GRAPH SIGNAL PROCESSING: A STATISTICAL VIEWPOINT

Size: px

Start display at page:

Download "GRAPH SIGNAL PROCESSING: A STATISTICAL VIEWPOINT"

Andrew Kelly
5 years ago
Views:

1 GRAPH SIGNAL PROCESSING: A STATISTICAL VIEWPOINT Cha Zhang Joint work with Dinei Florêncio and Philip A. Chou Microsoft Research

2 Outline Gaussian Markov Random Field Graph construction Graph transform Graph down-sampling Graph prediction Graph-based regularization Conclusions

3 Gaussian Markov Random Field (GMRF) A random vector x = x 1,, x n T is called a GMRF with respect to the graph G = V = 1,, n, E with mean μ and a precision matrix Q > 0 (positive definite), if and only if its density has the form: p x = 2π n 2 Q 1 2 exp 1 2 x μ T Q x μ and Q ij 0 i, j E for all i j.

4 Gaussian Markov Random Field (GMRF) It is defined on a graph It is a multivariate Gaussian distribution Regarding the precision matrix Q Q = Σ 1, Σ is the covariance matrix Conditional interpretation E x i x i = μ i 1 Q ii j:j~i Q ij x j μ j Prec x i x i = Q ii Corr x i, x j x ij = Q ij Q ii Q jj, i j

5 A Bijective Mapping Zero-mean GMRF a subset of bi-directed graphs Random variable node Q ij 0 edge Assign graph weights: W ij = Q ij, for i j n W ii = j=1 Q ij Mapping between W and Q is bijective Subset because Q > 0 1 Q 11 + Q 12 + Q 13 Q = Q 11 Q 12 Q 13 Q 21 Q 22 Q 23 Q 31 Q 32 Q 33 Q 12 Q Q 23 pseudograph Q 12 + Q 22 + Q 23 Q 11 + Q 12 + Q 13

6 Intrinsic GMRF (IGMRF) A random vector x = x 1,, x n T is called an intrinsic GMRF of k th order if it has density: p x = 2π n k 2 ( Q ) 1 2 exp 1 2 x μ T Q x μ Where Q is of rank n k, and denotes the generalized determinant (product of the non-zero eigenvalues). IGMRF can be interpreted as limits of proper GMRFs, thus many of the properties of GMRF holds for IGMRF. H. Rue and L. Held. Gaussian Markov Random Fields: Theory and Applications. Chapman and Hall/CRC, 2005

7 A Special GMRF Zero mean, and all rows of Q sum to zero For all i, j Q ij = 0 When Q is of rank n 1: IGMRF of first order No self-loops in the graph Simple Graph Local behavior E x i x i = 1 Q ii j:j~i Q ij x j, with j:j~i Q ij Q ii = 1 IGMRF of first order closely related to firstorder random walk

8 So For many bi-directed graphs, we can give them a GMRF (IGMRF) interpretation It s a Gaussian, isn t it too simple? Real-world signals are hardly Gaussian What can we do with such a simple model?

9 So For many bi-directed graphs, we can give them a GMRF (IGMRF) interpretation It s a Gaussian, isn t it too simple? Real-world signals are hardly Gaussian What can we do with such a simple model? A lot! Many are already using it without realizing it

10 Graph Construction Data-driven Collect enough examples to estimate μ and Q Q is unlikely sparse densely connected graph Intuitive model-based Widely used in 1D/2D regular grid signals IGMRF, no self-loops: nice local behavior Q identical to the Laplacian matrix 1 2 n (a) (b) (c) (d)

11 Graph Construction Model-constrained data-driven approach Maximum a-posterior estimation Q = arg max p(q x 1,, x N ) Q = arg max Q p Q n=1 p(x n Q): likelihood, GMRF model p Q : prior knowledge Application dependent For example, Q is sparse, know graph structure, etc. N p(x n Q) X. Dong, D. Thanou, P. Frossard, and P. Vandergheynst. Learning graphs from signal observations under smoothness prior. In submitted to: IEEE Transacctions on Signal Processing, 2014.

12 Graph Transform Karhunen-Loève transform (KLT) Σ = ΦΛΦ T Where Λ = diag λ 0,, λ n is the diagonal matrix of eigenvalues of Σ. Since: Q = Σ 1 = ΦΛ 1 Φ T The eigenvector matrix of Q is the same as the KLT, optimal for decorrelation. Since GMRF is defined on a graph, we call the eigenvector matrix of the precision matrix Q the Graph Transform.

