1 Numerical geometry of non-rigid shapes Lecture I – Introduction Numerical geometry of shapes...
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![Page 1: 1 Numerical geometry of non-rigid shapes Lecture I – Introduction Numerical geometry of shapes Lecture I – Introduction non-rigid Michael Bronstein.](https://reader038.fdocuments.us/reader038/viewer/2022110207/56649d385503460f94a11765/html5/thumbnails/1.jpg)
1Numerical geometry of non-rigid shapes Lecture I – Introduction
Numerical geometryof shapes
Lecture I – Introduction
non-rigid
Michael Bronstein
![Page 2: 1 Numerical geometry of non-rigid shapes Lecture I – Introduction Numerical geometry of shapes Lecture I – Introduction non-rigid Michael Bronstein.](https://reader038.fdocuments.us/reader038/viewer/2022110207/56649d385503460f94a11765/html5/thumbnails/2.jpg)
2Numerical geometry of non-rigid shapes Lecture I – Introduction
Welcome to non-rigid world
![Page 3: 1 Numerical geometry of non-rigid shapes Lecture I – Introduction Numerical geometry of shapes Lecture I – Introduction non-rigid Michael Bronstein.](https://reader038.fdocuments.us/reader038/viewer/2022110207/56649d385503460f94a11765/html5/thumbnails/3.jpg)
3Numerical geometry of non-rigid shapes Lecture I – Introduction
Non-rigid shapes everywhere
Articulatedshapes
Volumetricmedical data
Computer graphics models
Two-dimensional shapes
![Page 4: 1 Numerical geometry of non-rigid shapes Lecture I – Introduction Numerical geometry of shapes Lecture I – Introduction non-rigid Michael Bronstein.](https://reader038.fdocuments.us/reader038/viewer/2022110207/56649d385503460f94a11765/html5/thumbnails/4.jpg)
4Numerical geometry of non-rigid shapes Lecture I – Introduction
Auguste Rodin
Non-rigid shapes in art
![Page 5: 1 Numerical geometry of non-rigid shapes Lecture I – Introduction Numerical geometry of shapes Lecture I – Introduction non-rigid Michael Bronstein.](https://reader038.fdocuments.us/reader038/viewer/2022110207/56649d385503460f94a11765/html5/thumbnails/5.jpg)
5Numerical geometry of non-rigid shapes Lecture I – Introduction
Rock
Paper
Scissors
じゃんけんぽん
Jan-ken-pon (Rock-paper-scissors )
![Page 6: 1 Numerical geometry of non-rigid shapes Lecture I – Introduction Numerical geometry of shapes Lecture I – Introduction non-rigid Michael Bronstein.](https://reader038.fdocuments.us/reader038/viewer/2022110207/56649d385503460f94a11765/html5/thumbnails/6.jpg)
6Numerical geometry of non-rigid shapes Lecture I – Introduction
Hands
Rock
Paper
Scissors
じゃんけんぽん
![Page 7: 1 Numerical geometry of non-rigid shapes Lecture I – Introduction Numerical geometry of shapes Lecture I – Introduction non-rigid Michael Bronstein.](https://reader038.fdocuments.us/reader038/viewer/2022110207/56649d385503460f94a11765/html5/thumbnails/7.jpg)
7Numerical geometry of non-rigid shapes Lecture I – Introduction
Invariant similarity
SIMILARITY
TRANSFORMATION
![Page 8: 1 Numerical geometry of non-rigid shapes Lecture I – Introduction Numerical geometry of shapes Lecture I – Introduction non-rigid Michael Bronstein.](https://reader038.fdocuments.us/reader038/viewer/2022110207/56649d385503460f94a11765/html5/thumbnails/8.jpg)
8Numerical geometry of non-rigid shapes Lecture I – Introduction
Deformation-invariant similarity
Define a class of deformations
Find properties of the shape which are invariant under the class of
deformations and discriminative (uniquely describe the shape)
Define a shape distance based on these properties
![Page 9: 1 Numerical geometry of non-rigid shapes Lecture I – Introduction Numerical geometry of shapes Lecture I – Introduction non-rigid Michael Bronstein.](https://reader038.fdocuments.us/reader038/viewer/2022110207/56649d385503460f94a11765/html5/thumbnails/9.jpg)
