Wavelet modeling

From yet another perspective, images are formed as a superposition of local responses from some kind of sensor elements. Moreover, they exhibit such phenomena at multiple scales [59]. These local dependencies at multiple scales are well captured, mathematically as well as empirically, by the wavelet-based models [45,102]. Some limitations of these methods stem from the choice of the particular wavelet decomposition basis as well as the parametric models typically imposed on the wavelet coefficients.

Although these models may seem diverse, there exist many theoretical connections between them at a high level. For instance, some wavelet-based image processing techniques relate to regularity-based schemes in certain Besov spaces [26], and some statistical schemes relying on MRFs relate to variational schemes via the Gibbs formula in statistical mechanics [26].

The fundamental concept in this dissertation, the idea of nonparametric modeling of Markov PDFs, is not entirely new. In the past, however, such approaches involve supervision or training data where many observations from the unknown MRF are available a priori [131,50,172]. The novelty in this dissertation, though, is that we derive the MRF model unsupervisedly from the given input data itself and process the images based on this model. In this way, we are able to design unsupervised adaptive algorithms for many classic image-processing problems. Furthermore, we have applied these algorithms to many new relevant applications to produce results that compete with, and often further, the current state-of-the-art. During the process of applying the nonparametric MRF model for image processing, we have also tried to provide some new theoretical insights into statistical and information-theoretic image processing.

Popat and Picard [131] were the first to employ nonparametric MRF image models. They model the Markov PDFs via clustering-based nonparametric density estimation, unlike the kernel-based Parzen-window scheme underlying the proposed approach. They exploit their model for image restoration, image compression, and texture classification. Their learning approach, however, relies on training data, which limits its practical use. In contrast, the proposed method learns the Markov statistics of the image directly from the input data.

Learning Markov statistics nonparametrically entails estimation of PDFs in high-dimensional spaces. For instance, for a first-order local neighborhood having 6 voxels, i.e., 2 neighbors along each cardinal axis, we need to estimate PDFs on a $7$D space (center voxel along with its 6 neighbors). Lee et al. [91] as well as deSilva and Carlsson [40] analyze the statistics of 3 $\times$ 3 pixel neighborhoods, in $2$D images, in the corresponding $9$D spaces, and find the data to be concentrated in clusters and low-dimensional manifolds exhibiting nontrivial topologies. If we consider the neighborhood intensities as observations derived from a MRF, then the inherent structure of their distribution closely relates to the regularity captured by the Markov PDFs.

The literature on texture modeling also sheds light on the proposed modeling scheme. Elfadel and Picard [52] demonstrate the explicit connection between co-occurrence matrices for image intensities and the Gibbs PDFs for MRFs. Specifically, the nonlinear Gibbs energy is equivalent to a linear combination of co-occurrence measures over the Markov neighborhood. The proposed modeling technique employs Parzen-window density estimation, a generalization of co-occurrences, to estimate the Markov PDFs. Some texture-synthesis algorithms rely on learning Markov statistics from a sample texture image to construct new images having the same Markov statistics as the input texture [41,189,50,172]. Levina [98] proves that the empirically-learned Markov statistics converge asymptotically to the true texture statistics. This proof of convergence is also applicable towards the nonparametric learning of the Markov statistics in the proposed method. Paget [122] presents a nonparametric multiscale MRF framework to learn Markov statistics from a sample texture for synthesizing novel texture images.