Markov Random Fields

Transcription

1 Markov Random Fields Umamahesh Srinivas ipal Group Meeting February 25, 2011

2 Outline 1 Basic graph-theoretic concepts 2 Markov chain 3 Markov random field (MRF) 4 Gauss-Markov random field (GMRF), and applications 5 Other popular MRFs 02/25/2011 ipal Group Meeting 2

3 References 1 Charles Bouman, Markov random fields and stochastic image models. Tutorial presented at ICIP Mario Figueiredo, Bayesian methods and Markov random fields. Tutorial presented at CVPR /25/2011 ipal Group Meeting 3

4 Basic graph-theoretic concepts A graph G = (V, E) is a finite collection of nodes (or vertices) V = {n 1, n 2,..., n N } and set of edges E ( ) V 2 We consider only undirected graphs Neighbor: Two nodes n i, n j V are neighbors if (n i, n j ) E Neighborhood of a node: N (n i ) = {n j : (n i, n j ) E} Neighborhood is a symmetric relation: n i N (n j ) n j N (n i ) Complete graph: n i V, N (n i ) = {(n i, n j ), j = {1, 2,..., N}\{i}} Clique: a complete subgraph of G. Maximal clique: Clique with maximal number of nodes; cannot add any other node while still retaining complete connectedness. 02/25/2011 ipal Group Meeting 4

5 Basic graph-theoretic concepts A graph G = (V, E) is a finite collection of nodes (or vertices) V = {n 1, n 2,..., n N } and set of edges E ( ) V 2 We consider only undirected graphs Neighbor: Two nodes n i, n j V are neighbors if (n i, n j ) E Neighborhood of a node: N (n i ) = {n j : (n i, n j ) E} Neighborhood is a symmetric relation: n i N (n j ) n j N (n i ) Complete graph: n i V, N (n i ) = {(n i, n j ), j = {1, 2,..., N}\{i}} Clique: a complete subgraph of G. Maximal clique: Clique with maximal number of nodes; cannot add any other node while still retaining complete connectedness. 02/25/2011 ipal Group Meeting 4

6 Basic graph-theoretic concepts A graph G = (V, E) is a finite collection of nodes (or vertices) V = {n 1, n 2,..., n N } and set of edges E ( ) V 2 We consider only undirected graphs Neighbor: Two nodes n i, n j V are neighbors if (n i, n j ) E Neighborhood of a node: N (n i ) = {n j : (n i, n j ) E} Neighborhood is a symmetric relation: n i N (n j ) n j N (n i ) Complete graph: n i V, N (n i ) = {(n i, n j ), j = {1, 2,..., N}\{i}} Clique: a complete subgraph of G. Maximal clique: Clique with maximal number of nodes; cannot add any other node while still retaining complete connectedness. 02/25/2011 ipal Group Meeting 4

7 Illustration V = {1, 2, 3, 4, 5, 6} E = {(1, 2), (1, 3), (2, 4), (2, 5), (3, 4), (3, 6), (4, 6), (5, 6)} N (4) = {2, 3, 6} Examples of cliques: {(1), (3, 4, 6), (2, 5)} Set of all cliques: V E {3, 4, 6} 02/25/2011 ipal Group Meeting 5

8 Separation Let A, B, C be three disjoint subsets of V C separates A from B if any path from a node in A to a node in B contains some node in C Example: C = {1, 4, 6} separates A = {3} from B = {2, 5} 02/25/2011 ipal Group Meeting 6

9 Markov chains Graphical model: Associate each node of a graph with a random variable (or a collection thereof) Homogeneous 1-D Markov chain: p(x n x i, i < n) = p(x n x n 1 ) Probability of a sequence given by: N p(x) = p(x 0 ) p(x n x n 1 ) n=1 02/25/2011 ipal Group Meeting 7

10 2-D Markov chains Advantages: Simple expressions for probability Simple parameter estimation Disadvantages: No natural ordering of image pixels Anisotropic model behavior 02/25/2011 ipal Group Meeting 8

11 Random fields on graphs Consider a collection of random variables x = (x 1, x 2,..., x N ) with associated joint probability distribution p(x) Let A, B, C be three disjoint subsets of V. Let x A denote the collection of random variables in A. Conditional independence: A B C A B C p(x A, x B x C) = p(x A x C)p(x B x C) Markov random field: undirected graphical model in which each node corresponds to a random variable or a collection of random variables, and the edges identify conditional dependencies. 02/25/2011 ipal Group Meeting 9