13 Graph Transform and DCT When Q is the Laplacian of a regular grid graph 1D signal Graph transform = 1D DCT 2D signal on regular grid 2D DCT is one of the eigenvector matrices of Q Thus under such an image model, 1D DCT and 2D DCT are optimal for decorrelation First known proof of 2D DCT optimality! C. Zhang and D. Florêncio. Analyzing the optimality of predictive transform coding using graph-based models. IEEE Signal Processing Letters, 20(1), Jan

14 Graph Down-sampling Given random vector x = x 1,, x n T on graph G = V = 1,, n, E, keep x 1 = x 1,, x n k T, remove x 2 = x n k+1,, x n T, what is the graph for x 1?

15 Graph Down-sampling Given GMRF random vector x = x 1,, x n T on graph G = V = 1,, n, E, keep x 1 = x 1,, x n k T, remove x 2 = x n k+1,, x n T, what is the graph for x 1? μ = μ 1 μ 2, Σ = Σ 11 Σ 12 Σ 21 Σ 22 = Q 1 = Q 11 Q 12 Q 21 Q 22 1 Q 1 = Σ 1 11 = Q 11 Q 12 Q 1 22 Q 21 Identical to Kron Reduction, a well-know technique for graph down-sampling originally derived for electronic networks. F. Dörfler and F. Bullo. Kron reduction of graphs with applications to electrical networks. Circuits and Systems I: Regular Papers, IEEE Transactions on, 60(1): , 2013.

16 Graph Prediction Random vector x = x 1,, x n, x n+1,, x m T follows GMRF with precision matrix Q. x 1 = x 1,, x n T unknown x 2 = x n+1,, x m T known

17 Graph Prediction Random vector x = x 1,, x n, x n+1,, x m T follows GMRF with precision matrix Q. x 1 = x 1,, x n T unknown x 2 = x n+1,, x m T known Partition Q: Q = Q 11 Q 12 Q 21 Q 22 x 1 x 2 is also a GMRF with respect to its own subgraph with mean μ x1 x 2 and precision matrix Q x1 x 2 > 0: μ x1 x 2 = μ x1 Q 1 11 Q 12 x 2 μ x2 Q x1 x 2 = Q 11 H. Rue and L. Held. Gaussian Markov Random Fields: Theory and Applications. Chapman and Hall/CRC, 2005

18 Predictive Transform Coding In the above formulation: x 1 : pixels to be encoded x 2 : know pixels nearby New conditional mean: best prediction μ x1 x 2 = μ x1 Q 1 11 Q 12 x 2 μ x2 New best decorrelation transform after applying best prediction: Eigenvector matrix of Q x1 x 2 = Q 11

19 Motion Prediction Reference block and current block, both zero mean GMRF, with Q ref and Q c Form a 3D GMRF model Precision matrix: Q = Q c + I I I Q ref + I Regarding x 1 x 2 : μ x1 x 2 = (Q c +I) 1 x 2 Q x1 x 2 = Q c + I Reference block Current block

20 Motion Prediction When Q c is the Laplacian matrix Q c = δl New mean: μ x1 x 2 = (δl + I) 1 x 2 Low pass filter on x 2 (Partial) reason that half-pixel prediction is better! New precision matrix: Q x1 x 2 = δl + I Its eigenvector matrix is same as the eigenvector matrix of L First known proof that DCT is still optimal for residue compression in motion compensation!

21 Intra-Frame Predictive Coding Zero mean GMRF New mean: μ x1 x 2 = Q 1 11 Q 12 x 2 Optimal prediction given the GMRF model New precision matrix: Q x1 x 2 = Q 11 DCT no longer optimal! First general analysis of the optimal transform for intraframe predictive coding C. Yeo, Y. H. Tan, Z. Li, and S. Rahardja. Modedependent transforms for coding directional intra prediction residuals. IEEE Trans. on Circuits and Systems for Video Technology, 22(4): , 2012.

22 Graph-Based Regularization Estimate a signal x on graph, such that a cost function is minimized, subject to certain regularization x = arg min x f(x) + λs p x A popular smoothness term (p-dirichlet form): S p x 1 p i j:j~i W ij x i x j 2 p 2

23 A Prob. Viewpoint on Regularization Target function: x = arg min x = arg max x Mapping to MAP: f(x) + λs p x x f e λ S p x Likelihood: e f x λ Prior: e S p x When p = 2 Laplacian GMRF prior When p 2 Generalized GMRF prior

24 Conclusions GMRF provides a statistical viewpoint for GSP Provides valuable insights for Graph construction Graph transform Graph down-sampling Graph prediction Graph-based regularization Always be aware of the statistical implications!

GAUSSIAN PROCESS TRANSFORMS

GAUSSIAN PROCESS TRANSFORMS Philip A. Chou Ricardo L. de Queiroz Microsoft Research, Redmond, WA, USA pachou@microsoft.com) Computer Science Department, Universidade de Brasilia, Brasilia, Brazil queiroz@ieee.org)