1 Multi Scale Markov Random Field Image Segmentation Taha hamedani

Renormalization Group Approach(RGA) Based on renormalization group ideas from statistical physics The RGA consists of two major steps: a renormalization step and a processing step. In the former step, we iteratively build a sequence of coarser and coarser grids : and a corresponding sequence of energy functions 2

The Hamiltonians are obtained from the original energy U iteratively as follows: for each m; 1 < m < M, we choose a conditional probability : The energy function of coarser grids are computed by the next formula: 3

coarse-to-fine processing starting at the bottom level: First we solve the problem at the coarsest level then we move upwards transmitting the obtained information at level m to the level m-1 above. 4

At level m-1, the processing is done under a constraint introduced by the level m below First, we find the global minimum of, then we are looking for the global minimum of but searching only in a smaller subspace of configurations constrained by the minimum via the equation: 5

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MRF Multi scale 7

To generate the multiscale version of the monogrid model: divide the initial grid into blocks of size n x n (2 x 2 here) associate the same label to the pixels of a block coarser scales are defined similarly with blocks 8

Neighborhood system 9

Equavalent model These sites form a coarse grid isomorphic to the corresponding scale To each block, we associate a unique site having the common label of the corresponding block. The isomorphism is just a projection of the coarse label field to the fine grid. 10

Energy function of course grid 11

Multiscale relaxation scheme Using a top-down strategy in the pyramid: 1.The problem is solved at a higher level 2.2. The lower level is initialized by the projection of the resulting labeling. 12

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Advantages vs Drawbacks Advs. At coarser grids, the state space has only a few elements => fast convergence properties on a sequential machine. For deterministic relaxation methods (ICM... ), the final result is improved Drawbacks. The multiscale scheme demands usually more iterations than the monogrid version. On a SIMD machine (CM200 for ex.), the multiscale scheme may be slower than the monogrid one (the rapidity depends only on the Virtual Processor Ratio) 14

Supervised multiscale segmentation 15

New neighboring system Keeping the interactions at each level, we introduce a new interaction between two neighbor grids: Each site interacts with its ancestor and its descendants. We consider only the first and second order cliques. 16

Energy function For the cliques which are located on the same level, the potential is not changed. For the cliques which site astride two neighbor levels, we will favor similar classes: 17

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