A Weighted Kernel PCA Approach to Graph-Based Image Segmentation

Transcription

1 A Weighted Kernel PCA Approach to Graph-Based Image Segmentation Carlos Alzate Johan A. K. Suykens ESAT-SCD-SISTA Katholieke Universiteit Leuven Leuven, Belgium January 25, 2007 International Conference on Embeddings of Graphs and Groups into Hilbert and Banach spaces with applications Lausanne, Switzerland, January 22-26, 2007

2 Outline 1 Graph-Based Image Segmentation Graph Partitioning Spectral Clustering Implicit Embedding 2 Weighted Kernel PCA LS-SVM Approach to Kernel PCA Introducing Weights Out-of-Sample Extension 3 Experimental Results 4 Conclusions

3 Cut Criteria Graph representation Data undirected graph G = (V, E). Each edge has a weight representing pairwise similarity. The objective is to minimize the cost of cutting the graph into k disjoint sets A 1,..., A k. The Binary Cut cut(a, B) = a A,b B w(a, b) where w(a, b) is the weight between node a and b.

4 Cut Criteria The k-way Cut kcut(a 1,..., A k ) = k cut(a i, A i ) i=1 Size of the clusters not taken into account. Bias towards small sets! The k-way Normalized Cut NCut(A 1,..., A k ) = k i=1 cut(a i, A i ) Vol(A i ) where Vol(A) = i A d i and d i = j w(i, j).

5 Cut Criteria k-way NCut Consider: f (k) {0, 1} N as the cluster indicator vector for the k-th cluster. g (i) = D 1/2 f (i) / D 1/2 f (i) 2, i = 1,..., k. The k-way NCut becomes: NCut(A 1,..., A k ) = k tr(g T LG) where L = D 1/2 WD 1/2 is the normalized Laplacian and G = [g (1),..., g (k) ]. Minimizing the NCut is an NP-hard problem.

6 Spectral Relaxation K-way NCut relaxation Relaxing the discrete constraint on the cluster indicator matrix G leads to the k-way NCut spectral relaxation: max NCut(A 1,..., A k ) G = tr( G T L G) such that G T G = I k The solution comes from Fan s theorem: G = U L R 1 where U L is any orthonormal basis of the k-th principal subspace of L and R 1 is an arbitrary rotation matrix.

7 Markov Random Walks Markov Random Walks Probabilistic interpretation of graph partitioning. P = D 1 W. P ij probability of moving from node i to node j in one step. Eigenvectors of P related to the eigenvectors of L by: U P = D 1/2 U L Piecewise constant eigenvectors.

8 Implicit Embedding Positive definite kernels A Mercer kernel is a function K : X X R which for all patterns {x i } N i=1, x i X and {c i } N i=1, c i R satisfies: N i=1 N c i c j K(x i, x j ) 0. j=1 Typically (but not necessarily) X = R d. For any Mercer kernel K, there exists a Reproducing Kernel Hilbert Space (RKHS) H and a feature map ϕ : X H such that K(x i, x j ) = ϕ(x i ), ϕ(x j ) H Implicit embedding of the graph into a RKHS by using a Mercer kernel K(x i, x j ) as similarity between vertices x i and x j.

9 Image Segmentation

10 Image Segmentation

11 Image Segmentation

12 Image Segmentation

13 Kernel PCA LS-SVM Approach to Kernel PCA Clear primal optimization problem to which kernel PCA is the dual. Underlying loss function is explicit L 2. max p(w, e) w,e = γ 1 2 et e 1 2 wt w such that e = Φ c w Φ c is the N n h feature matrix: ϕ(x 1 ) T ˆµ T ϕ ϕ(x 2 ) T ˆµ T ϕ Φ c =. ϕ(x N ) T ˆµ T ϕ Dual Eigendecomposition of the centered kernel matrix Ω c : Ω c α = λα

14 Weighted Kernel PCA Weighted Kernel PCA Introducing a weighting matrix V into the formulation: max p(w, e) w,e = γ 1 2 et Ve 1 2 wt w such that e = Φw Dual Non-symmetric eigenvalue problem: VΩα = λα Φ = [ϕ(x 1 ) T ; ϕ(x 2 ) T ;... ; ϕ(x N ) T ], V = V T > 0. Equivalence If V = D 1 then weighted kernel PCA is equivalent to the random walks algorithm.

15 Out-of-sample extension Clustering New Points No straightforward extensions for out-of-sample data points in the spectral clustering framework. Extensions can be done via approximation techniques such as Nyström [Bengio et al., 2003].

16 Out-of-sample extension Clustering New Points No approximations needed! Cluster indicators for new points can be obtained naturally from the projections onto the eigenvectors. z (m) (x) = N i=1 where k is the number of clusters. α (m) i K(x i, x), m = 1,..., k 1 Allows to train using a subsampled version of the image, inferring the clusters of the remaining pixels. The clustering model can be trained, validated and tested.

17 Segmentation Results Remarks Every pixel in the image becomes a vertex in the graph: huge kernel matrix (but generally sparse). Composite RBF kernel function as similarity measure: ( K ij = exp f (x i) f (x j ) 2 2 σf 2 s(x i) s(x j ) 2 ) 2 σs 2, where f (x i ) is the pixel feature information of the i-th pixel (e.g. grayscale value, RBG, HSV for color) and s(x i ) is the spatial information.

18 Subsampling Full Training 40 60

19 Validation set projections Model selection

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22 Conclusions Weighted kernel PCA view of graph-based clustering. Out-of-sample extension based on the projections onto the eigenvectors. Reduced computation times due to subsampling.