12 Random fields on graphs Consider a collection of random variables x = (x 1, x 2,..., x N ) with associated joint probability distribution p(x) Let A, B, C be three disjoint subsets of V. Let x A denote the collection of random variables in A. Conditional independence: A B C A B C p(x A, x B x C) = p(x A x C)p(x B x C) Markov random field: undirected graphical model in which each node corresponds to a random variable or a collection of random variables, and the edges identify conditional dependencies. 02/25/2011 ipal Group Meeting 9

13 Markov properties Pairwise Markovianity: (n i, n j ) / E x i and x j are independent when conditioned on all other variables Local Markovianity: p(x i, x j x \{i,j} ) = p(x i x \{i,j} )p(x j x \{i,j} ) Given its neighborhood, a variable is independent on the rest of the variables p(x i x V\{i} ) = p(x i x N (i) ) Global Markovianity: Let A, B, C be three disjoint subsets of V. If C separates A from B p(x A, x B x C ) = p(x A x C )p(x B x C ), then p( ) is global Markov w.r.t. G. 02/25/2011 ipal Group Meeting 10

14 Markov properties Pairwise Markovianity: (n i, n j ) / E x i and x j are independent when conditioned on all other variables Local Markovianity: p(x i, x j x \{i,j} ) = p(x i x \{i,j} )p(x j x \{i,j} ) Given its neighborhood, a variable is independent on the rest of the variables p(x i x V\{i} ) = p(x i x N (i) ) Global Markovianity: Let A, B, C be three disjoint subsets of V. If C separates A from B p(x A, x B x C ) = p(x A x C )p(x B x C ), then p( ) is global Markov w.r.t. G. 02/25/2011 ipal Group Meeting 10

15 Markov properties Pairwise Markovianity: (n i, n j ) / E x i and x j are independent when conditioned on all other variables Local Markovianity: p(x i, x j x \{i,j} ) = p(x i x \{i,j} )p(x j x \{i,j} ) Given its neighborhood, a variable is independent on the rest of the variables p(x i x V\{i} ) = p(x i x N (i) ) Global Markovianity: Let A, B, C be three disjoint subsets of V. If C separates A from B p(x A, x B x C ) = p(x A x C )p(x B x C ), then p( ) is global Markov w.r.t. G. 02/25/2011 ipal Group Meeting 10

16 Hammersley-Clifford Theorem Consider a random field x on a graph G, such that p(x) > 0. Let C denote the set of all maximal cliques of the graph. If the field has the local Markov property, then p(x) can be written as a Gibbs distribution: { p(x) = 1 Z exp } V C (x C ), C C where Z, the normalizing constant, is called the partition function; V C (x C ) are the clique potentials If p(x) can be written in Gibbs form for the cliques of some graph, then it has the global Markov property. Fundamental consequence: every Markov random field can be specified via clique potentials. 02/25/2011 ipal Group Meeting 11

17 Hammersley-Clifford Theorem Consider a random field x on a graph G, such that p(x) > 0. Let C denote the set of all maximal cliques of the graph. If the field has the local Markov property, then p(x) can be written as a Gibbs distribution: { p(x) = 1 Z exp } V C (x C ), C C where Z, the normalizing constant, is called the partition function; V C (x C ) are the clique potentials If p(x) can be written in Gibbs form for the cliques of some graph, then it has the global Markov property. Fundamental consequence: every Markov random field can be specified via clique potentials. 02/25/2011 ipal Group Meeting 11

18 Hammersley-Clifford Theorem Consider a random field x on a graph G, such that p(x) > 0. Let C denote the set of all maximal cliques of the graph. If the field has the local Markov property, then p(x) can be written as a Gibbs distribution: { p(x) = 1 Z exp } V C (x C ), C C where Z, the normalizing constant, is called the partition function; V C (x C ) are the clique potentials If p(x) can be written in Gibbs form for the cliques of some graph, then it has the global Markov property. Fundamental consequence: every Markov random field can be specified via clique potentials. 02/25/2011 ipal Group Meeting 11

19 Regular rectangular lattices V = {(i, j), i = 1,..., M, j = 1,..., N} Order-K neighborhood system: N K (i, j) = {(m, n) : (i m) 2 + (j n) 2 K} 02/25/2011 ipal Group Meeting 12

20 Auto-models Only pair-wise interactions In terms of clique potentials: C > 2 V C ( ) = 0 Simplest possible neighborhood models 02/25/2011 ipal Group Meeting 13

