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Gromov-Hausdor! stable signatures forshapes using persistence
joint with F. Chazal, D. Cohen-Steiner, L. Guibas and S. Oudot
Facundo Memoli
ABabcdfghiejkl
2
Goal
• Shape discrimination is a very important problem in several fields.
• Isometry invariant shape discrimination has been approached with dif-ferent tools, mostly via computation and comparison of invariant signa-tures, [HK03,Osada-02,Fro90,SC-00].
• The Gromov-Hausdor! distance (and certain variants) provides a rig-orous and well motivated framework for studying shape matching underinvariances [MS04,MS05,M07,M08].
• However, its direct computation leads to NP hard problems (BQAP:bottleneck quadratic assignment problems).
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• Most of the e!ort has gone into finding lower bounds for the GH distancethat use informative invariant signatures and lead to easier optimizationproblems [M07,M08].
• Using persistent topology [ELZ00], we obtain a new family of sig-natures and prove that they are stable w.r.t the GH distance: i.e., weobtain lower bounds for the GH distance!
• These lower bounds:
– perform very well in practical application of shape discrimination.
– lead to BAPs (bottleneck assignment problems) which can be solvedin polynomial time.
!
"""""#
0 d!12 d!
13 d!14 . . .
d!12 0 d!
23 d!24 . . .
d!13 d!
23 0 d!34 . . .
d!14 d!
24 d!34 0 . . .
......
......
. . .
$
%%%%%&
!
"""""#
0 d12 d13 d14 . . .d12 0 d23 d24 . . .d13 d23 0 d34 . . .d14 d24 d34 0 . . ....
......
.... . .
$
%%%%%&
visual summary
!
"""""#
0 d!12 d!
13 d!14 . . .
d!12 0 d!
23 d!24 . . .
d!13 d!
23 0 d!34 . . .
d!14 d!
24 d!34 0 . . .
......
......
. . .
$
%%%%%&
!
"""""#
0 d12 d13 d14 . . .d12 0 d23 d24 . . .d13 d23 0 d34 . . .d14 d24 d34 0 . . ....
......
.... . .
$
%%%%%&
!
0
0
1
!
0
0
1
visual summary
!
"""""#
0 d!12 d!
13 d!14 . . .
d!12 0 d!
23 d!24 . . .
d!13 d!
23 0 d!34 . . .
d!14 d!
24 d!34 0 . . .
......
......
. . .
$
%%%%%&
!
"""""#
0 d12 d13 d14 . . .d12 0 d23 d24 . . .d13 d23 0 d34 . . .d14 d24 d34 0 . . ....
......
.... . .
$
%%%%%&
!
0
0
1
!
0
0
1
?visual summary
!
"""""#
0 d!12 d!
13 d!14 . . .
d!12 0 d!
23 d!24 . . .
d!13 d!
23 0 d!34 . . .
d!14 d!
24 d!34 0 . . .
......
......
. . .
$
%%%%%&
!
"""""#
0 d12 d13 d14 . . .d12 0 d23 d24 . . .d13 d23 0 d34 . . .d14 d24 d34 0 . . ....
......
.... . .
$
%%%%%&
!
0
0
1
!
0
0
1
?
visual summary
shapes/spaces signatures (persistence diagrams)
M, dGH D, dB
• M: collection of all shapes (finite metric spaces).
• D: collection of all signatures (persistence diagrams).
Dh, h H
shapes/spaces signatures (persistence diagrams)
M, dGH D, dB
X, Y
• M: collection of all shapes (finite metric spaces).
• D: collection of all signatures (persistence diagrams).
Dh, h H
shapes/spaces signatures (persistence diagrams)
M, dGH D, dB
X, Y
• M: collection of all shapes (finite metric spaces).
• D: collection of all signatures (persistence diagrams).
h HDh X , Dh Y
Dh, h H
shapes/spaces signatures (persistence diagrams)
M, dGH D, dB
X, Y
dGH X, Y
• M: collection of all shapes (finite metric spaces).
