Unsupervised segmentation of textures

Having obtained an initial segmentation, the texture model parameters can be estimated from the larger patches that result from this initialization step. This segmentation result is then iteratively refined in the next stage using a Markovian prior for the labels field, where in each iteration the texture model parameters are re-estimated for each patch.

More specifically, the estimated parametric models of the individual textures are employed to construct an energy function. This energy function is used in a simulated annealing algorithm to obtain maximum a-posteriori (MAP) estimates of the label field. The label field estimation is iteratively performed until the algorithm converges. The optimal estimate of the label field at site s is given by:

$$f^{*}=\min_{f\in F} \{U_s(f\vert d)\}=\min_{f\in F} \{U_s(d\vert f)+U_s(f)\}$$

where f is the label and d is the data field. The energy U_s(d|f) is evaluated at each site s using the following procedure:

• Let W(s) be some window centered at s. Let F^d(s) be the magnitude of the DFT of the observed image, evaluated in that window.
  
• Using the texture model parameters that correspond to the current label f of the site, we synthesize the texture, and evaluate the magnitude of the synthesized field DFT, F^f(s), in the same window W(s).
  
• The conditional energy is then given by:
  

  $$U_s(d\vert f)=\Vert F^{d}(s)-F^{f}(s)\Vert _{2}$$

  
  

Note that by adopting our estimate of the conditional energy,phase errors due to possible translation discrepancies are eliminated. Let C be the set of cliques corresponding to some neighborhood system. Here the field is assumed to be a first order Markov random field. The regularization energy at site s is then defined using a Potts model:

$$U_s(f)=-K\sum _{<r,s>\in C}\delta (f_{r},f_{s})$$

  

Figure: Unsupervised segmentation of a satelite image. From top to the bottom : Original image; Sea; Urban region.

  

Figure: Unsupervised segmentation of a medical image. From top to the bottom : Original image; Final segmentation result.