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Numerical geometry of non-rigid shapes: deformation-invariant similarities. Michael M. Bronstein. Department of Computer Science Technion – Israel Institute of Technology Ph.D under the supervision of Prof. Ron Kimmel. Co-authors. Alex Bronstein. Alfred Bruckstein. Ron Kimmel.

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slide1

Numerical geometry of non-rigid shapes:

deformation-invariant similarities

Michael M. Bronstein

Department of Computer Science

Technion – Israel Institute of Technology

Ph.D under the supervision of Prof. Ron Kimmel

slide2

Co-authors

Alex Bronstein

Alfred Bruckstein

Ron Kimmel

Irad Yavneh

BBK = Bronstein, Bronstein, Kimmel

BBBK = Bronstein, Bronstein, Bruckstein, Kimmel

BBKY = Bronstein, Bronstein, Kimmel, Yavneh

slide3

Similarity between non-rigid shapes

(deformation-invariant distance)

Main problems

ANALYSIS

SYNTHESIS

Non-rigid correspondence and calculus of shapes

slide4

Intrinsic vs. extrinsic similarity

EXTRINSIC

SIMILARITY

INTRINSIC

SIMILARITY

  • Similarity = isometry
  • and are isometric if
  • and are -isometric if
slide5

3D example: expression-invariant face recognition

  • Facial expressions are approximate isometries of the facial surface
  • Identity = intrinsic geometry
  • Expression = extrinsic geometry

BBK, IJCV, 2005

slide6

2D example: articulated shapes

  • Articulated shape – shape consisting of rigid disjoints parts and non-
  • rigid joints,
  • -articulated shape if the joints size is bounded
  • -articulation: a deformation mapping isometrically rigid parts and
  • preserving the joints size:

H. Ling, D. Jacobs, CVPR 2005

BBBK, AMDO 2006

slide7

Articulations versus isometries

  • -articulation is an -isometry, but not necessarily vice versa

-articulated shape

-articulation

-isometry

  • A composition of two -articulations is an -articulation
  • A composition of two -isometries is a -isometry

BBBK, IJCV, submitted

slide9

Isometry-invariant distance construction

Let be a class of non-rigid shapes. A distance

measuring dissimilarity of shapes should satisfy:

  • Non-negativity:
  • Symmetry:
  • Triangle inequality:
  • Similarity: if then and are -isometric
  • if and are -isometric, then
  • Consistency to sampling: if is a finite -covering of , then
  • Efficiency: can be efficiently approximated numerically

BBK, PNAS, 2006

slide10

Canonical forms distance

  • Embed and into a common metric space by
  • minimum-distortion embeddings and .
  • Compare the images (canonical forms) as rigid objects

A. Elad, R. Kimmel, CVPR 2001

slide11

Canonical forms distance (cont.)

  • Satisfies the metric axioms only approximately
  • Approximately consistent to sampling
  • Efficient computation using multidimensional scaling (MDS)

Given a sampling the minimum-distortion embedding is found by optimizing over the images and not on itself

A. Elad, R. Kimmel, CVPR 2001

slide12

Multigrid MDS

Time (sec)

Stress

Complexity (MFLOPs)

Convergence of our MG MDS algorithm

(x10 faster than state-of-the-art)

BBKY, NLAA 2006

slide13

How to choose the embedding space ?

  • Schwartz et al. 1989:
  • Elad & Kimmel 2001:
  • Elad & Kimmel 2002:
  • BBK 2005:
  • Walter & Ritter 2002:

Euclidean

Spherical

Hyperbolic

Problem: using non-Euclidean embedding spaces, it is possible to reduce the representation error, but not avoid it completely.

slide14

Gromov-Hausdorff distance

Allow for arbitrary embedding space

where are isometric embeddings.

  • Satisfies the metric axioms with
  • Consistent to sampling: if is an -covering of , then
  • Computation: intractable

M. Gromov, 1981

slide15

Gromov-Hausdorff distance (cont.)

For compact surfaces, there exists an equivalent definition in terms of metric distortions:

where:

slide16

Gromov-Hausdorff distance (cont.)

measures how isometrically can be embedded into

slide17

Gromov-Hausdorff distance (cont.)

measures how isometrically can be embedded into

slide18

Gromov-Hausdorff distance (cont.)

measures how far and are from being one the inverse of the other

slide19

Computing the Gromov-Hausdorff distance

Mémoli & Sapiro (2005)

  • Drop the terms ,
  • Replace with a simpler expression
  • Probabilistic bound on the error
  • Combinatorial problem

F. Mémoli, G. Sapiro, Foundations Comp. Math, 2005

slide20

Computing the Gromov-Hausdorff distance (cont.)

