Markov Random Fields (MRF)

Markov Random Fields (MRF)

Spring 2009

Ben-Gurion University of the Negev

Sensor Fusion Spring 2009

Instructor

• Dr. H. B Mitchell

email: [email protected]


Markov Random Field

MRF: A probabilistic model defined by local conditional probabilities. In image fusion it provides a convenient way to exploit pixel

dependencies in fusion process. Notation:

is the conditional probability of gray-level G(m,n) at pixel (m,n) given the gray-levels in the neighborhood of (m,n)

Neighborhood of (m,n)

Center pixel (m,n)


MRF Fusion of Multiple Thresholded Images

Multiple thresholding algorithms. Experiments show that different thresholding react differently to

different pictures:



Experiments show that different thresholding react differently to different pictures:

MRF provides a way of fusing them together taking into account context



Given thresholded images Seek a binary image such that

Theory of MRF suggests can find by minimizing a sum of local energy functions:



The local energy has following form

Split this into spatial context and inter-image context:


MRF Fusion: Spatial Context

Spatial context is

Write it as a sum of number of times B(m,n) is different from B(p,q):


MRF Fusion: Inter-Image Context

Inter-image context is

Write it as a sum of number of times B(m,n) is different from



The formula:

means the inter-image context does not depend on how the accuracy of the thresholding algorithm varies with the pixel gray-levels.

We correct for this by rewriting the inter-image context as

where



We use the same considerations to calculate the weights

where


Algorithm

Solve MRF equations iteratively

Initialization. Set spatial context to zero:

Iterations. For each iteration update by minimizing Stop. Stop when difference between solution obtained at kth

iteration and (k+1)th iteration is sufficiently small.

Documents

Markov Random Fields (MRF)