• D: collection of all signatures (persistence diagrams).
h HDh X , Dh Y
Dh, h H
shapes/spaces signatures (persistence diagrams)
M, dGH D, dB
X, Y
dGH X, Y
• M: collection of all shapes (finite metric spaces).
• D: collection of all signatures (persistence diagrams).
h HDh X , Dh Y
Dh, h H
shapes/spaces signatures (persistence diagrams)
M, dGH D, dB
X, Y
dGH X, Y
• M: collection of all shapes (finite metric spaces).
• D: collection of all signatures (persistence diagrams).
h HDh X , Dh Y
dB Dh X , Dh Y
Dh, h H
shapes/spaces signatures (persistence diagrams)
M, dGH D, dB
X, Y
dGH X, Y
• M: collection of all shapes (finite metric spaces).
• D: collection of all signatures (persistence diagrams).
h HDh X , Dh Y
dB Dh X , Dh Y
Dh, h H
!
"""""#
0 d12 d13 d14 . . .d12 0 d23 d24 . . .d13 d23 0 d34 . . .d14 d24 d34 0 . . ....
......
.... . .
$
%%%%%&
!
"""""#
0 d!12 d!
13 d!14 . . .
d!12 0 d!
23 d!24 . . .
d!13 d!
23 0 d!34 . . .
d!14 d!
24 d!34 0 . . .
......
......
. . .
$
%%%%%&
Shapes as metric spaces
!
"""""#
0 d12 d13 d14 . . .d12 0 d23 d24 . . .d13 d23 0 d34 . . .d14 d24 d34 0 . . ....
......
.... . .
$
%%%%%&
!
"""""#
0 d!12 d!
13 d!14 . . .
d!12 0 d!
23 d!24 . . .
d!13 d!
23 0 d!34 . . .
d!14 d!
24 d!34 0 . . .
......
......
. . .
$
%%%%%&
Shapes as metric spaces
then use Gromov-Hausdor! distance..
Choice of the metric: geodesic vs Euclidean
!
"""""#
0 d12 d13 d14 . . .d12 0 d!
23 d24 . . .d13 d23 0 d34 . . .d14 d24 d34 0 . . ....
......
.... . .
$
%%%%%&!
!
"""""#
0 d!12 d!
13 d!14 . . .
d!12 0 d!
23 d!24 . . .
d!13 d!
23 0 d!34 . . .
d!14 d!
24 d!34 0 . . .
......
......
. . .
$
%%%%%&
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Invariance to isometric deformations (change in pose)
!
"""""#
0 d12 d13 d14 . . .d12 0 d!
23 d24 . . .d13 d23 0 d34 . . .d14 d24 d34 0 . . ....
......
.... . .
$
%%%%%&!
!
"""""#
0 d!12 d!
13 d!14 . . .
d!12 0 d!
23 d!24 . . .
d!13 d!
23 0 d!34 . . .
d!14 d!
24 d!34 0 . . .
......
......
. . .
$
%%%%%&
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Invariance to isometric deformations (change in pose)
geodesic distance remains approximately constant
!
"""""#
0 d12 d13 d14 . . .d12 0 d!
23 d24 . . .d13 d23 0 d34 . . .d14 d24 d34 0 . . ....
......
.... . .
$
%%%%%&!
!
"""""#
0 d!12 d!
13 d!14 . . .
d!12 0 d!
23 d!24 . . .
d!13 d!
23 0 d!34 . . .
d!14 d!
24 d!34 0 . . .
......
......
. . .
$
%%%%%&
8
Invariance to isometric deformations (change in pose)
geodesic distance remains approximately constant
9
Definition [Correspondences]
For finite sets A and B, a subset C ! A " B is a correspondence (between Aand B) if and and only if
• # a $ A, there exists b $ B s.t. (a, b) $ R
• # b $ B, there exists a $ A s.t. (a, b) $ R
Let C(A, B) denote all possible correspondences between sets A and B.