9Numerical geometry of non-rigid shapes Lecture I – Introduction
Rigid Elastic
TopologicalInelastic
Invariance
![Page 10: 1 Numerical geometry of non-rigid shapes Lecture I – Introduction Numerical geometry of shapes Lecture I – Introduction non-rigid Michael Bronstein.](https://reader038.fdocuments.us/reader038/viewer/2022110207/56649d385503460f94a11765/html5/thumbnails/10.jpg)
10Numerical geometry of non-rigid shapes Lecture I – Introduction
Invariant correspondence
CORRESPONDENCE
TRANSFORMATION
![Page 11: 1 Numerical geometry of non-rigid shapes Lecture I – Introduction Numerical geometry of shapes Lecture I – Introduction non-rigid Michael Bronstein.](https://reader038.fdocuments.us/reader038/viewer/2022110207/56649d385503460f94a11765/html5/thumbnails/11.jpg)
11Numerical geometry of non-rigid shapes Lecture I – Introduction
Analysis and synthesis
Elephant image: courtesy M. Kilian and H. Pottmann
SYNTHESISANALYSIS
![Page 12: 1 Numerical geometry of non-rigid shapes Lecture I – Introduction Numerical geometry of shapes Lecture I – Introduction non-rigid Michael Bronstein.](https://reader038.fdocuments.us/reader038/viewer/2022110207/56649d385503460f94a11765/html5/thumbnails/12.jpg)
12Numerical geometry of non-rigid shapes Lecture I – Introduction
Landscape
“HORSE”
Image processing Geometry processing
Pattern recognition
Computervision
Computergraphics2D world 3D world
![Page 13: 1 Numerical geometry of non-rigid shapes Lecture I – Introduction Numerical geometry of shapes Lecture I – Introduction non-rigid Michael Bronstein.](https://reader038.fdocuments.us/reader038/viewer/2022110207/56649d385503460f94a11765/html5/thumbnails/13.jpg)
13Numerical geometry of non-rigid shapes Lecture I – Introduction
In a nutshell
Analysis and synthesis of non-rigid shapes
Archetype problems: shape similarity and correspondence
Metric geometry as a common denominator
Tools from geometry, algebra, optimization, numerical analysis,
statistics,
and multidimensional data analysis
Practical numerical methods
Applications in computer vision, pattern recognition, computer
graphics,
and geometry processing
![Page 14: 1 Numerical geometry of non-rigid shapes Lecture I – Introduction Numerical geometry of shapes Lecture I – Introduction non-rigid Michael Bronstein.](https://reader038.fdocuments.us/reader038/viewer/2022110207/56649d385503460f94a11765/html5/thumbnails/14.jpg)
14Numerical geometry of non-rigid shapes Lecture I – Introduction
Additional reading
Excerpts from the book
On paperOnline
tosca.cs.technion.ac.il/book
ProblemsSolutions
Lecture slides
Software
Links
Tutorials
Data
Springer, October 2008
![Page 15: 1 Numerical geometry of non-rigid shapes Lecture I – Introduction Numerical geometry of shapes Lecture I – Introduction non-rigid Michael Bronstein.](https://reader038.fdocuments.us/reader038/viewer/2022110207/56649d385503460f94a11765/html5/thumbnails/15.jpg)
15Numerical geometry of non-rigid shapes Lecture I – Introduction
Raffaello Santi, School of Athens, Vatican
![Page 16: 1 Numerical geometry of non-rigid shapes Lecture I – Introduction Numerical geometry of shapes Lecture I – Introduction non-rigid Michael Bronstein.](https://reader038.fdocuments.us/reader038/viewer/2022110207/56649d385503460f94a11765/html5/thumbnails/16.jpg)
16Numerical geometry of non-rigid shapes Lecture I – Introduction
Metric model
Shape = metric space , where is a metric
Shape similarity = similarity of metric spaces
![Page 17: 1 Numerical geometry of non-rigid shapes Lecture I – Introduction Numerical geometry of shapes Lecture I – Introduction non-rigid Michael Bronstein.](https://reader038.fdocuments.us/reader038/viewer/2022110207/56649d385503460f94a11765/html5/thumbnails/17.jpg)
