Image Restoration using Markov Random Fields

Transcription

1 Image Restoration using Markov Random Fields Based on the paper Stochastic Relaxation, Gibbs Distributions and Bayesian Restoration of Images, PAMI, 1984, Geman and Geman. and the book Markov Random Field Modeling in Computer Vision, by Stan Z. Li AJIT RAJWADE, CAP

2 Overview What are MRFs? (Locality Principle) Image Restoration approach using MRFs. Gibbs Random Fields Equivalence between MRF and GRF Concept of Metropolis Sampler and Gibbs Sampler. Discontinuities: Line Processes Results Limitations 2

3 Markov Random Fields: Intro Parameterized image model, where parameters have much lower dimensionality than the image. Consider image X with pixels (x1,x2,..xn). Basic concept: to factorize the global joint distribution of all the pixel values into a product of local factors. Within each local model, each entity depends only upon a set of neighboring entities. 3

4 MRF Intro To represent these relationships, use a graphical model. Each pixel in an image: NODE in the graph. Nodes arranged on a 2D lattice. Neighborhood relations: EDGES in the graph. Each node bears a label (value of intensity, gradient, segment label etc.) 4

5 Neighborhood Systems A neighborhood system defines what nodes will be treated as neighbors. Example: Neighborhood Systems of order one (4 neighbors) and two (8 neighbors). 5

6 Cliques Neighborhood systems have associated cliques. Cliques are a subset of nodes in which every pair of nodes are neighbors. Cliques for neighborhoods of order one and two: 6

7 MRF Definition Markov Model: Conditional probability of the value of a signal at time t depends only on the k previous values. k = order of the Markov Model. Markov Random Field: An extension of above concept to 2D. The conditional probability of a pixel label depends only on the labels of its neighbors. 7

8 MRF Definition Let be a set of sites (nodes of a graph), and be a neighborhood system. Let be a set of random variables indexed by S, and be a set of labels. Let be the set of all possible configurations of X. Then X is an MRF w.r.t. if and only if: 8

9 Image Restoration Approach Let G = a degraded image. Given G, restore the original image X. Bayesian approach: find mode of the posterior distribution, i.e. Problem: computationally prohibitive even for small images with very few intensity levels. 9

10 Gibbs Random Fields A set of random variables X are called a GRF, if its realizations follow a Gibbs distribution, i.e. is called an energy functional, given as follows: Here is called as the clique potential. The clique potential encodes the relationship between nodes in a clique (example, difference in intensity values). 10

11 Gibbs Random Fields Z is a normalizing constant given as 11

12 MRF = GRF The Hammersley-Clifford theorem states that: If X is an MRF on a set of nodes S with a neighborhood system N, then it is also a GRF. Incidentally, the converse is also true. Practical use: it allows us to specify the joint probability of the MRF for the original (denoised) image. 12

13 MRF = GRF But look at the formula for joint probability: Calculating Z is impossible. The trick is we need not compute Z for the required MAP estimate. 13

14 MRF = GRF Recall P ( X = ω G = g) = P( G = g X = ω) P( X = ω) / P( G = g) We have We also have the prior The likelihood does not have a term for Z. 14

15 MRF = GRF Taking logarithms, we get 15

16 Image Restoration Framework Formulate task as a Bayesian estimation problem. Denoised image is given as ω = arg maxω( P( X = ω G = g)) Expand posterior as P ( X = ω G = g) = P( G = g X = ω) P( X = ω) / P( G = g) Prior is given by the GRF with the prior energy. 16

17 Image Restoration Framework The likelihood noise statistics. is given by knowledge of Total posterior energy is given as: U ( X = ω G = g) = U(G = g X = ω) + U ( X = ω) Minimizing posterior energy = maximizing posterior probability (MAP). Energy minimization done using a sampling scheme combined with simulated annealing. 17

18 Computing MAP Estimate Solution: use Metropolis or Gibbs sampler. Both samplers work by generating a sequence of configurations starting from the initial configuration. It can be proved that this sampling scheme results in the required Gibbs distribution. And that the final outcome is not dependent upon. 18

19 Metropolis Sampler Let the initial (possibly unlikely) configuration =. Generate a new configuration starting from the previous one through some stochastic process. Determine energy change If the energy decreases, accept and go to step (2). Else accept with a probability and return to step (2). This is repeated for a large number of iterations 19

20 Metropolis Sampler After many iterations of the Metropolis sampler, the generated samples will obey a Gibbs distribution. It is a stochastic algorithm, i.e. it sometimes allows changes that increase the energy and hence avoids getting trapped in local minima. Disadvantage: could be very slow if there are many rejections. 20

21 Gibbs Sampler The Gibbs sampler also starts from some random configuration. A new sample is generated by updating the image, site by site (raster scan), using the conditional probability of the pixel value at that site, given its neighbors, i.e. from the following: The Gibbs Sampler also yields samples from the Gibbs distribution at equilibrium (after many iterations), i.e. 21

22 Gibbs Sampler No question of rejections, so faster. Implementation could be done in parallel (as nonneighboring nodes can be updated independently). The temperature parameter T is decreased during successive iterations (annealing). 22

23 Preserving Discontinuities In image smoothing, one also wants to preserve discontinuities in an image. That means we want to prevent interaction between two nodes in a clique if an edge is present in between them (or the difference in their intensities exceeds some threshold). To incorporate such constraints, we include a line process value between every pair of (neighboring) pixels. 23

24 Preserving Discontinuities 24

25 MRF Example: Ising Model Consider a model in which L = 2 (binary image X) and having an energy function The corresponding conditional distribution at site i is then given as: P( X i = x i X Ni ) = exp( x ij ( α + βv 1+ exp( ( α + βv i, j i, j )) )) v = i j x +, i, j+ 1 xi+ 1, j 25

26 Application: Texture Synthesis Generate a random binary image. Run several Gibb s sampler iterations on it, making use of previously mentioned conditional probability. Different values of produce different textures. 26

27 Incorporation of Line Processes The energy for an MRF can be written as: With line processes, the energy is written as: Here is the line process field potential. 27

28 Line Processes In the paper by Geman and Geman, different configurations of line processes have been assigned different energy values: 28

29 Results Original Image Degraded Image 29

30 Results Image restored without line processes Restored with line processes 30

31 Limitations Parameters of the MRF (weighting factors in the clique potential functions, i.e. and ) were chosen on an ad hoc basis. Parameter estimation is a non-trivial task. 31