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Markov Random Fields
Umamahesh Srinivas
iPAL Group Meeting
February 25, 2011
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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
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References
1 Charles Bouman, Markov random fields and stochastic imagemodels. Tutorial presented at ICIP 1995
2 Mario Figueiredo, Bayesian methods and Markov random fields.Tutorial presented at CVPR 1998
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Basic graph-theoretic concepts
A graph G = (V, E) is a finite collection of nodes (or vertices)V = {n1, n2, . . . , nN} and set of edges E ⊂
(V2
)We consider only undirected graphs
Neighbor: Two nodes ni, nj ∈ V are neighbors if (ni, nj) ∈ E
Neighborhood of a node: N (ni) = {nj : (ni, nj) ∈ E}
Neighborhood is a symmetric relation: ni ∈ N (nj)⇔ nj ∈ N (ni)
Complete graph:∀ni ∈ V, N (ni) = {(ni, nj), j = {1, 2, . . . , N}\{i}}
Clique: a complete subgraph of G.
Maximal clique: Clique with maximal number of nodes; cannot addany other node while still retaining complete connectedness.
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Basic graph-theoretic concepts
A graph G = (V, E) is a finite collection of nodes (or vertices)V = {n1, n2, . . . , nN} and set of edges E ⊂
(V2
)We consider only undirected graphs
Neighbor: Two nodes ni, nj ∈ V are neighbors if (ni, nj) ∈ E
Neighborhood of a node: N (ni) = {nj : (ni, nj) ∈ E}
Neighborhood is a symmetric relation: ni ∈ N (nj)⇔ nj ∈ N (ni)
Complete graph:∀ni ∈ V, N (ni) = {(ni, nj), j = {1, 2, . . . , N}\{i}}
Clique: a complete subgraph of G.
Maximal clique: Clique with maximal number of nodes; cannot addany other node while still retaining complete connectedness.
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Basic graph-theoretic concepts
A graph G = (V, E) is a finite collection of nodes (or vertices)V = {n1, n2, . . . , nN} and set of edges E ⊂
(V2
)We consider only undirected graphs
Neighbor: Two nodes ni, nj ∈ V are neighbors if (ni, nj) ∈ E
Neighborhood of a node: N (ni) = {nj : (ni, nj) ∈ E}
Neighborhood is a symmetric relation: ni ∈ N (nj)⇔ nj ∈ N (ni)
Complete graph:∀ni ∈ V, N (ni) = {(ni, nj), j = {1, 2, . . . , N}\{i}}
Clique: a complete subgraph of G.
Maximal clique: Clique with maximal number of nodes; cannot addany other node while still retaining complete connectedness.
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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}
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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 Bcontains some node in C
Example: C = {1, 4, 6} separates A = {3} from B = {2, 5}
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Markov chains
Graphical model: Associate each node of a graph with a randomvariable (or a collection thereof)
Homogeneous 1-D Markov chain:
p(xn|xi, i < n) = p(xn|xn−1)
Probability of a sequence given by:
p(x) = p(x0)
N∏n=1
p(xn|xn−1)
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2-D Markov chains
Advantages:
Simple expressions for probability
Simple parameter estimation
Disadvantages:
No natural ordering of image pixels
Anisotropic model behavior
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Random fields on graphs
Consider a collection of random variables x = (x1, x2, . . . , xN ) withassociated joint probability distribution p(x)
Let A,B,C be three disjoint subsets of V. Let xA denote thecollection of random variables in A.
Conditional independence: A ⊥⊥ B | C
A ⊥⊥ B | C ⇔ p(xA,xB |xC) = p(xA|xC)p(xB |xC)
Markov random field: undirected graphical model in which eachnode corresponds to a random variable or a collection of randomvariables, and the edges identify conditional dependencies.
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Random fields on graphs
Consider a collection of random variables x = (x1, x2, . . . , xN ) withassociated joint probability distribution p(x)
Let A,B,C be three disjoint subsets of V. Let xA denote thecollection of random variables in A.