17Numerical geometry of non-rigid shapes Lecture I – Introduction
Isometries
Two metric spaces and are equivalent if there exists a
distance-preserving map (isometry) satisfying
Self-isometries of form an isometry group
Such and are called isometric, denoted
![Page 18: 1 Numerical geometry of non-rigid shapes Lecture I – Introduction Numerical geometry of shapes Lecture I – Introduction non-rigid Michael Bronstein.](https://reader038.fdocuments.us/reader038/viewer/2022110207/56649d385503460f94a11765/html5/thumbnails/18.jpg)
18Numerical geometry of non-rigid shapes Lecture I – Introduction
Euclidean metric
Shape is a subset of the Euclidean embedding space
Restricted Euclidean metric
for all
![Page 19: 1 Numerical geometry of non-rigid shapes Lecture I – Introduction Numerical geometry of shapes Lecture I – Introduction non-rigid Michael Bronstein.](https://reader038.fdocuments.us/reader038/viewer/2022110207/56649d385503460f94a11765/html5/thumbnails/19.jpg)
19Numerical geometry of non-rigid shapes Lecture I – Introduction
Euclidean isometries
Isometry group in the Euclidean space consists of rigid
motions
Two shapes differing by a Euclidean isometry are congruent
Rotation Translation Reflection
![Page 20: 1 Numerical geometry of non-rigid shapes Lecture I – Introduction Numerical geometry of shapes Lecture I – Introduction non-rigid Michael Bronstein.](https://reader038.fdocuments.us/reader038/viewer/2022110207/56649d385503460f94a11765/html5/thumbnails/20.jpg)
20Numerical geometry of non-rigid shapes Lecture I – Introduction
Geodesic metric
Given a path on , define its length
The length can be induced by the Euclidean metric
Geodesic (intrinsic) metric
Geodesic = minimum-length path
Technical condition: is a smooth submanifold of
![Page 21: 1 Numerical geometry of non-rigid shapes Lecture I – Introduction Numerical geometry of shapes Lecture I – Introduction non-rigid Michael Bronstein.](https://reader038.fdocuments.us/reader038/viewer/2022110207/56649d385503460f94a11765/html5/thumbnails/21.jpg)
21Numerical geometry of non-rigid shapes Lecture I – Introduction
Riemannian view
Define a Euclidean tangent space at every point
Define an inner product (Riemannian metric) on the tangent space
Measure the length of a curve using the Riemannian metric
Bernhard Riemann(1826-1866)
![Page 22: 1 Numerical geometry of non-rigid shapes Lecture I – Introduction Numerical geometry of shapes Lecture I – Introduction non-rigid Michael Bronstein.](https://reader038.fdocuments.us/reader038/viewer/2022110207/56649d385503460f94a11765/html5/thumbnails/22.jpg)
22Numerical geometry of non-rigid shapes Lecture I – Introduction
Nash embedding theorem
John Forbes Nash
Embedding theorem (1956): Any smooth
Riemannian manifold can be realized as
an embedded surface in Euclidean space
of sufficiently high yet finite dimension
Technical conditions:
Manifold is
For -dimensional manifold,
embedding
space dimension isPractically: intrinsic and extrinsic views are
equivalent!Nash, 1956
![Page 23: 1 Numerical geometry of non-rigid shapes Lecture I – Introduction Numerical geometry of shapes Lecture I – Introduction non-rigid Michael Bronstein.](https://reader038.fdocuments.us/reader038/viewer/2022110207/56649d385503460f94a11765/html5/thumbnails/23.jpg)
23Numerical geometry of non-rigid shapes Lecture I – Introduction
Uniqueness of the embedding
Nash theorem guarantees existence but not uniqueness of
embedding
Embedding is clearly defined up to a congruence (Euclidean
isometry)
IN OTHER WORDS:
Do isometric yet incongruent shapes exist?
Are there cases of non-trivial non-uniqueness?
