localized in frequency Stable under some non-isometric
deformation Some multi-scale properties
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WKS(E;x) = limT!1
1
T
Z T
0
jÃE(x; t)j2 dt =1X
n=0
Án(x)2fE(¸k)
2
Bad properties:
[Similar to HKS] Can filter out large-scale features
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1X
n=0
f(¸n)Án(x)2
Can you learn the function f?http://arxiv.org/pdf/1110.5015.pdf
Considerations:
Collection of shapes Potential transformations/noise
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Spin imageshttp://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=00765655
Bin points using: ® = distance to normal line
¯ = distance to tangent plane
Can use low-rank approximation!
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Shape contexthttp://cg.tuwien.ac.at/hostings/cescg/CESCG-2003/MKoertgen/paper.pdf
Bin directions y-x for each x
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Integral volumehttp://graphics.stanford.edu/~niloy/research/global_registration/paper_docs/global_registration_sgp_05_poster.pdf
Vr(p) =
Z
Br(p)\SdV
Related to mean curvature Integral invariants robust
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Shape Diameter Functionhttp://www.cs.jhu.edu/~misha/ReadingSeminar/Papers/Shapira08.pdf
Weighted average distance along the surface Lightweight version of medial axis distance
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Signature of Histograms of OrienTationshttp://www.vision.deis.unibo.it/fede/papers/eccv10.pdf
Bin nearby normals in a canonical orientation
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Structural indexing Point signatures Point fingerprints Intrinsic shape signature Multi-scale surface descriptors Slippage Spherical harmonics RIFT HMM …
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http://graphics.stanford.edu/~niloy/research/structure/paper_docs/structure_sig_08.pdf
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Segmentation Symmetry detection Global shape description Retrieval Recognition Feature extraction Alignment …
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Segmentation Symmetry detection Global shape description Retrieval Recognition Feature extraction Alignment …
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Compression Reconstruction Classification
Analysis Alignment Matching
http://www0.cs.ucl.ac.uk/staff/n.mitra/research/symmetry_survey/symmetrySurvey_12.html http://www.cs.princeton.edu/courses/archive/fall03/cs597D/lectures/symmetry.pdf
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http://graphics.stanford.edu/~niloy/research/structure/paper_docs/structure_sig_08.pdf
Continuous
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Ex. 1: Discrete extrinsic symmetries
Partial and Approximate Symmetry Detection for 3D Geometry Mitra, Guibas, Pauly 2006
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Ex. 1: Discrete extrinsic symmetries
Partial and Approximate Symmetry Detection for 3D Geometry Mitra, Guibas, Pauly 2006
Compute simple curvature features to help pair similar
points.
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Ex. 1: Discrete extrinsic symmetries
Partial and Approximate Symmetry Detection for 3D Geometry Mitra, Guibas, Pauly 2006
Pairs of points with similar signatures vote for different
transformations.
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Ex. 1: Discrete extrinsic symmetries
Partial and Approximate Symmetry Detection for 3D Geometry Mitra, Guibas, Pauly 2006
Use mean shift clustering to find prominent
transformations.
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Ex. 1: Discrete extrinsic symmetries
Partial and Approximate Symmetry Detection for 3D Geometry Mitra, Guibas, Pauly 2006
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Ex. 1: Discrete extrinsic symmetries
Discovering Structural Regularity in 3D Geometry Pauly et al. 2008
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Ex. 2: Discrete intrinsic symmetries
Partial Intrinsic Reflectional Symmetry of 3D Shapes Xu et al. 2009
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Ex. 2: Discrete intrinsic symmetries
Partial Intrinsic Reflectional Symmetry of 3D Shapes Xu et al. 2009
IRSA Transform (“Intrinsic Reflectional Symmetry Axis”)
Want T: MM (or parts thereof) preserving geodesic distances; fixed points are symmetry axis Sample potential axes; voting scheme for IRSA transform
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Ex. 2: Discrete intrinsic symmetries
Global Intrinsic Symmetries of Shapes Ovsjanikov, Sun, and Guibas 2008
Intrinsic symmetries become extrinsic in GPS space!
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Ex. 3: Continuous intrinsic symmetries
On Discrete Killing Fields and Patterns on Surfaces Ben Chen et al. 2010
Flows of Killing vector fields (KVFs) generate isometries DEC framework for finding approximate KVFs
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Ex. 3: Continuous intrinsic symmetries
Discovery of Intrinsic Primitives on Triangle Meshes Solomon et al. 2011
Approximate KVFs can be used to find nearly symmetric pieces
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Feature points
A Concise and Provably Informative Multi-Scale Signature Based on Heat Diffusion Sun, Ovsjanikov, and Guibas 2009
Maxima of kt(x,x) for large t.
