In many situations involving Markov modeling, the Markov PDFs or the associated Gibbs PDFs are described parametrically. This means that the functional forms for the PDFs must be known a priori. These forms, typically, correspond to a parameterized family of PDFs, e.g., Gaussian. Fixing the parameter values chooses one particular member of this family. The parameters for these Markov PDFs, however, are unknown. In order to choose a suitable model for the data, we need to optimally estimate the parameters from the data.
Typically, these parameterized families of PDFs are relatively simple and have limited expressive power to accurately capture the structure and variability in image data [188,79,91]. As a result, in many instances, the data do not comply well with such parametric MRF models. This chapter proposes a method [9,5] of modeling the Markov PDFs nonparametrically and using data-driven strategies, in order to capture the properties underlying the data more accurately. In this way, the model is able to adapt to the data. As we saw in the previous chapter, with sufficient data, the nonparametric estimates can come very close to the underlying models. This chapter introduces the mathematics and engineering underpinning the proposed data-driven nonparametric MRF modeling scheme. The following chapters exploit this model for solving many classic image-processing problems dealing with image restoration and segmentation. The results demonstrate the success of this adaptive-MRF model, confirming that the model indeed adaptively captures the regularity in a wide-spectrum of images for a variety of applications.