Riemannian
manifold
Embedded surface
![Page 24: 1 Numerical geometry of non-rigid shapes Lecture I – Introduction Numerical geometry of shapes Lecture I – Introduction non-rigid Michael Bronstein.](https://reader038.fdocuments.us/reader038/viewer/2022110207/56649d385503460f94a11765/html5/thumbnails/24.jpg)
24Numerical geometry of non-rigid shapes Lecture I – Introduction
Bending
Shapes with incongruent isometries are called bendable
Plane is the simplest example of a bendable surface
Shapes that do not have incongruent isometries are called rigid
Extrinsic geometry of a rigid shape is fully determined by the
intrinsic one
![Page 25: 1 Numerical geometry of non-rigid shapes Lecture I – Introduction Numerical geometry of shapes Lecture I – Introduction non-rigid Michael Bronstein.](https://reader038.fdocuments.us/reader038/viewer/2022110207/56649d385503460f94a11765/html5/thumbnails/25.jpg)
25Numerical geometry of non-rigid shapes Lecture I – Introduction
Rigidity conjecture
Leonhard Euler(1707-1783)
In practical applications shapes
are represented as polyhedra
(triangular meshes), so…
If the faces of a polyhedron were made of
metal plates and the polyhedron edges
were replaced by hinges, the polyhedron
would be rigid.
Do non-rigid shapes really exist?
![Page 26: 1 Numerical geometry of non-rigid shapes Lecture I – Introduction Numerical geometry of shapes Lecture I – Introduction non-rigid Michael Bronstein.](https://reader038.fdocuments.us/reader038/viewer/2022110207/56649d385503460f94a11765/html5/thumbnails/26.jpg)
26Numerical geometry of non-rigid shapes Lecture I – Introduction
Rigidity conjecture timeline
Euler’s Rigidity Conjecture: every polyhedron is rigid1766
1813
1927
1974
1977
Cauchy: every convex polyhedron is rigid
Connelly finally disproves Euler’s conjecture
Cohn-Vossen: all surfaces with positive Gaussian
curvature are rigid
Gluck: almost all simply connected surfaces are rigid
![Page 27: 1 Numerical geometry of non-rigid shapes Lecture I – Introduction Numerical geometry of shapes Lecture I – Introduction non-rigid Michael Bronstein.](https://reader038.fdocuments.us/reader038/viewer/2022110207/56649d385503460f94a11765/html5/thumbnails/27.jpg)
27Numerical geometry of non-rigid shapes Lecture I – Introduction
Connelly sphere
Isocahedron
Rigid polyhedron
Connelly sphere
Non-rigid polyhedron
Connelly, 1978
![Page 28: 1 Numerical geometry of non-rigid shapes Lecture I – Introduction Numerical geometry of shapes Lecture I – Introduction non-rigid Michael Bronstein.](https://reader038.fdocuments.us/reader038/viewer/2022110207/56649d385503460f94a11765/html5/thumbnails/28.jpg)
28Numerical geometry of non-rigid shapes Lecture I – Introduction
“Almost rigidity”
Most of the shapes (especially, polyhedra) are rigid
This may give the impression that the world is more rigid than non-rigid
This is true if isometry is considered in the strict sense:
if exists such that
Many objects have some elasticity and therefore can bend almost
isometrically
No known results about “almost rigidity” of shapes
![Page 29: 1 Numerical geometry of non-rigid shapes Lecture I – Introduction Numerical geometry of shapes Lecture I – Introduction non-rigid Michael Bronstein.](https://reader038.fdocuments.us/reader038/viewer/2022110207/56649d385503460f94a11765/html5/thumbnails/29.jpg)
29Numerical geometry of non-rigid shapes Lecture I – Introduction
Rock-paper-scissors again
INTRINSICALLY
SIMILAR
EXTRINSICALLY
SIMILAR
Invariant to
inelastic deformations
Invariant to
rigid motions
![Page 30: 1 Numerical geometry of non-rigid shapes Lecture I – Introduction Numerical geometry of shapes Lecture I – Introduction non-rigid Michael Bronstein.](https://reader038.fdocuments.us/reader038/viewer/2022110207/56649d385503460f94a11765/html5/thumbnails/30.jpg)
30Numerical geometry of non-rigid shapes Lecture I – Introduction
Extrinsic vs. intrinsic similarity
INTRINSIC SIMILARITY
isometry w.r.t.
geodesic metric
EXTRINSIC SIMILARITY
isometry w.r.t.
