1 Michael M. Bronstein Partial similarity of objects 17 December 2006 Partial similarity of objects,...

1Michael M. Bronstein Partial similarity of objects

17 December 2006

Partial similarity of objects, or how to compare a centaur to a horse

Michael M. Bronstein

Department of Computer ScienceTechnion – Israel Institute of Technology

2Michael M. Bronstein Partial similarity of objects

17 December 2006

Co-authors

Ron KimmelAlex Bronstein

BBK = Bronstein, Bronstein, KimmelBBBK = Bronstein, Bronstein, Bruckstein, Kimmel

Alfred Bruckstein

3Michael M. Bronstein Partial similarity of objects

17 December 2006

Intrinsic vs. extrinsic similarity

INTRINSIC

SIMILARITY

EXTRINSIC

SIMILARITY

4Michael M. Bronstein Partial similarity of objects

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Non-rigid objects: basic terms

Isometry – deformation that preserves the geodesic distances

is -isometrically embeddable into if

and are -isometric if , and is

-surjective

5Michael M. Bronstein Partial similarity of objects

17 December 2006

Examples of near-isometric shapes

6Michael M. Bronstein Partial similarity of objects

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Canonical forms and MDS

A. Elad, R. Kimmel, CVPR 2001

Embed and into a common metric space by

minimum-distortion embeddings and .

Compare the images (canonical forms) as rigid objects

Efficient computation using multidimensional scaling (MDS)

7Michael M. Bronstein Partial similarity of objects

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Generalized MDS

Generalized MDS: embed one surface into another

Measure of similarity: embedding error

Related to the Gromov-Hausdorff distance

F. Memoli, G. Sapiro, 2005BBBK, PNAS, 2006

8Michael M. Bronstein Partial similarity of objects

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Semantic definition of partial similarity

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

Example: Jacobs et al.

9Michael M. Bronstein Partial similarity of objects

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More precise definitions

Part: subset with restricted metric

(technically, the set of all parts of is a

-algebra)

Dissimilarity: intrinsic distance criterion defined on the set of parts

(Gromov-Hausdorff distance)

Partiality: size of the object parts cropped off,

where is the measure of area on

10Michael M. Bronstein Partial similarity of objects

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Full versus partial similarity

Full similarity

Full similarity: and are -isometric

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

which are -isometric, and

Partial similarity

BBBK, IJCV, submitted

11Michael M. Bronstein Partial similarity of objects

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Multicriterion optimization

BBBK, IJCV, submitted

UTOPIA

Minimize the vector objective function over

Competing criteria – impossible to minimize and simultaneously

ATTAINABLE CRITERIA

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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

13Michael M. Bronstein Partial similarity of objects

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Fuzzy computation

Optimization over subsets turns into an NP-hard

combinatorial

problem when discretized

Fuzzy optimization: optimize over membership functions

BBBK, IJCV, submitted

Crisp part Fuzzy part

14Michael M. Bronstein Partial similarity of objects

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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, 1979BBBK, IJCV, submitted

15Michael M. Bronstein Partial similarity of objects

17 December 2006BBBK, IJCV, submitted

Example II – mythological creatures

Large Gromov-Hausdorff distanceSmall Salukwadze distance

Large Gromov-Hausdorff distanceLarge Salukwadze distance

16Michael M. Bronstein Partial similarity of objects

17 December 2006

Example II – mythological creatures (cont.)

BBBK, IJCV, submitted

17Michael M. Bronstein Partial similarity of objects

17 December 2006BBBK, IJCV, submitted

Example II – mythological creatures (cont.)

Gromov-Hausdorff distance Salukwadze distance(using L1-norm)

18Michael M. Bronstein Partial similarity of objects

17 December 2006

19Michael M. Bronstein Partial similarity of objects

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Example II – 3D partially missing objects

BBBK, ScaleSpace, submitted

Pareto frontiers, representing partial dissimilarities between partially missing objects

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

0.01

0.02

0.03

0.04

0.05

0.06

0.07

20Michael M. Bronstein Partial similarity of objects

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Example II – 3D partially missing objects

Salukwadze distance between partially missing objects(using L1-norm)

BBBK, ScaleSpace, submitted

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Partial similarity of strings