9
0 1 1 0 0 1 1
1 1 0 1 0 1 1
1 0 1 0 0 1 0
0 1 0 1 1 0 1
1 0 1 1 0 1 0
0 1 1 0 0 1 1
1 1 0 1 0 1 1
1 0 1 0 1 1 0
0 0 0 0 0 0 0
1 0 1 1 0 1 0
A
B
Definition [Correspondences]
For finite sets A and B, a subset C ! A " B is a correspondence (between Aand B) if and and only if
• # a $ A, there exists b $ B s.t. (a, b) $ R
• # b $ B, there exists a $ A s.t. (a, b) $ R
Let C(A, B) denote all possible correspondences between sets A and B.
9
0 1 1 0 0 1 1
1 1 0 1 0 1 1
1 0 1 0 0 1 0
0 1 0 1 1 0 1
1 0 1 1 0 1 0
0 1 1 0 0 1 1
1 1 0 1 0 1 1
1 0 1 0 1 1 0
0 0 0 0 0 0 0
1 0 1 1 0 1 0
A
B
Definition [Correspondences]
For finite sets A and B, a subset C ! A " B is a correspondence (between Aand B) if and and only if
• # a $ A, there exists b $ B s.t. (a, b) $ R
• # b $ B, there exists a $ A s.t. (a, b) $ R
Let C(A, B) denote all possible correspondences between sets A and B.
9
0 1 1 0 0 1 1
1 1 0 1 0 1 1
1 0 1 0 0 1 0
0 1 0 1 1 0 1
1 0 1 1 0 1 0
0 1 1 0 0 1 1
1 1 0 1 0 1 1
1 0 1 0 1 1 0
0 0 0 0 0 0 0
1 0 1 1 0 1 0
A
B
Definition [Correspondences]
For finite sets A and B, a subset C ! A " B is a correspondence (between Aand B) if and and only if
• # a $ A, there exists b $ B s.t. (a, b) $ R
• # b $ B, there exists a $ A s.t. (a, b) $ R
Let C(A, B) denote all possible correspondences between sets A and B.
Definition. [BBI] For finite metric spaces X, dX and Y, dY , define theGromov-Hausdor! distance between them by
dGH X,Y12
minC
maxx,y , x ,y C
dX x, x dY y, y
Definition. [BBI] For finite metric spaces X, dX and Y, dY , define theGromov-Hausdor! distance between them by
dGH X,Y12
minC
maxx,y , x ,y C
dX x, x dY y, y
11
Construction of our signatures
• Our signatures take the form of persistence diagrams: we capture cer-tain topological and metric information from the shape.
• First example: construction based on Rips filtrations: Let X, dX be ashape.
– Let Kd X be the d-dimensional full simplicial complex on X.
– To each ! x0, x1, . . . , xk Kd X assign its filtration time
F ! :12
maxi,j
dX xi, xj
– This gives rise to a filtration Kd X , F .
– Apply persistence algorithm [ELZ00] to summarize topologicalinformation in the filtration and obtain persistence diagram.
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!
0
0
1
• Persistence diagrams are colored multi-subsets of the extended realplane.. can also be represented as barcodes.
• Let D denote the collection of all persistence diagrams. Compare two dif-ferent persistence diagrams with bottleneck distance view D, dBas a metric space.
12
!
0
0
1
• Persistence diagrams are colored multi-subsets of the extended realplane.. can also be represented as barcodes.
• Let D denote the collection of all persistence diagrams. Compare two dif-ferent persistence diagrams with bottleneck distance view D, dBas a metric space.
13
Example: Rips filtration on a torus
13
Example: Rips filtration on a torus
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Our signatures: more richness
• Let’s assume that in there is also a function defined on the shape: X, dX , fX .Then, we redefine the filtration values of ! x0, x1, . . . , xk
F ! max12
maxi,j
dX xi, xj ,maxi
fX xi
• Again, this gives rise to a filtration: Kd X , F use persistencealgorithm to obtain a persistence diagram.
• This increases discrimination power!