21 Gauss-Markov Random Fields (GMRF) Joint probability function (assuming zero mean): { 1 p(x) = exp 1 } (2π) n/2 Σ 1/2 2 xt Σ 1 x Quadratic form in the exponent: x T Σ 1 x = x i x i Σ 1 i,j auto-model i j The neighborhood system is determined by the potential matrix Σ 1 Local conditionals are univariate Gaussian 02/25/2011 ipal Group Meeting 14

22 Gauss-Markov Random Fields (GMRF) Joint probability function (assuming zero mean): { 1 p(x) = exp 1 } (2π) n/2 Σ 1/2 2 xt Σ 1 x Quadratic form in the exponent: x T Σ 1 x = x i x i Σ 1 i,j auto-model i j The neighborhood system is determined by the potential matrix Σ 1 Local conditionals are univariate Gaussian 02/25/2011 ipal Group Meeting 14

23 Gauss-Markov Random Fields Specification via clique potentials: ( ) 2 ( ) 2 V C (x C ) = 1 αi C x i = 1 αi C x i 2 2 i C as long as i / C αi C = 0 The exponent of the GMRF density becomes: ( ) 2 V C (x C ) = 1 αi C x i 2 C C C C i V ( ) = 1 αi C αj C x i x j = xt Σ 1 x. i V j V C C i V 02/25/2011 ipal Group Meeting 15

24 GMRF: Application to image processing Classical image smoothing prior Consider an image to be a rectangular lattice with first-order pixel neighborhoods Cliques: pairs of vertically or horizontally adjacent pixels Clique potentials: squares of first-order differences (approximation of continuous derivative) V {(i,j),(i,j 1)} (x i,j, x i,j 1 ) = 1 2 (x i,j x i,j 1 ) 2 Resulting Σ 1 : block-tridiagonal with tridiagonal blocks 02/25/2011 ipal Group Meeting 16

25 Bayesian image restoration with GMRF prior Observation model: y = Hx + n, n N(0, σ 2 I) Smoothing GMRF prior: p(x) exp{ 1 2 xt Σ 1 x} MAP estimate: ˆx = [σ 2 Σ 1 + H T H] 1 H T y 02/25/2011 ipal Group Meeting 17

26 Bayesian image restoration with GMRF prior Figure: (a) Original image, (b) Blurred and slightly noisy image, (c) Restored version of (b), (d) No blur, severe noise, (e) Restored version of (d). Deblurring: good Denoising: oversmoothing; edge discontinuities smoothed out How to preserve discontinuities? Other prior models Hidden/latent binary random variables Robust potential functions (e.g. L 2 vs. L 1-norm) 02/25/2011 ipal Group Meeting 18

27 Bayesian image restoration with GMRF prior Figure: (a) Original image, (b) Blurred and slightly noisy image, (c) Restored version of (b), (d) No blur, severe noise, (e) Restored version of (d). Deblurring: good Denoising: oversmoothing; edge discontinuities smoothed out How to preserve discontinuities? Other prior models Hidden/latent binary random variables Robust potential functions (e.g. L 2 vs. L 1-norm) 02/25/2011 ipal Group Meeting 18

28 Compound GMRF Insert binary variables v to turn off clique potentials Modified clique potentials: V (x i,j, x i,j 1, v i,j ) = 1 2 (1 v i,j)(x i,j x i,j 1 ) 2 Intuitive explanation: v = 0 clique potential is quadratic ( on ) v = 1 V C ( ) = 0 no smoothing; image has an edge at this location. Can choose separate latent variables v and h for vertical and horizontal edges respectively { p(x h, v) exp 1 } 2 xt Σ 1 (h, v)x MAP estimate: ˆx = [σ 2 Σ 1 (h, v) + H T H] 1 H T y 02/25/2011 ipal Group Meeting 19

29 Compound GMRF Insert binary variables v to turn off clique potentials Modified clique potentials: V (x i,j, x i,j 1, v i,j ) = 1 2 (1 v i,j)(x i,j x i,j 1 ) 2 Intuitive explanation: v = 0 clique potential is quadratic ( on ) v = 1 V C ( ) = 0 no smoothing; image has an edge at this location. Can choose separate latent variables v and h for vertical and horizontal edges respectively { p(x h, v) exp 1 } 2 xt Σ 1 (h, v)x MAP estimate: ˆx = [σ 2 Σ 1 (h, v) + H T H] 1 H T y 02/25/2011 ipal Group Meeting 19

