Introduction

This dissertation is about processing digital images. An image is, essentially, data that are acquired to measure some physical properties of a natural process. Image processing, broadly speaking, deals with the transformation and representation of the information contained in image data. We use the term image to mean any scalar or vector-valued function defined on an $n$-dimensional ($n$D) domain. Digital images consist of discrete samples on dense Cartesian grids. We can find several examples of digital images in our day-to-day lives such as digital photographs and videos. Black-and-white photographs consist of scalar data on a 2D grid, while color photographs contain 3D data (the RGB color) on a 2D grid. Color videos are 3D data on a 3D grid where the third grid dimension constitutes time. In the field of medical imaging, magnetic resonance (MR) images can contain scalar, vector, or tensor data on 3D grids. Image processing subsumes a gamut of domains and applications ranging from the low-level tasks of image modeling, restoration, segmentation, registration, and compression to the high-level tasks of recognition and interpretation [65,81,25]. Image processing has applications in many fields including computer vision, robotics, and medicine.

The information contained in images manifests itself, virtually always, in some patterns evident in the image data. We refer to these patterns as the regularity in the data. Describing this regularity in a way that is both general and powerful is one of the key problems in image processing. Typically, we capture this regularity in geometric or statistical terms. We refer to the process of describing regularity in images as image modeling. Indeed, the use of the term modeling is synonymous with its colloquial meaning of a schematic description of a system that accounts for its known/inferred properties and is used for further study of its characteristics, e.g., an atomic model, an economic model, etc. In this dissertation, we use the term in the statistical sense of a generative model. Thus, given an image model, we can generate image data that conform to, or are derived from, the model.

Typical image-modeling and processing techniques rely on a wide variety of mathematical principles in the fields of linear systems, variational calculus, probability and statistics, information theory, etc. In this dissertation, we desire algorithms that learn the physical model that generated the data through statistical inference methodologies. Observing that the image data always lie on a discrete Cartesian grid, we can model the regularity or the local statistical dependencies in the data through an underlying grid of random variables or a Markov random field (MRF). Theoretical and applied research over the last few decades has firmly established MRFs as powerful tools for statistical image modeling and processing.

This dissertation deals with several classic problems concerning restoration and segmentation. Image restoration deals with processing corrupted or degraded image data in order to obtain the uncorrupted image. This is typically performed by assuming certain models of the uncorrupted images or the degradation. For instance, image models try to capture the regularity in uncorrupted images. The literature presents different kinds of image models that suit best for different kinds of data. In practice, virtually all image data are degraded to an extent and many image-processing algorithms explicitly account for such degradations. Image segmentation is the process of dividing an image into partitions, or segments, where some semantics are associated with each segment.

Many image-processing strategies, including those for restoration and segmentation, make strong statistical or geometric assumptions about the properties of the signal or degradation. As a result, they break down when images exhibit properties that do not adhere to the underlying assumptions and lack the generality to be easily applied to diverse image collections. Strategies incorporating specific models work best when the data conform to that model and poorer otherwise. Models imposing stronger constraints (more restrictive) typically give better results with data conforming to those constraints as compared with weaker more-general models. However, schemes with restrictive models also fare much poorer when the data do not satisfy the model. As we shall see, many image-processing applications are not inherently conducive to strict models and, therefore, there is a need for generic image models and the associated image-processing algorithms. This dissertation presents a very general image model that adapts its specifications based on the observed data. Subsequently, the dissertation presents effective algorithms for image restoration and segmentation that easily apply to a wide spectrum of images.

One way of capturing image regularity is by incorporating a priori information in the image model itself. Some approaches rely on training data to extract prior information that is, in turn, transfused into the model specification. This allows us to learn complex models to which the data truly conform. Effective training sets, however, are not readily available for most applications and, therefore, this calls for unsupervised approaches [74]. Unsupervised approaches do not use training exemplars for learning properties about the data. However, they typically encode prior information via parametric statistical or geometric models that define the model structure. To refrain from imposing ill-fitting models on the data, unsupervised approaches need to learn the optimal parameter values from the data. As an alternative, unsupervised approaches can also rely on nonparametric modeling approaches where even the model structure, together with the associated internal parameters, is determined from the data. In these ways, unsupervised approaches need to be adaptive [74]. Adaptive methods automatically adjust their behavior in accordance with the perceived environment by adjusting their internal parameters. They do not impose a priori models but rather adapt their behavior, as well as the underlying model, to the data. Therefore, adaptive methods have the potential for being easily applicable to a wide spectrum of image data.

This dissertation uses a statistical MRF model to build adaptive algorithms for image processing. Broadly speaking, a statistical model is a set of probability density functions (PDFs) on the sample space associated with the data. Parametric statistical modeling parameterizes this set using a few control variables. An inherent difficulty with this approach is to find suitable parameter values such that the model is well-suited for the data. For instance, most parametric PDFs are unimodal whereas typical practical problems involve multimodal PDFs. Nonparametric statistical modeling [48,171,156] fundamentally differs from this approach by not imposing strong parametric models on the data. It provides the power to model and learn arbitrary (smooth) PDFs via data-driven strategies. As we shall see in this dissertation, such nonparametric schemes--that adapt the model to best capture the characteristics of the data and then process the data based on that model--can form powerful tools in formulating unsupervised adaptive image-processing methods.

We exploit the adaptive-MRF model to tackle several classic problems in image processing, medical image analysis, and computer vision. We enforce optimality criteria based on fundamental information-theoretic concepts that help us analyze the functional dependence, information content, and uncertainty in the data. In this way, information theory forms an important statistical tool in the design of unsupervised adaptive algorithms. The adaptive-MRF model allows us to statistically infer the structure underlying corrupted data. Learning this structure allows us to restore images without enforcing strong models on the signal. The restoration proceeds by improving the predictability of pixel intensities from their neighborhoods, by decreasing their joint entropy. When the noise model is known, e.g., MR images exhibit Rician noise, Bayesian reconstruction strategies coupled with MRFs can prove effective. We employ this model for optimal brain tissue classification in MR images. The method relies on maximizing the mutual information between the classification labels and image data, to capture their mutual dependency. This general formulation enables the method to easily adapt to various kinds of MR images, implicitly handling the noise, partial-voluming effects, and inhomogeneity. We use a similar strategy for unsupervised texture segmentation, observing that textures are precisely defined by the regularity in their Markov statistics.