1 michael m. bronstein partial similarity of objects 17 december 2006 partial similarity of objects,...
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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
17 December 2006
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
17 December 2006
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
17 December 2006
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
17 December 2006
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
17 December 2006
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
17 December 2006
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
17 December 2006
Multicriterion optimization
BBBK, IJCV, submitted
UTOPIA
Minimize the vector objective function over
Competing criteria – impossible to minimize and simultaneously
ATTAINABLE CRITERIA
12Michael M. Bronstein Partial similarity of objects
17 December 2006
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
17 December 2006
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
17 December 2006
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
17 December 2006
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
17 December 2006
Example II – 3D partially missing objects
Salukwadze distance between partially missing objects(using L1-norm)
BBBK, ScaleSpace, submitted
21Michael M. Bronstein Partial similarity of objects
17 December 2006
Partial similarity of strings