Numerical geometry

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Numerical geometry of shapes
non-rigid
Lecture I Introduction
Michael Bronstein
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Welcome to non-rigid world
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Non-rigid shapes everywhere
Computer graphics models
Volumetric medical data
Articulated shapes
Two-dimensional shapes
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Non-rigid shapes in art
Auguste Rodin
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???????
Rock
Scissors
Paper
Jan-ken-pon (Rock-paper-scissors )
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???????
Hands
Rock
Scissors
Paper
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Invariant similarity
SIMILARITY
TRANSFORMATION
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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

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Invariance
Topological
Inelastic
Rigid
Elastic
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Invariant correspondence
CORRESPONDENCE
TRANSFORMATION
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Analysis and synthesis
SYNTHESIS
ANALYSIS
Elephant image courtesy M. Kilian and H.
Pottmann
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Landscape
HORSE
Pattern recognition
Computer vision
Computer graphics
2D world
3D world
Image processing
Geometry processing
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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

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Additional reading
Excerpts from the book
Problems
Solutions
Tutorials
Lecture slides
Data
Software
Links
On paper
Online tosca.cs.technion.ac.il/book
Springer, October 2008
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Raffaello Santi, School of Athens, Vatican
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Metric model

• Shape metric space , where
  is a metric
• Shape similarity similarity of metric spaces

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Isometries

• Two metric spaces and are
  equivalent if there exists a
• distance-preserving map (isometry)
  satisfying

• Such and are called
  isometric, denoted

• Self-isometries of form an isometry
  group

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Euclidean metric

• Shape is a subset of the Euclidean embedding
  space
• Restricted Euclidean metric
• for all

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Euclidean isometries

• Isometry group in the Euclidean
  space consists of rigid motions

Rotation
Translation
Reflection

• Two shapes differing by a Euclidean isometry are
  congruent

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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

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Riemannian view
Bernhard Riemann (1826-1866)

• 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

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Nash embedding theorem

• Technical conditions
• Manifold is
• For -dimensional manifold, embedding
• space dimension is

John Forbes Nash
Practically intrinsic and extrinsic views are
equivalent!
Nash, 1956
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Uniqueness of the embedding

• Nash theorem guarantees existence but not
  uniqueness of embedding
• Embedding is clearly defined up to a congruence
  (Euclidean isometry)

Riemannian manifold
Embedded surface

• Are there cases of non-trivial non-uniqueness?

• IN OTHER WORDS
• Do isometric yet incongruent shapes exist?

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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

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Rigidity conjecture
In practical applications shapes are represented
as polyhedra (triangular meshes), so
Leonhard Euler (1707-1783)
Do non-rigid shapes really exist?
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Rigidity conjecture timeline
Eulers Rigidity Conjecture every polyhedron is
rigid
1766

Cauchy every convex polyhedron is rigid
1813

Cohn-Vossen all surfaces with positive Gaussian
curvature are rigid
1927
Gluck almost all simply connected surfaces are
rigid
1974
Connelly finally disproves Eulers conjecture
1977
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Connelly sphere


Isocahedron Rigid polyhedron
Connelly sphere Non-rigid polyhedron
Connelly, 1978
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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

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Rock-paper-scissors again
INTRINSICALLY SIMILAR
EXTRINSICALLY SIMILAR
Invariant to inelastic deformations
Invariant to rigid motions
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Extrinsic vs. intrinsic similarity
INTRINSIC SIMILARITY isometry w.r.t. geodesic
metric
EXTRINSIC SIMILARITY isometry w.r.t. Euclidean
metric
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Extrinsic vs. intrinsic similarity
RIGID MOTION
EXTRINSIC SIMILARITY CONGRUENCE
For rigid shapes, intrinsic similarity
extrinsic similarity (since all the isometries
are congruences)
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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

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Iterative closest point (ICP) algorithms

• 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

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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
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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
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Iterative closest point algorithm
Closest point correspondence
Optimal alignment
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And now, intrinsic similarity
INTRINSIC SIMILARITY
EXTRINSIC SIMILARITY
Part of the same metric space
Two different metric spaces
SOLUTION Find a representation of
and in a common metric space
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Canonical forms
Isometric embedding
Elad Kimmel, 2003
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Canonical form distance
?
INTRINSIC SIMILARITY
Compute canonical forms
EXTRINSIC SIMILARITY OF CANONICAL FORMS
INTRINSIC SIMILARITY
Elad Kimmel, 2003
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Examples of canonical forms
Elad Kimmel, 2003
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Expression-invariant face recognition
Images Leonid Larionov
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Is geometry sensitive to expressions?
x
y
y
x
Euclidean distances
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Is geometry sensitive to expressions?
x
y
y
x
Geodesic distances
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Extrinsic vs. intrinsic
Distance distortion distribution

• Extrinsic geometry sensitive to expressions
• Intrinsic geometry insensitive to expressions

Bronstein, Bronstein Kimmel, 2003
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Isometric model of expressions

• Expressions are approximately inelastic
  deformations of the facial surface
• Identity intrinsic geometry
• Expression extrinsic geometry

Bronstein, Bronstein Kimmel, 2003
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Canonical forms of faces
Bronstein, Bronstein Kimmel, 2005
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Telling identical twins apart
Extrinsic similarity
Intrinsic similarity
Bronstein, Bronstein Kimmel, 2005
Michael
Alex
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Telling identical twins apart
Michael
Alex
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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

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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)

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References
S. Rusinkiewicz and M. Levoy, Efficient variants
of the ICP algorithm, Proc. 3D Digital 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 cloud data 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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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 representation
• of faces, Trans. Image Processing (2007)