Kernel on Bag of Paths For Measuring Shapes Similarity Suard Fr´ed´eric, Rakotomamonjy Alain and Bensrhair Abdelaziz Laboratoire Informatique, Traitement de l’Information, Syst`emes EA4051 Institut National des Sciences Appliqu´ees (INSA) de Rouen, Universit´e de Rouen, Email: [email protected] http://asi.insa-rouen.fr/∼fsuard/

Introduction Graph kernel

Context :

Wallraven [6] : building a kernel on set ⇔ computing a similarity between each element of sets :

• shape recognition, • structured representation,

XX

′

K(G, G ) =

• application to shock graph retrieval.

h′

h

X

• Kashima [4]: K(G1, G2) =

KL(h, h′)

p1(h1)p2(h2)KL(h1, h2)

h1 ,h2 ∈V1⋆ ,V2⋆

Using graph for comparison :

– p1, p2 : probabilities to generate random walk, – positive definite. 1 1 X • Mean kernel : K(G1, G2) = K(P1, P2) = N1 N 2

• shape represented with skeleton, • skeleton comparison ⇔ graph comparison, • graph kernel : kernel on set approach,

X

KL(hi, hj )

i:hi ∈P1 j:hj ∈P2

• graph : set of subgraphs, set of paths.

– all paths are compared, – positive definite. 1 ˆ ˆ • Max matching kernel : K(G1, G2) = K(P1, P2) = [K(P 1 , P2 ) + K(P2 , P1 )] 2 – best comparison retained, X 1 ˆ 1 , P2 ) = max KL(hi, hj ) – K(P j:hj ∈P2 |P1|

Shape representation with graph

i:hi ∈P1

How to build a graph ?

X 1 1 ˆ 1 , P2 ) = • Matching kernel : K(P |P1| |P2|

1. original image,

X

KdL (hi, hj )

i:hi ∈P1 j:hj ∈P2

2. binary mask,

– Haasdonk [3] : max KL(hi, hj ) approximated with

3. morphological skeleton,

j:hj ∈P2

4. transform skeleton into graph :

X

KdL (hi, hj )

j:hj ∈P2

 2



– KdL (h1, h2) = exp − dL(h2σ1,h2 2) ,

• vertices : skeleton branch ending or branches intersection, • edges.

– dL(h1, h2)2 = KL(h1, h1) + KL(h2, h2) − 2KL(h1, h2). – positive definite for all σ > 0. • Path level set kernel : K(G1, G2) = K(P1, P2) = hbP1 , bP2 i ·

→

→

1

4

3

αiP1 αjP2 KL(hi, hj )

i:hi ∈P1 j:hj ∈P2

→

2

X

X

– Desobry [2] : comparing the probability distribution of each set of paths using one-class SVM, 1 1 X 2 – optimization problem : min kf kH + max(0, b − f (xi)) − b f ∈H,b 2 νn i X P X P 1 – fP1 (h) = αi KL(hi, h) − bP1 and fP2 (h) = αj 2 KL(hj , h) − bP2 i

5. label graph : edge :

j

– positive definite.

• length (L), • skeleton length (s), • orientation (θ),

Shock graph mining

• distance between skeleton and edge (e). vertex : • distance to the center of mass (ρ), • Rudger tools database :25 objects, 8 classes,

• distance to the nearest shape edge (E).

• shape retrieval : ranking for each shape query, • vertex label : E, ρ, • edge label : L. → compare graph formulations

Set of paths

→ path length influence

Performance of each graph kernel according to path length :

• V : set of vertices,

1

• E : set of edges, • labeled graph : G = (V, E), Recognition rate for best match

0.95

• path : h = {v1, · · · , vn} - generated using random walk or shortest path, - threshold for path length. ⇒ h is a subgraph of G • set of m paths : P = {h1, h2, ..., hm} ⇒ G is represented with a bag of paths. Graph

L=0

L=1

L=2

Bag of paths

0.9

0.85

0.8

Path Max Matching Mean Kashima

0.75

0.7

0.65

0.6

→

→

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Maximal length of path

• Sebastian [5], Demirci [1] : 96%, • path-level set, max matching kernel : 100%, • path length : 2 or 3 for best performance. →

→

Query result for each object of the database :

Similarity between path : ′

KL(h, h ) =

  0

, if path lengths are different ′

  Kv (l(v1), l(v1))

n Y

′ , vi′))Kv (l(vi), l(vi′)) , otherwise Ke(l(vi−1, vi), l(vi−1

i=2

• Kv kernel on vertex label, • Ke kernel on edge label.

References

[1] F. Demirci, A. Shokoufandeh, L. Bretzner, and S. Dickinson. Object recognition as many-to-many feature matching. Internation Journal of Computer Vision, 69(2):203–222, 2006. [2] F. Desobry, M. Davy, and W.J. Fitzgerald. A class of kernels for sets of vectors. In Proceedings of the 13th European Symposium on Artificial Neural Networks, 2005. [3] B. Haasdonk and C. Bahlmann. Learning with distance substitution kernels. In Springer, editor, Pattern Recognition - Proc. of the 26th DAGM Symposium, 2004. [4] H. Kashima, K. Tsuda, and A. Inokuchi. Marginalized kernels between labeled graphs. In Proceedings of the Twentieh International Conference on Machine Learning, 2003.

Conclusion Conclusions :

Perspectives :

[5] T. Sebastian, P. Klein, and B. Kimia. Recognition of shapes by editing shock graphs. IEEE Trans. on Pattern Analysis and Machine Intelligence, 26(5):550–571, 2001.

• Graph kernel for measuring graph similarity,

• Add other information : local histograms, texture,

• New graph kernel formulations,

• Explore other ways for graph designing : points of interest.

[6] C. Wallraven, B.Caputo, and A. Graf. Recognition with local features: the kernel recipe. In Proceedings of International Conference on Computer Vision, pages 257–264, 2003.

• Application to shock graph mining.