Conditional independence: A ⊥⊥ B | C
A ⊥⊥ B | C ⇔ p(xA,xB |xC) = p(xA|xC)p(xB |xC)
Markov random field: undirected graphical model in which eachnode corresponds to a random variable or a collection of randomvariables, and the edges identify conditional dependencies.
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Markov properties
Pairwise Markovianity:
(ni, nj) /∈ E ⇒ xi and xj are independent when conditioned on allother variables
p(xi, xj |x\{i,j}) = p(xi|x\{i,j})p(xj |x\{i,j})
Local Markovianity:
Given its neighborhood, a variable is independent on the rest of thevariables
p(xi|xV\{i}) = p(xi|xN (i))
Global Markovianity:
Let A,B,C be three disjoint subsets of V. If
C separates A from B ⇒ p(xA,xB |xC) = p(xA|xC)p(xB |xC),
then p(·) is global Markov w.r.t. G.
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Markov properties
Pairwise Markovianity:
(ni, nj) /∈ E ⇒ xi and xj are independent when conditioned on allother variables
p(xi, xj |x\{i,j}) = p(xi|x\{i,j})p(xj |x\{i,j})
Local Markovianity:
Given its neighborhood, a variable is independent on the rest of thevariables
p(xi|xV\{i}) = p(xi|xN (i))
Global Markovianity:
Let A,B,C be three disjoint subsets of V. If
C separates A from B ⇒ p(xA,xB |xC) = p(xA|xC)p(xB |xC),
then p(·) is global Markov w.r.t. G.
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Markov properties
Pairwise Markovianity:
(ni, nj) /∈ E ⇒ xi and xj are independent when conditioned on allother variables
p(xi, xj |x\{i,j}) = p(xi|x\{i,j})p(xj |x\{i,j})
Local Markovianity:
Given its neighborhood, a variable is independent on the rest of thevariables
p(xi|xV\{i}) = p(xi|xN (i))
Global Markovianity:
Let A,B,C be three disjoint subsets of V. If
C separates A from B ⇒ p(xA,xB |xC) = p(xA|xC)p(xB |xC),
then p(·) is global Markov w.r.t. G.
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Hammersley-Clifford Theorem
Consider a random field x on a graph G, such that p(x) > 0. Let Cdenote the set of all maximal cliques of the graph.
If the field has the local Markov property, then p(x) can be writtenas a Gibbs distribution:
p(x) =1
Zexp
{−∑C∈C
VC(xC)
},
where Z, the normalizing constant, is called the partition function;VC(xC) 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 specifiedvia clique potentials.
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Hammersley-Clifford Theorem
Consider a random field x on a graph G, such that p(x) > 0. Let Cdenote the set of all maximal cliques of the graph.
If the field has the local Markov property, then p(x) can be writtenas a Gibbs distribution:
p(x) =1
Zexp
{−∑C∈C
VC(xC)
},
where Z, the normalizing constant, is called the partition function;VC(xC) 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 specifiedvia clique potentials.
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Hammersley-Clifford Theorem
Consider a random field x on a graph G, such that p(x) > 0. Let Cdenote the set of all maximal cliques of the graph.
If the field has the local Markov property, then p(x) can be writtenas a Gibbs distribution:
p(x) =1
Zexp
{−∑C∈C
VC(xC)
},
where Z, the normalizing constant, is called the partition function;VC(xC) 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 specifiedvia clique potentials.