Euclidean metric
![Page 31: 1 Numerical geometry of non-rigid shapes Lecture I – Introduction Numerical geometry of shapes Lecture I – Introduction non-rigid Michael Bronstein.](https://reader038.fdocuments.us/reader038/viewer/2022110207/56649d385503460f94a11765/html5/thumbnails/31.jpg)
31Numerical geometry of non-rigid shapes Lecture I – Introduction
Extrinsic vs. intrinsic similarity
RIGID
MOTION
EXTRINSIC SIMILARITY
= CONGRUENCE
For rigid shapes, intrinsic similarity = extrinsic similarity
(since all the isometries are congruences)
![Page 32: 1 Numerical geometry of non-rigid shapes Lecture I – Introduction Numerical geometry of shapes Lecture I – Introduction non-rigid Michael Bronstein.](https://reader038.fdocuments.us/reader038/viewer/2022110207/56649d385503460f94a11765/html5/thumbnails/32.jpg)
32Numerical geometry of non-rigid shapes Lecture I – Introduction
Extrinsic similarity
Given two shapes and , find the degree of their incongruence
Compare and as subsets of the Euclidean space
Invariance to Euclidean isometry where
Euclidean isometries = rotation, translation, (reflection):
is a rotation matrix,
is a translation vector
![Page 33: 1 Numerical geometry of non-rigid shapes Lecture I – Introduction Numerical geometry of shapes Lecture I – Introduction non-rigid Michael Bronstein.](https://reader038.fdocuments.us/reader038/viewer/2022110207/56649d385503460f94a11765/html5/thumbnails/33.jpg)
33Numerical geometry of non-rigid shapes Lecture I – Introduction
Given two shapes and , find the best rigid motion
bringing as close as possible to :
is some shape-to-shape distance
Minimum = extrinsic dissimilarity of and
Minimizer = best rigid alignment between and
ICP is a family of algorithms differing in
The choice of the shape-to-shape distance
The choice of the numerical minimization algorithm
Iterative closest point (ICP) algorithms
![Page 34: 1 Numerical geometry of non-rigid shapes Lecture I – Introduction Numerical geometry of shapes Lecture I – Introduction non-rigid Michael Bronstein.](https://reader038.fdocuments.us/reader038/viewer/2022110207/56649d385503460f94a11765/html5/thumbnails/34.jpg)
34Numerical geometry of non-rigid shapes Lecture I – Introduction
Shape-to-shape distance
Hausdorff distance: distance between subsets of a metric space
where ,
Non-symmetric version of Hausdorff distance
where is closest-point correspondence
![Page 35: 1 Numerical geometry of non-rigid shapes Lecture I – Introduction Numerical geometry of shapes Lecture I – Introduction non-rigid Michael Bronstein.](https://reader038.fdocuments.us/reader038/viewer/2022110207/56649d385503460f94a11765/html5/thumbnails/35.jpg)
35Numerical geometry of non-rigid shapes Lecture I – Introduction
Iterative closest point algorithm
Initialize
Find the closest point correspondence
Minimize the misalignment between corresponding points
Update
Iterate until convergence…Chen & Medioni, 1991; Besl & McKay, 1992
![Page 36: 1 Numerical geometry of non-rigid shapes Lecture I – Introduction Numerical geometry of shapes Lecture I – Introduction non-rigid Michael Bronstein.](https://reader038.fdocuments.us/reader038/viewer/2022110207/56649d385503460f94a11765/html5/thumbnails/36.jpg)
36Numerical geometry of non-rigid shapes Lecture I – Introduction
Iterative closest point algorithm
Closest point correspondenceOptimal alignment
![Page 37: 1 Numerical geometry of non-rigid shapes Lecture I – Introduction Numerical geometry of shapes Lecture I – Introduction non-rigid Michael Bronstein.](https://reader038.fdocuments.us/reader038/viewer/2022110207/56649d385503460f94a11765/html5/thumbnails/37.jpg)
37Numerical geometry of non-rigid shapes Lecture I – Introduction
And now, intrinsic similarity…
INTRINSIC SIMILARITYEXTRINSIC SIMILARITY
Part of the same metric space Two different metric spaces
SOLUTION: Find a representation of and
in a common metric space