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Feature curves
Separatrix Persistence: Extraction of Salient Edges on Surfaces Using Topological Methods Weinkauf and Gunther 2009
Filter out extraneous feature curves
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Feature curves
Separatrix Persistence: Extraction of Salient Edges on Surfaces Using Topological Methods Weinkauf and Gunther 2009
Morse-Smale Complex: Topological skeleton of critical points and separatricies
x is in the descending manifold of critical point p if there exists a gradient flow curve connecting p to x
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Feature curves
Separatrix Persistence: Extraction of Salient Edges on Surfaces Using Topological Methods Weinkauf and Gunther 2009
Morse-Smale Complex: Topological skeleton of critical points and separatricies
x is in the ascending manifold of critical point p if there exists a gradient flow curve connecting x to p
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Feature curves
Separatrix Persistence: Extraction of Salient Edges on Surfaces Using Topological Methods Weinkauf and Gunther 2009
Morse-Smale Complex: Topological skeleton of critical points and separatricies
Separatrix: intersection of one ascending and one descending manifold
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Feature curves
Separatrix Persistence: Extraction of Salient Edges on Surfaces Using Topological Methods Weinkauf and Gunther 2009
1. Build combinatorial Morse-Smale complex.
2. Apply persistence to simplify.
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http://www.cs.rutgers.edu/~decarlo/pubs/sg03.pdf
Use curvature to choose better contour lines
Suggestive contour generator:
Points with zero/increasing curvature in view direction
Feature curves
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“Feature patches” Segmentation
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“Feature patches” Segmentation
Simplest strategy: Cluster feature points (k-means, mean shift, etc.); use standard vision techniques for continuous regions
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“Feature patches” Segmentation
Supervised learning problem
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“Feature patches” Segmentation
Variational Shape Approximation Cohen-Steiner, Alliez, and Desbrun 2004
Partition Proxies Simplified
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“Feature patches” Segmentation
Variational Shape Approximation Cohen-Steiner, Alliez, and Desbrun 2004
Partition Proxies Simplified
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“Feature patches” Segmentation
Variational Shape Approximation Cohen-Steiner, Alliez, and Desbrun 2004
Partition Proxies
Flood using priority queue
(Xi,Ni) minimizing fixed functional
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(Gromov-)Hausdorff distance
dH(X;Y ) = max
½supx2X
infy2Y
d(x; y); supy2Y
infx2X
d(x; y)
¾
dGH(X;Y ) = minf;g isometries
dH(f(x); g(y))
Easy to compute
Hard to compute Less hard to approximate
Related to Gromov-Wasserstein distance
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Perceptual distancehttp://liris.cnrs.fr/guillaume.lavoue/travaux/conference/SPIE-2006.pdf
Combine simple metrics (curvature, edge length distortion, etc.) with user studies.
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Segmentation Symmetry detection Global shape description Retrieval Recognition Feature extraction Alignment …
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Intermission http://en.wikipedia.org/wiki/Let%27s_All_Go_to_the_Lobby
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http://www.cs.princeton.edu/~funk/sig11.pdf
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Which points map to which?
http://graphics.stanford.edu/projects/lgl/papers/ommg-opimhk-10/ommg-opimhk-10.pdf http://www.cs.princeton.edu/~funk/sig11.pdf
http://gfx.cs.princeton.edu/pubs/Lipman_2009_MVF/mobius.pdf
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MOSTLY
BORROWED
Maks Ovsjanikov and Mirela Ben-Chen, CS 468
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Local vs. global Refinement or alignment?
Rigid vs. deformable Rotation/translation or stretching?
Pair vs. collection Two shapes or many shapes?
http://graphics.stanford.edu/courses/cs468-10-fall/LectureSlides/11_shape_matching.pdf
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Local vs. global Refinement or alignment?
Rigid vs. deformable Rotation/translation or stretching?
Pair vs. collection Two shapes or many shapes?
http://graphics.stanford.edu/courses/cs468-10-fall/LectureSlides/11_shape_matching.pdf
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Local vs. global Refinement or alignment?
Rigid vs. deformable Rotation/translation or stretching?