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Regular rectangular lattices
V = {(i, j), i = 1, . . . ,M, j = 1, . . . , N}
Order-K neighborhood system:
NK(i, j) = {(m,n) : (i−m)2 + (j − n)2 ≤ K}
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Auto-models
Only pair-wise interactions
In terms of clique potentials: |C| > 2⇒ VC(·) = 0
Simplest possible neighborhood models
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Gauss-Markov Random Fields (GMRF)
Joint probability function (assuming zero mean):
p(x) =1
(2π)n/2|Σ|1/2exp
{−1
2xTΣ−1x
}Quadratic form in the exponent:
xTΣ−1x =∑i
∑j
xixiΣ−1i,j ⇒ auto-model
The neighborhood system is determined by the potential matrixΣ−1
Local conditionals are univariate Gaussian
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Gauss-Markov Random Fields (GMRF)
Joint probability function (assuming zero mean):
p(x) =1
(2π)n/2|Σ|1/2exp
{−1
2xTΣ−1x
}Quadratic form in the exponent:
xTΣ−1x =∑i
∑j
xixiΣ−1i,j ⇒ auto-model
The neighborhood system is determined by the potential matrixΣ−1
Local conditionals are univariate Gaussian
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Gauss-Markov Random Fields
Specification via clique potentials:
VC(xC) =1
2
(∑i∈C
αCi xi
)2
=1
2
(∑i∈V
αCi xi
)2
as long as i /∈ C ⇒ αCi = 0
The exponent of the GMRF density becomes:
−∑C∈C
VC(xC) = −1
2
∑C∈C
(∑i∈V
αCi xi
)2
=1
2
∑i∈V
∑j∈V
(∑C∈C
αCi α
Cj
)xixj = −1
2xTΣ−1x.
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GMRF: Application to image processing
Classical image “smoothing” prior
Consider an image to be a rectangular lattice with first-order pixelneighborhoods
Cliques: pairs of vertically or horizontally adjacent pixels
Clique potentials: squares of first-order differences (approximation ofcontinuous derivative)
V{(i,j),(i,j−1)}(xi,j , xi,j−1) =1
2(xi,j − xi,j−1)2
Resulting Σ−1: block-tridiagonal with tridiagonal blocks
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Bayesian image restoration with GMRF prior
Observation model:
y = Hx + n, n ∼ N(0, σ2I)
Smoothing GMRF prior: p(x) ∝ exp{− 12xTΣ−1x}
MAP estimate:
x̂ = [σ2Σ−1 + HTH]−1HTy
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Bayesian image restoration with GMRF prior
Figure: (a) Original image, (b) Blurred and slightly noisy image, (c) Restoredversion 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. L2 vs. L1-norm)
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Bayesian image restoration with GMRF prior
Figure: (a) Original image, (b) Blurred and slightly noisy image, (c) Restoredversion 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. L2 vs. L1-norm)
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Compound GMRFInsert binary variables v to “turn off” clique potentials
Modified clique potentials:
V (xi,j , xi,j−1, vi,j) =1
2(1− vi,j)(xi,j − xi,j−1)2
Intuitive explanation:
v = 0⇒ clique potential is quadratic (“on”)
v = 1⇒ VC(·) = 0→ no smoothing; image has an edge at thislocation.
Can choose separate latent variables v and h for vertical andhorizontal edges respectively
p(x|h,v) ∝ exp
{−1
2xTΣ−1(h,v)x
}MAP estimate:
x̂ = [σ2Σ−1(h,v) + HTH]−1HTy
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Compound GMRFInsert binary variables v to “turn off” clique potentials
Modified clique potentials:
V (xi,j , xi,j−1, vi,j) =1
2(1− vi,j)(xi,j − xi,j−1)2
Intuitive explanation:
v = 0⇒ clique potential is quadratic (“on”)
v = 1⇒ VC(·) = 0→ no smoothing; image has an edge at thislocation.
Can choose separate latent variables v and h for vertical andhorizontal edges respectively
p(x|h,v) ∝ exp
{−1
2xTΣ−1(h,v)x
}MAP estimate:
x̂ = [σ2Σ−1(h,v) + HTH]−1HTy
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Compound GMRFInsert binary variables v to “turn off” clique potentials
Modified clique potentials:
V (xi,j , xi,j−1, vi,j) =1
2(1− vi,j)(xi,j − xi,j−1)2
Intuitive explanation:
v = 0⇒ clique potential is quadratic (“on”)
v = 1⇒ VC(·) = 0→ no smoothing; image has an edge at thislocation.
