Multiscale Markov Models

Abstract

The pixel-wise Markov image models introduced in the previous chapters have a limited ability in dealing with texture images. In order to describe large scale behavior in texture images, it is necessary to increase the size of the neighborhood. However, this dramatically increases the number of parameters and makes the computation for the parameter estimation quite difficult. Considering that the scales of the texture elements can be diverse, one possible way of dealing with various scales of texture elements is to express image data hierarchically using a multiscale transform. Laplacian pyramids [27] and wavelet multiresolution representations [124] are techniques that can decompose an image into multiple scales. Successive filtering and decimation operations in multiscale transforms provide a pyramid structure for image data. As a result, we have a sequence of image data for various scales and frequency bands. Specifically, coarse image components with smaller image resolution are located at the higher level and fine image components capturing detail can be found at the lower levels of the image pyramid. This multiscale image structure can provide useful information for the analysis and representation of the image data, especially for texture images. Processing successively from coarse to fine levels of the pyramid, image features at different scales and frequency bands are captured and class labels are assigned accordingly to large regions with low frequency and then refined to small image blocks and eventually to pixels with higher frequency information. This multiscale processing also allows us to save computations. That is, coarsening the image data for a multiscale image representation may smooth out local minima and result in faster convergence with a reduced sensitivity to an initial class label. Then, the optimal class labels at the current level of the pyramid to the next finer level can be obtained by searching in the vicinity of the class labels obtained at the previous coarser level. Since there exist strong inter scale and intra scale dependencies among vertically and horizontally connected neighboring data in the image pyramid, the multiscale images can be described more efficiently via Markov modeling. Adopting Markov models for the multiscale image data, the following issues arise: (i) how to describe various inter scale and intra scale interactions in terms of the Markov models, (ii) how to formulate the optimization problem for the class labels in the image pyramid, (iii) how to obtain the optimal class labels defined in (ii). These issues are treated in this chapter.