BBK (2006)

  • Generalized MDS problem
  • Continuous optimization
  • Deterministic approximation (exact up to numerical accuracy / local
  • convergence)

BBK, PNAS, 2006

slide21

Generalized multidimensional scaling (GMDS)

G

MDS:

MDS:

  • The distances have no analytic expression and must be
  • approximated numerically
  • Points represented in barycentric coordinates
  • Optimization with a modified line search
  • Multiresolution scheme to prevent local convergence
  • -norm can be used instead of

BBK, PNAS, 2006

slide22

Gromov-Hausdorff distance via GMDS

  • Sampling: ,
  • Optimization over images and

BBK, PNAS, 2006

slide23

Gromov-Hausdorff vs. canonical forms

GROMOV-HAUSDORFF

CANONICAL FORMS

  • Two stages: embedding and
  • comparison
  • Embedding error is a problem
  • degrading accuracy
  • Many points (~1000) are
  • required for accurate comparison
  • Computational core: MDS
  • One stage: generalized
  • embedding
  • Embedding error is the
  • measure of similarity
  • Few points (~100) are required
  • to compute accurate distortion
  • Computational core: GMDS
slide24

Example I – 3D objects

BBK, SIAM J. Sci. Comp, 2006

slide25

Example I – 3D objects (cont.)

Canonical forms distance

(MDS, 500 points)

Gromov-Hausdorff distance

(GMDS, 50 points)

BBK, SIAM J. Sci. Comp, 2006

slide27

Example II – Articulated shapes (cont.)

Gromov-Hausdorff distance between articulated shapes

BBBK, IJCV, submitted

slide28

Face recognition project

  • Authentication of car driver based on 3D face recognition
  • Collaboration with General Motors
  • Current accuracy: <3% error rate

Raja

Giryes

Alon

Salzman

Daniel

Vainsencher

Vladimir

Zdornov

Yaron

Honen

slide29

Partial similarity between non-rigid shapes,

or how to compare a centaur and a horse?

slide30

Example from real life

Can we compare parts of objects?

Conclusion: objects may have similar parts, while being dissimilar.

Illustration: Herluf Bidstrup

slide31

Semantic definition of partial similarity

Two objects are partially similar if they have “large” “similar” “parts”.

Example: Jacobs et al.

slide32

More precise definitions

  • Part: subset with restricted metric
  • (technically, the set of all parts of is a
  • -algebra)
  • Dissimilarity: Gromov-Hausdorff distance defined on the set of parts,
  • Partiality: size of the object parts cropped off,
  • where is the measure of area on
slide33

Full versus partial similarity

  • Full similarity: and are -isometric
  • Partial similarity: and are -isometric, i.e., have parts
  • which are -isometric, and

Partial similarity

Full similarity

BBBK, IJCV, submitted

slide34

Multicriterion optimization

  • Minimize the vector objective function over
  • Competing criteria – impossible to minimize and simultaneously

ATTAINABLE CRITERIA

UTOPIA

BBBK, IJCV, submitted

slide35

Vector optimality

  • No total order relation in - impossible to say which point is “better”
  • Partial order: only when both criteria are better
slide36

Scalar versus vector optimality

Multicriterion optimization

Traditional (scalar) optimization

V. Pareto, 1901

slide37

Pareto optimum

  • Pareto optimum: point at which no criterion can be improved without
  • compromising the other
  • Pareto frontier: set of all Pareto optima, acting as a set-valued
  • criterion of partial dissimilarity
  • Only partial order relation exists between set-valued distances: not
  • always possible to compare

BBBK, IJCV, submitted

slide38

Fuzzy computation

  • Optimization over subsets turns into an NP-hard combinatorial
  • problem when discretized
  • Fuzzy optimization: optimize over membership functions

Crisp part

Fuzzy part

BBBK, IJCV, submitted

slide39

Salukwadze distance

  • The set-valued distance can be converted into a scalar valued one by
  • selecting a single point on the Pareto frontier.
  • Naïve selection: fixed value of or .
  • Smart selection: closest to the utopia point (Salukwadze optimum)

Salukwadze distance:

M. E. Salukwadze, 1979

BBBK, IJCV, submitted

slide40

Example II – mythological creatures

Large Gromov-Hausdorff distance

Small Salukwadze distance

Large Gromov-Hausdorff distance

Large Salukwadze distance

BBBK, IJCV, submitted

slide42

Example II – mythological creatures (cont.)

Gromov-Hausdorff distance

Salukwadze distance

(using L1-norm)

BBBK, IJCV, submitted

slide44

Example II – 3D partially missing objects

0.07

0.06

0.05

0.04

0.03

0.02

0.01

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Pareto frontiers, representing partial dissimilarities between

partially missing objects

BBBK, ScaleSpace, submitted

slide45

Example II – 3D partially missing objects

Salukwadze distance between partially missing objects

(using L1-norm)

BBBK, ScaleSpace, submitted