Pair vs. collection Two shapes or many shapes?
http://graphics.stanford.edu/courses/cs468-10-fall/LectureSlides/11_shape_matching.pdf
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Iterative Closest Point (ICP)
Repeat: 1.For each xi in X, find
closest yi in Y. 2.Find rigid deformation
(R,T) minimizing X
i
k(Rxi + T )¡ yik
http://graphics.stanford.edu/courses/cs468-10-fall/LectureSlides/11_shape_matching.pdf http://www.gris.uni-tuebingen.de/people/staff/bokeloh/gallery/bunny_res1.png
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Iterative Closest Point (ICP)
Repeat: 1.For each xi in X, find
closest yi in Y. 2.Find rigid deformation
(R,T) minimizing X
i
k(Rxi + T )¡ yik
http://graphics.stanford.edu/courses/cs468-10-fall/LectureSlides/11_shape_matching.pdf http://www.gris.uni-tuebingen.de/people/staff/bokeloh/gallery/bunny_res1.png
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Selection of sample points One or both surfaces? How many?
Matching points on the surfaces Closest? Approximate nearest? Normal lines? Compatible normal/curvature/color?
Weighting correspondences Distance? Compatibility? Scanner certainty?
www.math.tau.ac.il/~dcor/Graphics/adv-slides/ICP.ppt
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Reject outlier pairs Too far? Inconsistent with neighbors? Incompatible descriptors?
Modified error metric Allow affine transformations? Nonrigid motion?
Optimization technique Avoid local minima?
www.math.tau.ac.il/~dcor/Graphics/adv-slides/ICP.ppt
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http://gmsv.kaust.edu.sa/people/faculty/pottmann/pottmann_pdf/registration.pdf
Align shapes in arbitrary positions Starting point for ICP
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http://graphics.stanford.edu/courses/cs468-10-fall/LectureSlides/11_shape_matching.pdf
Exhaustive search Normalization Random sampling Invariance
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http://graphics.stanford.edu/courses/cs468-10-fall/LectureSlides/11_shape_matching.pdf http://vis.lbl.gov/~scrivelli/DShop_research.html
Exhaustive search
Sample possible alignments Keep best post-ICP (Slow, only for rigid!)
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Find canonical alignment
e.g. using PCA; reduces number of starting points http://graphics.stanford.edu/courses/cs468-10-fall/LectureSlides/11_shape_matching.pdf
Normalization
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Random sampling
RANSAC: Random Sample Consensus
http://upload.wikimedia.org/wikipedia/commons/d/de/Fitted_line.svg
Repeat: 1.Guess minimum
number of points to determine parameters
2.Check if model works for other points
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Random sampling
4-Points Congruent Sets for Robust Pairwise Surface Registration Aiger, Mitra, and Cohen-Or 2008
RANSAC with sets of four near-coplanar points. Affine maps preserve ||c-e||/||c-d||, so sample points e’ with these ratios (n2 time), then match those.
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Invariance: Already done!
1.Find interesting points. 2.Match feature vectors on those points. 3.Compute the aligning transformation.
http://graphics.stanford.edu/courses/cs468-10-fall/LectureSlides/11_shape_matching.pdf
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Needs a deformation model
Elastic, thin shell, volumetric, ARAP, bending, …
igl.ethz.ch/projects/ARAP/
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Needs a deformation model
Elastic, thin shell, volumetric, ARAP, bending, …
igl.ethz.ch/projects/ARAP/
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http://www.hao-li.com/publications/papers/sgp2008GCO.pdf
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Generalized Multi-Dimensional Scaling (GMDS)
Generalized Multidimensional Scaling Bronstein, Bronstein, Kimmel 2006
Embed samples of one surface directly over another by minimizing a “generalized stress” involving geodesics.
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Non-Rigid Registration Under Isometric Deformations Huang et al. 2008
Alternate between matching feature points using descriptors and moving other points in rigid clusters; isometry assumption helps prune bad matches.
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Deformation-Driven Shape Correspondence Zheng et al. 2008
1.Extract feature points (part extrema)
2.Combinatorial search matching features; evaluate by deforming the mesh
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One Point Isometric Matching with the Heat Kernel Ovsjanikov et al. 2010
HKMp(x; t) = kt(p;x)
How much heat diffuses from p to x in time t?
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One Point Isometric Matching with the Heat Kernel Ovsjanikov et al. 2010
HKMp(x; t) = kt(p;x)
Theorem: Only have to match one point!
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Möbius Voting for Surface Correspondence Lipman and Funkhouser 2009
isometries µ conformal mapsHard! Easier
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Möbius Voting for Surface Correspondence Lipman and Funkhouser 2009
1. Map surfaces to complex plane
2. Select three points 3. Map plane to itself
matching these points 4.Vote for pairings using
distortion metric to weight
5. Return to 2
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Blended Intrinsic Maps Kim, Lipman, and Funkhouser 2011
Different simple maps might be good in different places.
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Blended Intrinsic Maps Kim, Lipman, and Funkhouser 2011
Combine good parts of different maps!