![Page 38: 1 Numerical geometry of non-rigid shapes Lecture I – Introduction Numerical geometry of shapes Lecture I – Introduction non-rigid Michael Bronstein.](https://reader038.fdocuments.us/reader038/viewer/2022110207/56649d385503460f94a11765/html5/thumbnails/38.jpg)
38Numerical geometry of non-rigid shapes Lecture I – Introduction
Canonical forms
Isometric embedding
Elad & Kimmel, 2003
![Page 39: 1 Numerical geometry of non-rigid shapes Lecture I – Introduction Numerical geometry of shapes Lecture I – Introduction non-rigid Michael Bronstein.](https://reader038.fdocuments.us/reader038/viewer/2022110207/56649d385503460f94a11765/html5/thumbnails/39.jpg)
39Numerical geometry of non-rigid shapes Lecture I – Introduction
Canonical form distance
Compute canonical formsEXTRINSIC SIMILARITY OF CANONICAL FORMS
INTRINSIC SIMILARITY
= INTRINSIC SIMILARITY
Elad & Kimmel, 2003
![Page 40: 1 Numerical geometry of non-rigid shapes Lecture I – Introduction Numerical geometry of shapes Lecture I – Introduction non-rigid Michael Bronstein.](https://reader038.fdocuments.us/reader038/viewer/2022110207/56649d385503460f94a11765/html5/thumbnails/40.jpg)
40Numerical geometry of non-rigid shapes Lecture I – Introduction
Examples of canonical forms
Elad & Kimmel, 2003
![Page 41: 1 Numerical geometry of non-rigid shapes Lecture I – Introduction Numerical geometry of shapes Lecture I – Introduction non-rigid Michael Bronstein.](https://reader038.fdocuments.us/reader038/viewer/2022110207/56649d385503460f94a11765/html5/thumbnails/41.jpg)
41Numerical geometry of non-rigid shapes Lecture I – Introduction
Expression-invariant face recognition
Images: Leonid Larionov
![Page 42: 1 Numerical geometry of non-rigid shapes Lecture I – Introduction Numerical geometry of shapes Lecture I – Introduction non-rigid Michael Bronstein.](https://reader038.fdocuments.us/reader038/viewer/2022110207/56649d385503460f94a11765/html5/thumbnails/42.jpg)
42Numerical geometry of non-rigid shapes Lecture I – Introduction
Is geometry sensitive to expressions?
x
x’
y
y’
Euclidean distances
![Page 43: 1 Numerical geometry of non-rigid shapes Lecture I – Introduction Numerical geometry of shapes Lecture I – Introduction non-rigid Michael Bronstein.](https://reader038.fdocuments.us/reader038/viewer/2022110207/56649d385503460f94a11765/html5/thumbnails/43.jpg)
43Numerical geometry of non-rigid shapes Lecture I – Introduction
Is geometry sensitive to expressions?
x
x’
y
y’
Geodesic distances
![Page 44: 1 Numerical geometry of non-rigid shapes Lecture I – Introduction Numerical geometry of shapes Lecture I – Introduction non-rigid Michael Bronstein.](https://reader038.fdocuments.us/reader038/viewer/2022110207/56649d385503460f94a11765/html5/thumbnails/44.jpg)
44Numerical geometry of non-rigid shapes Lecture I – Introduction
Extrinsic vs. intrinsic
Distance distortion distribution
Extrinsic geometry sensitive to expressions
Intrinsic geometry insensitive to expressionsBronstein, Bronstein & Kimmel, 2003
![Page 45: 1 Numerical geometry of non-rigid shapes Lecture I – Introduction Numerical geometry of shapes Lecture I – Introduction non-rigid Michael Bronstein.](https://reader038.fdocuments.us/reader038/viewer/2022110207/56649d385503460f94a11765/html5/thumbnails/45.jpg)
45Numerical geometry of non-rigid shapes Lecture I – Introduction
Isometric model of expressions
Expressions are approximately inelastic deformations of the facial
surface
Identity = intrinsic geometry
Expression = extrinsic geometryBronstein, Bronstein & Kimmel, 2003
![Page 46: 1 Numerical geometry of non-rigid shapes Lecture I – Introduction Numerical geometry of shapes Lecture I – Introduction non-rigid Michael Bronstein.](https://reader038.fdocuments.us/reader038/viewer/2022110207/56649d385503460f94a11765/html5/thumbnails/46.jpg)
46Numerical geometry of non-rigid shapes Lecture I – Introduction
Canonical forms of faces
Bronstein, Bronstein & Kimmel, 2005