• We denote by H a family of maps that attach a function to a given finitemetric space.
• Then, for each h H, we denote by Dh X the persistence diagram arisingfrom the filtration above. This constitutes our family projection onto D.
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Example (Eccentricity). To each finite metric space X, dX one can assignthe eccentricity function:
eccX x maxx X
dX x, x .
shapes/spaces signatures (persistence diagrams)
M, dGH D, dB
h H
shapes/spaces signatures (persistence diagrams)
M, dGH D, dB
X, Y
h H
shapes/spaces signatures (persistence diagrams)
M, dGH D, dB
X, Y
h H
Dh X , Dh Y
shapes/spaces signatures (persistence diagrams)
M, dGH D, dB
X, Y
dGH X, Y
h H
Dh X , Dh Y
shapes/spaces signatures (persistence diagrams)
M, dGH D, dB
X, Y
dGH X, Y
h H
Dh X , Dh Y
shapes/spaces signatures (persistence diagrams)
M, dGH D, dB
X, Y
dGH X, Y
h H
Dh X , Dh Y
dB Dh X , Dh Y
Theorem (stability of our signatures). For all X, Y M,
dGH X, Y suph H
C h dB Dh X , Dh Y .
Remark.
• Proof relies on properties of the GH distance and new results on the sta-bility of persistence diagrams [CCGGO09].
• For a given h, the computation leads to a BAP which can be solved inpolynomial time.
• There are adaptations one can do in practice to speed up, see paper.
• One can obtain more generality and discrimination power by working inthe class of mm-spaces: shapes are represented as triples X, dX , µX
where µX are weights assigned to each point see [M07] and paper.
• Our results include stability of Rips persistence diagrams.
17
Theorem (stability of our signatures). For all X, Y M,
dGH X, Y suph H
C h dB Dh X , Dh Y .
Remark.
• Proof relies on properties of the GH distance and new results on the sta-bility of persistence diagrams [CCGGO09].
• For a given h, the computation leads to a BAP which can be solved inpolynomial time.
• There are adaptations one can do in practice to speed up, see paper.
• One can obtain more generality and discrimination power by working inthe class of mm-spaces: shapes are represented as triples X, dX , µX
where µX are weights assigned to each point see [M07] and paper.
• Our results include stability of Rips persistence diagrams.
17
Theorem (stability of our signatures). For all X, Y M,
dGH X, Y suph H
C h dB Dh X , Dh Y .
Remark.
• Proof relies on properties of the GH distance and new results on the sta-bility of persistence diagrams [CCGGO09].
• For a given h, the computation leads to a BAP which can be solved inpolynomial time.
• There are adaptations one can do in practice to speed up, see paper.
• One can obtain more generality and discrimination power by working inthe class of mm-spaces: shapes are represented as triples X, dX , µX
where µX are weights assigned to each point see [M07] and paper.
• Our results include stability of Rips persistence diagrams.
17
Theorem (stability of our signatures). For all X, Y M,
dGH X, Y suph H
C h dB Dh X , Dh Y .
Remark.
• Proof relies on properties of the GH distance and new results on the sta-bility of persistence diagrams [CCGGO09].
• For a given h, the computation leads to a BAP which can be solved inpolynomial time.
• There are adaptations one can do in practice to speed up, see paper.
• One can obtain more generality and discrimination power by working inthe class of mm-spaces: shapes are represented as triples X, dX , µX
where µX are weights assigned to each point see [M07] and paper.
• Our results include stability of Rips persistence diagrams.
17
Some experiments
• Sumner database: 62 shapes total, 6 classes. Used graph estimate of thegeodesic distance. Number of vertices ranged from 7K to 30K.
• Subsampled shapes and retained subsets of 300 points (farthest point sam-pling). Normalized distance matrices.
• Used the mm-space representation of shapes: weights were based on Voronoiregions.
• Used several functions ! h for ! in a finite subset of scales.
• Obtained 4% (or 2%) classification error in a 1-nn classification problem.