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Blended Intrinsic Maps Kim, Lipman, and Funkhouser 2011
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Blended Intrinsic Maps Kim, Lipman, and Funkhouser 2011
Find groups of consistent/similar maps by clustering in a similarity matrix.
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Blended Intrinsic Maps Kim, Lipman, and Funkhouser 2011
Weight maps at each vertex based on deviation from isometry. Output weighted geodesic centroid.
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Map representations
f ± Á
f :M2 !R
f ± Á Á
Functional Maps: A Flexible Representation of Maps Between Shapes Ovsjanikov et al. 2012 (to appear)
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Map representations
f ± Á
Soft Maps Between Surfaces Solomon et al. 2012 (to appear … shortly!)
Exploring Collections of 3D Models using Fuzzy Correspondences
Kim et al. 2012 (to appear)
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http://www.stanford.edu/~huangqx/siga11_jointseg.pdf
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One surface Two surfaces
…
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Sceneshttp://graphics.stanford.edu/~mdfisher/Data/GraphKernel.pdf
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Databaseshttp://people.cs.umass.edu/~kalo/papers/ShapeSynthesis/index.html
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Motions of one objecthttp://ars.sciencedirect.com/content/image/1-s2.0-S0097849311000501-gr9.jpg
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You can learn about one shape using its
relationship to other shapes.
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Function Key features Deformation model Usability Structure Symmetries Missing information …
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“There are manifoldnesses in which the determination of position requires not a finite number, but . . . a continuous manifoldness of determinations of quantity. Such manifoldnesses are, for example, the possible determinations of a function for a given region, the possible shapes of a solid figure, and so on.” - Riemann (via Clifford)
http://www.ics.uci.edu/~fowlkes/papers/f-ma240.pdf
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Learn shape space from exampleshttp://graphics.ethz.ch/Downloads/Publications/Papers/2011/Mar11/Mar11.pdf
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Joint Shape Segmentation with Linear Programming Huang, Koltun, and Guibas 2011
Individual Joint
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Joint Shape Segmentation with Linear Programming Huang, Koltun, and Guibas 2011
maxsegmentations S1;S2
[score(S1) + score(S2) + consistency(S1; S2)]
Initial Segments
Create small discrete pieces by cutting surface in different ways.
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Initial Segments
Joint Shape Segmentation with Linear Programming Huang, Koltun, and Guibas 2011
maxsegmentations S1;S2
[score(S1) + score(S2) + consistency(S1; S2)]
A segmentation consists of decisions about whether to include each piece, where each point is covered once.
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Joint Shape Segmentation with Linear Programming Huang, Koltun, and Guibas 2011
maxsegmentations S1;S2
[score(S1) + score(S2) + consistency(S1; S2)]
Unary segmentation scores measure segment quality with area. Binary consistency scores match segments (many-to-one) using geometric similarity and adjacency.
Pairwise and then joint
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Pairwise and then joint
Joint Shape Segmentation with Linear Programming Huang, Koltun, and Guibas 2011
maxsegmentations S1;S2
[score(S1) + score(S2) + consistency(S1; S2)]
Unary segmentation scores measure segment quality with area. Binary consistency scores match segments (many-to-one) using geometric similarity and adjacency.
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Joint Shape Segmentation with Linear Programming Huang, Koltun, and Guibas 2011
Rand index (smaller is better)
JointAll uses the dog’s neck to help segment the geometry of the human.
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Joint Shape Segmentation with Linear Programming Huang, Koltun, and Guibas 2011
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An Optimization Approach to Improving Collections of Shape Maps Nguyen et al. 2011
Shape collections indicate which maps make sense.
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An Optimization Approach to Improving Collections of Shape Maps Nguyen et al. 2011
Maps are edges in a graph of shapes. Cycles are self maps after composition.
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An Optimization Approach to Improving Collections of Shape Maps Nguyen et al. 2011
Cycle consistency measured by displacement around loop.
B
Econs(°) =1
jAjX
p2AdA(p;m°(p))
CA
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An Optimization Approach to Improving Collections of Shape Maps Nguyen et al. 2011
Iterate: 1. Compute error of each
three-cycle. 2. Assign errors to edges in
map graph by solving an LP distributing cycle error.
3. Replace bad edges with composition.
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Exploration of Continuous Variability in Collections of 3D Shapes Ovsjanikov et al. 2011
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http://www.stanford.edu/~huangqx/siga11_jointseg.pdf
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Summarized approaches to
Local descriptors Shape understanding Correspondence Shape collections
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Summarized approaches to
Local descriptors Shape understanding Correspondence Shape collections
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