Stochastic Image Processing

A neighborhood system provides a convenient means for representing non-causal dependence of a pixel in a 2-D image lattice. A neighborhood system η is a set of neighborhoods η _s ⊂ Ω which satisfies the following conditions.

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.

The multiscale Markov models introduced in Chapter 4 can be used to extract large scale image features by parameterizing inter scale and intra scale interactions in an image. Since there are multiple scales with different parameter values for different scales, the parameter estimation process in the multi-scale approach becomes a major computational burden. One way to alleviate this complexity is to adopt a block-wise image modeling paradigm. That is, dividing the image space into non-overlapping blocks, a random variable is assigned for each image block to represent the class label. Then, a representative feature for each image block is extracted and is treated as the observed image data. The collection of all block features constitute the realization of the random field Y. Also, the class label assigned for each image block is a realization of the unobservable random field X. Since the representative feature values for image blocks normally have spatial continuity, the random field for the block class labels can be modeled as an MRF.