Some experiments
• Sumner database: 62 shapes total, 6 classes. Used graph estimate of thegeodesic distance. Number of vertices ranged from 7K to 30K.
• Subsampled shapes and retained subsets of 300 points (farthest point sam-pling). Normalized distance matrices.
• Used the mm-space representation of shapes: weights were based on Voronoiregions.
• Used several functions ! h for ! in a finite subset of scales.
• Obtained 4% (or 2%) classification error in a 1-nn classification problem.
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-0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04MDSCALE with M
camelcatelephant
faceheadhorse
C
10 20 30 40 50 60
10
20
30
40
50
60
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
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Discussion
• Summary of our proposal:
– Use the metric (or mm-space) representation of shapes.– Formulate the shape matching problem using the Gromov-Hausdor!
distance.– Compute our signatures for shapes.– Solve the BAP lower bounds: computationally easy! By our theorem,
the computed quantities give lower bounds for the GH distance.
• Implications and Future directions:
– We do not need a mesh– general: can be applied to any dataset.– We obtain stability of Rips persistence diagrams.– Richness of the family H? how close can I get to the GH distance?– Local signatures: more discrimination.– Extension to partial shape matching: which (local) signatures are
useful for this?
21
Acknowledgements
• ONR through grant N00014-09-1-0783
• NSF through grants ITR 0205671, FRG 0354543 and FODAVA 808515
• NIH through grant GM- 072970,
• DARPA through grant HR0011-05-1-0007.
• INRIA-Stanford associated TGDA team.
http://math.stanford.edu/~memoli
Bibliography
[BBI] Burago, Burago and Ivanov. A course on Metric Geometry. AMS, 2001
[HK-03] A. Ben Hamza, Hamid Krim: Probabilistic shape descriptor for tri-angulated surfaces. ICIP (1) 2005: 1041-1044
[Osada-02] Robert Osada, Thomas A. Funkhouser, Bernard Chazelle, DavidP. Dobkin: Matching 3D Models with Shape Distributions. Shape Mod-eling International 2001: 154-166
[SC-00] S. Belongie and J. Malik (2000). ”Matching with Shape Contexts”.IEEE Workshop on Contentbased Access of Image and Video Libraries(CBAIVL-2000).
[CCGMO09] Chazal, Cohen-Steiner, Guibas, Memoli, Oudot. Gromov-Hausdor!stable signatures for shapes using persistence. SGP 2009.
[CCGGO09] Chazal, Cohen-Steiner, Guibas, Glissee, Oudot. Proximity ofpersistence modules and their diagrams. Proc. 25th ACM Symp. Comp.Geom., 2009.
[M08] F. Memoli. Gromov-Hausdor! distances in Euclidean spaces. NORDIA-CVPR-2008.
[M07] F.Memoli. On the use of gromov-hausdor! distances for shape compar-ison. In Proceedings of PBG 2007, Prague, Czech Republic, 2007.
[MS04] F. Memoli and G. Sapiro. Comparing point clouds. In SGP ’04:Proceedings of the 2004 Eurographics/ACM SIGGRAPH symposium onGeometry processing, pages 32–40, New York, NY, USA, 2004. ACM.
[MS05] F. Memoli and G. Sapiro. A theoretical and computational frameworkfor isometry invariant recognition of point cloud data. Found. Comput.Math., 5(3):313–347, 2005.
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Let X1, X2 Z be two di!erent samples of the same shape Z, and Y anothershape then
dGH X1, Y dGH X2, Y dGH X1, X2 r1 r2
![Entropy, homology and semialgebraic geometryshmuel/QuantCourse /Gromov on Yomdin.… · 225 ENTROPY, HOMOLOGY AND SEMIALGEBRAIC GEOMETRY [after Y. Yomdin] by M. GROMOV Seminaire BOURBAKI](https://img.pdfslide.net/doc/110x75/5f03b8287e708231d40a6f89/entropy-homology-and-semialgebraic-shmuelquantcourse-gromov-on-yomdin-225.jpg)