Can choose separate latent variables v and h for vertical andhorizontal edges respectively
p(x|h,v) ∝ exp
{−1
2xTΣ−1(h,v)x
}MAP estimate:
x̂ = [σ2Σ−1(h,v) + HTH]−1HTy
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Compound GMRFInsert binary variables v to “turn off” clique potentials
Modified clique potentials:
V (xi,j , xi,j−1, vi,j) =1
2(1− vi,j)(xi,j − xi,j−1)2
Intuitive explanation:
v = 0⇒ clique potential is quadratic (“on”)
v = 1⇒ VC(·) = 0→ no smoothing; image has an edge at thislocation.
Can choose separate latent variables v and h for vertical andhorizontal edges respectively
p(x|h,v) ∝ exp
{−1
2xTΣ−1(h,v)x
}MAP estimate:
x̂ = [σ2Σ−1(h,v) + HTH]−1HTy
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Discontinuity-preserving restoration
Convex potentials:
Generalized Gaussians: V (x) = |x|p, p ∈ [1, 2]
Stevenson:
V (x) =
{x2, |x| < a
2a|x| − a2, |x| ≥ a
Green: V (x) = 2a2 log cosh(x/a)
Non-convex potentials:
Blake, Zisserman: V (x) = (min{|x|, a})2
Geman, McClure: V (x) = x2
x2+a2
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Ising model (2-D MRF)
V (xi, xj) = βδ(xi 6= xj), β is a model parameter
Energy function: ∑C∈C
VC(xC) = β(boundary length)
Longer boundaries less probable
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Application: Image segmentation
Discrete MRF used to model the segmentation field
Each class represented by a value Xs ∈ {0, . . . ,M − 1}
Joint distribution function:
P{Y ∈ dy,X = x} = p(y|x)p(x)
(Bayesian) MAP estimation:
X̂ = arg maxx
px|Y (x|Y ) = arg maxx
(log p(Y |x) + log(p(x)))
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Application: Image segmentation
Discrete MRF used to model the segmentation field
Each class represented by a value Xs ∈ {0, . . . ,M − 1}
Joint distribution function:
P{Y ∈ dy,X = x} = p(y|x)p(x)
(Bayesian) MAP estimation:
X̂ = arg maxx
px|Y (x|Y ) = arg maxx
(log p(Y |x) + log(p(x)))
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Application: Image segmentation
Discrete MRF used to model the segmentation field
Each class represented by a value Xs ∈ {0, . . . ,M − 1}
Joint distribution function:
P{Y ∈ dy,X = x} = p(y|x)p(x)
(Bayesian) MAP estimation:
X̂ = arg maxx
px|Y (x|Y ) = arg maxx
(log p(Y |x) + log(p(x)))
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MAP optimization for segmentation
Data model:py|x(y|x) =
∏s∈S
p(ys|xs)
Prior (Ising) model:
px(x) =1
Zexp{−βt1(x)},
where t1(x) is the number of horizontal and vertical neighbors of xhaving a different value
MAP estimate:
x̂ = arg minx{− log py|x(y|x) + βt1(x)}
Hard optimization problem
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MAP optimization for segmentation
Data model:py|x(y|x) =
∏s∈S
p(ys|xs)
Prior (Ising) model:
px(x) =1
Zexp{−βt1(x)},
where t1(x) is the number of horizontal and vertical neighbors of xhaving a different value
MAP estimate:
x̂ = arg minx{− log py|x(y|x) + βt1(x)}
Hard optimization problem
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Some proposed approachesIterated 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) Simulatedannealing, (d) Multi-resolution approach.
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Summary
Graphical models study probability distributions whose conditionaldependencies arise out of specific graph structures
Markov random field is an undirected graphical model with specialfactorization properties
2-D MRFs have been widely used as priors in image processingproblems
Choice of potential functions leads to different optimization problems
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