30 Compound GMRF Insert binary variables v to turn off clique potentials Modified clique potentials: V (x i,j, x i,j 1, v i,j ) = 1 2 (1 v i,j)(x i,j x i,j 1 ) 2 Intuitive explanation: v = 0 clique potential is quadratic ( on ) v = 1 V C ( ) = 0 no smoothing; image has an edge at this location. Can choose separate latent variables v and h for vertical and horizontal edges respectively { p(x h, v) exp 1 } 2 xt Σ 1 (h, v)x MAP estimate: ˆx = [σ 2 Σ 1 (h, v) + H T H] 1 H T y 02/25/2011 ipal Group Meeting 19

31 Compound GMRF Insert binary variables v to turn off clique potentials Modified clique potentials: V (x i,j, x i,j 1, v i,j ) = 1 2 (1 v i,j)(x i,j x i,j 1 ) 2 Intuitive explanation: v = 0 clique potential is quadratic ( on ) v = 1 V C ( ) = 0 no smoothing; image has an edge at this location. Can choose separate latent variables v and h for vertical and horizontal edges respectively { p(x h, v) exp 1 } 2 xt Σ 1 (h, v)x MAP estimate: ˆx = [σ 2 Σ 1 (h, v) + H T H] 1 H T y 02/25/2011 ipal Group Meeting 19

32 Discontinuity-preserving restoration Convex potentials: Generalized Gaussians: V (x) = x p, p [1, 2] Stevenson: V (x) = { x 2, 2a x a 2, x < a x a Green: V (x) = 2a 2 log cosh(x/a) Non-convex potentials: Blake, Zisserman: V (x) = (min{ x, a}) 2 Geman, McClure: V (x) = x2 x 2 +a 2 02/25/2011 ipal Group Meeting 20

33 Ising model (2-D MRF) V (x i, x j ) = βδ(x i x j ), β is a model parameter Energy function: V C (x C ) = β(boundary length) C C Longer boundaries less probable 02/25/2011 ipal Group Meeting 21

34 Application: Image segmentation Discrete MRF used to model the segmentation field Each class represented by a value X s {0,..., M 1} Joint distribution function: P {Y dy, X = x} = p(y x)p(x) (Bayesian) MAP estimation: ˆX = arg max x p x Y (x Y ) = arg max(log p(y x) + log(p(x))) x 02/25/2011 ipal Group Meeting 22

35 Application: Image segmentation Discrete MRF used to model the segmentation field Each class represented by a value X s {0,..., M 1} Joint distribution function: P {Y dy, X = x} = p(y x)p(x) (Bayesian) MAP estimation: ˆX = arg max x p x Y (x Y ) = arg max(log p(y x) + log(p(x))) x 02/25/2011 ipal Group Meeting 22

36 Application: Image segmentation Discrete MRF used to model the segmentation field Each class represented by a value X s {0,..., M 1} Joint distribution function: P {Y dy, X = x} = p(y x)p(x) (Bayesian) MAP estimation: ˆX = arg max x p x Y (x Y ) = arg max(log p(y x) + log(p(x))) x 02/25/2011 ipal Group Meeting 22

37 MAP optimization for segmentation Data model: p y x (y x) = s S p(y s x s ) Prior (Ising) model: p x (x) = 1 Z exp{ βt 1(x)}, where t 1 (x) is the number of horizontal and vertical neighbors of x having a different value MAP estimate: Hard optimization problem ˆx = arg min{ log p y x (y x) + βt 1 (x)} x 02/25/2011 ipal Group Meeting 23

38 MAP optimization for segmentation Data model: p y x (y x) = s S p(y s x s ) Prior (Ising) model: p x (x) = 1 Z exp{ βt 1(x)}, where t 1 (x) is the number of horizontal and vertical neighbors of x having a different value MAP estimate: Hard optimization problem ˆx = arg min{ log p y x (y x) + βt 1 (x)} x 02/25/2011 ipal Group Meeting 23

39 Some proposed approaches Iterated conditional modes: iterative minimization w.r.t. each pixel Simulated annealing: Generate samples from prior distribution Multi-scale resolution segmentation Figure: (a) Synthetic image with three textures, (b) ICM, (c) Simulated annealing, (d) Multi-resolution approach. 02/25/2011 ipal Group Meeting 24

40 Summary Graphical models study probability distributions whose conditional dependencies arise out of specific graph structures Markov random field is an undirected graphical model with special factorization properties 2-D MRFs have been widely used as priors in image processing problems Choice of potential functions leads to different optimization problems 02/25/2011 ipal Group Meeting 25