Conclusions

This dissertation describes an adaptive image model that relies on the assumption of image data being derived from a stationary and ergodic MRF. We empirically infer the model underlying the data using principles from nonparametric density estimation. The density estimation schemes based on kernel smoothing help to compensate for the sparsity of data in the high-dimensional spaces. We use this model for processing images based on optimal information-theoretic measures and Bayesian decision theory.

We applied the adaptive algorithms for many different tasks concerning image restoration and segmentation. The generic theme underlying the restoration methods was to increase the predictability of pixel intensities from their neighborhoods by reducing the entropy of the pixel-intensity PDFs conditioned on the values of their neighbors. We found that the algorithms perform well on a wide spectrum of images with little parameter tuning. For denoising MR images, we exploited the knowledge of the statistical properties of Rician noise to empirically estimate the uncorrupted-signal Markov statistics from the corrupted-signal Markov statistics. This is essentially involves deconvolving a PDF for which we used the EM algorithm. Subsequently, following the empirical-Bayes approach, we employed the inferred corrupted-signal Markov statistics as a prior in a Bayesian decision-theoretic denoising framework. For the segmentation applications, i.e., MR brain tissue classification and texture segmentation, the key idea was to formulate the problem as one to maximize the mutual information between the Markov PDFs of the data and the unobserved segmentation labels. We had to impose a higher degree of smoothness on the estimated Markov PDFs for regularizing the region boundaries.

This dissertation makes the following contributions to the field of statistical image processing. It presents novel variations on standard MRF-based deterministic algorithms for image restoration, in the form of UINTA and the MRI-denoising method. It provides high-level arguments for the convergence of UINTA, although not a rigorous mathematical proof, and a proof of convergence of the MRI-denoising algorithm. It describes the equivalence between the mode-seeking mean-shift procedure and reducing Shannon's entropy on a nonparametric Parzen-window PDF, thereby providing further insights into the behavior of these algorithms. It exploits the adaptive-MRF model for unsupervised MR brain tissue classification using unimodal and multimodal MR data. This method tries to implicitly handle the noise, inhomogeneity, and partial voluming in the data with reasonable success. It applies these concepts for the classical image-processing tasks of restoration and denoising. The resulting algorithms often perform better than the current state-of-the-art.

There exist several other works where the key ideas relate to the methods in this dissertation. The idea of nonparametrically modeling image statistics is not entirely new. Popat and Picard [131] were the pioneers in employing nonparametric MRF image modeling. Their approach models the Markov PDFs via clustering-based nonparametric density estimation. Our approach, on the other hand, relies on kernel-smoothing approaches. 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 [50,172].

The NL-means algorithm for image denoising by Buades et al. [22,23] computes the denoised image intensity as a weighted average of a sample of image intensities, where the weights are derived from the neighborhoods of the pixels in the sample. The intensity updates in their method are based on the expectation of the conditional Markov PDF $P (X_t \vert {\bf y}_t)$ and closely resemble those in UINTA. While NL-means gets motivation from nonparametric regression theory, UINTA is motivated by information-theoretic concepts coupled with iterative MRF-based image processing.

The MRI-denoising strategy applies Robbins' empirical-Bayes approach for a nonparametric Parzen-window representation of the prior PDF. It also relates to the approach by Cordy and Thomas [33] that deconvolves PDFs by corrupted with i.i.d. additive Gaussian noise employing the EM algorithm for deconvolving PDFs. Snyder et al. [158], similar to our approach, use kernel density estimators for density deconvolution. The DUDE approach by Weismann et al. [175] focuses on discrete signal intensities and, subsequently, relies on inverting the channel-transition matrix (noise model) to give a closed form estimate for source statistics from the observed statistics. DUDE then follows an empirical-Bayes strategy for denoising.

Kim et al. [87] propose the mutual-information metric for texture segmentation using the intensity (or grayscale) histograms to distinguish between textures. The strategy in this dissertation can be viewed as an extension of the mutual-information metric that exploits the adaptive-MRF image model.

One of the limitations of the algorithms is that they are highly computationally expensive. Therefore, many of our applications are limited to $2$D images. We could alleviate this problem by exploiting parallelism in the algorithms or by developing effective fast approximations for the statistical-inference procedures. Parallelizing the algorithms to run on distributed-shared-memory multiprocessor machines or distributed clusters to obtain close-to-linear speedup is a nontrivial task, mainly because of the dynamically-changing Markov PDFs and the random memory-access patterns produced by the stochastic Parzen-window sampling schemes. Parallelization on commodity dual-processor or dual-core processors seems more straightforward, but produces limited gains. Delving into these issues would be an important part of future work.

Rapid advancements in technology, e.g., in medical-imaging and computer vision, will continue to generate new kinds of data along with the challenges of analyzing that data. Adaptive methods can play an important role in cases where accurate model formulation is difficult. Adaptive strategies using nonparametric statistics work best when sufficient data is available that allows empirical learning of the model. One may argue that this may not always be the case and a parametric model, if well designed, can effectively compensate for the scarcity of data. Nevertheless, one of the desirable properties of adaptive algorithms, which are designed to be more general purpose, is that their performance degrades gradually as the working conditions deviate from the optimal. This behavior also echoes in the philosophy behind the classic no-free-lunch theorem in optimization theory [182,48] that basically implies that one optimization strategy can perform better than another, on a specific problem, only if it incorporates prior information specific to that problem. Thus, specialized strategies, like those incorporating strong parametric or prior models, will show more drastic degradation in performance with sub-optimal working conditions. These advantages of adaptive strategies are corroborated by the results in this dissertation: in spite of scarce data in the high-dimensional feature spaces and the arguably-imperfect fit of the stationary-ergodic MRF model in some situations, the proposed algorithms behave robustly and perform well--many times better than the state of the art--for a wide spectrum of data and applications.