1. Markov Random Fields (MRF) Spring 2009 Ben-Gurion University of the Negev

2. Sensor Fusion Spring 2009 Instructor Dr. H. B Mitchell email: harveymitchell@walla.co.il

3. Sensor Fusion Spring 2009 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)

4. Sensor Fusion Spring 2009 MRF Fusion of Multiple Thresholded Images Multiple thresholding algorithms. Experiments show that different thresholding react differently to different pictures:

5. Sensor Fusion Spring 2009 MRF Fusion of Multiple Thresholded Images Experiments show that different thresholding react differently to different pictures: MRF provides a way of fusing them together taking into account context

6. Sensor Fusion Spring 2009 MRF Fusion of Multiple Thresholded Images Given thresholded images Seek a binary image such that Theory of MRF suggests can find by minimizing a sum of local energy functions:

7. Sensor Fusion Spring 2009 MRF Fusion of Multiple Thresholded Images The local energy has following form Split this into spatial context and inter-image context:

8. Sensor Fusion Spring 2009 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):

9. Sensor Fusion Spring 2009 MRF Fusion: Inter-Image Context Inter-image context is Write it as a sum of number of times B(m,n) is different from

10. Sensor Fusion Spring 2009 MRF Fusion: Inter-Image Context 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

11. Sensor Fusion Spring 2009 MRF Fusion: Inter-Image Context We use the same considerations to calculate the weights where

12. Sensor Fusion Spring 2009 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.