![Page 47: 1 Numerical geometry of non-rigid shapes Lecture I – Introduction Numerical geometry of shapes Lecture I – Introduction non-rigid Michael Bronstein.](https://reader038.fdocuments.us/reader038/viewer/2022110207/56649d385503460f94a11765/html5/thumbnails/47.jpg)
47Numerical geometry of non-rigid shapes Lecture I – Introduction
Telling identical twins apart
Extrinsic similarity Intrinsic similarity
MichaelAlexBronstein, Bronstein & Kimmel, 2005
![Page 48: 1 Numerical geometry of non-rigid shapes Lecture I – Introduction Numerical geometry of shapes Lecture I – Introduction non-rigid Michael Bronstein.](https://reader038.fdocuments.us/reader038/viewer/2022110207/56649d385503460f94a11765/html5/thumbnails/48.jpg)
48Numerical geometry of non-rigid shapes Lecture I – Introduction
Telling identical twins apart
MichaelAlex
![Page 49: 1 Numerical geometry of non-rigid shapes Lecture I – Introduction Numerical geometry of shapes Lecture I – Introduction non-rigid Michael Bronstein.](https://reader038.fdocuments.us/reader038/viewer/2022110207/56649d385503460f94a11765/html5/thumbnails/49.jpg)
49Numerical geometry of non-rigid shapes Lecture I – Introduction
![Page 50: 1 Numerical geometry of non-rigid shapes Lecture I – Introduction Numerical geometry of shapes Lecture I – Introduction non-rigid Michael Bronstein.](https://reader038.fdocuments.us/reader038/viewer/2022110207/56649d385503460f94a11765/html5/thumbnails/50.jpg)
50Numerical geometry of non-rigid shapes Lecture I – Introduction
Summary
Shape = metric space
Shape similarity = distance between metric spaces
Invariance = isometry
Definition of the metric determines the class of transformations to
which the similarity is invariant
Extrinsic similarity = congruence (Euclidean metric) computed using
ICP
Intrinsic similarity = congruence of canonical forms obtained by
isometric embedding
![Page 51: 1 Numerical geometry of non-rigid shapes Lecture I – Introduction Numerical geometry of shapes Lecture I – Introduction non-rigid Michael Bronstein.](https://reader038.fdocuments.us/reader038/viewer/2022110207/56649d385503460f94a11765/html5/thumbnails/51.jpg)
51Numerical geometry of non-rigid shapes Lecture I – Introduction
References
Metric geometry
Burago, Burago, Ivanov, A course on metric geometry, AMS (2001)
Rigidity
S. E. Cohn-Vossen, Nonrigid closed surfaces, Annals of Math. (1929)
R. Connelly, The rigidity of polyhedral surfaces, Math. Magazine (1979)
Iterative closest point algorithms
Y. Chen and G. Medioni, Object modeling by registration of multiple range images, Proc. Robotics and Automation (1991)
P. J. Besl and N. D. McKay, A method for registration of 3D shapes, Trans. PAMI(1992)
![Page 52: 1 Numerical geometry of non-rigid shapes Lecture I – Introduction Numerical geometry of shapes Lecture I – Introduction non-rigid Michael Bronstein.](https://reader038.fdocuments.us/reader038/viewer/2022110207/56649d385503460f94a11765/html5/thumbnails/52.jpg)
52Numerical geometry of non-rigid shapes Lecture I – Introduction
References
S. Rusinkiewicz and M. Levoy, Efficient variants of the ICP algorithm, Proc. 3DDigital Imaging and Modeling (2001)
N. Gelfand, N. J. Mitra, L. Guibas, and H. Pottmann, Robust global registration,Proc. SGP (2005)
H. Li and R. Hartley, The 3D-3D registration problem revisited, Proc. ICCV (2007)
N. J. Mitra, N. Gelfand, H. Pottmann, and L. Guibas, Registration of point clouddata from a geometric optimization perspective, Proc. SGP (2004)
Canonical forms
A. Elad and R. Kimmel, On bending invariant signatures for surfaces, Trans. PAMI (2003)
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53Numerical geometry of non-rigid shapes Lecture I – Introduction
References
Face recognition
A. M. Bronstein, M. M. Bronstein, R. Kimmel, Expression-invariant 3D face recognition, Proc. AVBPA (2003)
A. M. Bronstein, M. M. Bronstein, R. Kimmel, Three-dimensional face recognition, IJCV (2005)
A. M. Bronstein, M. M. Bronstein, R. Kimmel, Expression-invariant representationof faces, Trans. Image Processing (2007)