SPIE / IEEE Press, 1999. — 611 p.Science and engineering deal with temporal, spatial, and higher-dimensional processes that vary randomly from observation to observation. While one might study a specific observation for its algebraic or topological content, deterministic analysis does not provide a framework for understanding the ensemble of observations, nor does it provide a mechanism for prediction of future events. A system that functions over time needs to be designed in accordance with the manner in which input processes vary over time. System performance needs to be measured in terms of expected behavior and other statistical characteristics concerning operation on random inputs. There is an input random process, a transformation, and an output random process. System analysis begins with the input process and, based on the transformation, derives characteristics of the output process; system synthesis begins with an input process and a desired output process, and derives a transformation that estimates the desired output from the input. Image and (one-dimensional) signal processing concern the analysis and synthesis of linear and nonlinear systems that operate on spatial and temporal random functions. As areas of applied science, image and signal processing mark off a region within the overall domain of random processes that suits the set of applications they encompass. Because our major concern is algorithm development, our major interest is operator synthesis. We focus on three basic problems: representation, filter design, and modeling. These are not independent; they form a unity that is the key to algorithm development in a stochastic framework. The end goal is design of a filter (operator). If the input process is given in terms of a representation that is compatible with the form of the desired filter, then design is enhanced. Ultimately, the filter is to be used on some class of real-world images or signals. Therefore we need models that fit real processes and whose mathematical structure facilitates design of filters to extract desired structural information. My goal is not just to present the theory along with applications, but also to help students intuitively appreciate random functions. Were this a mathematics book, I would have taken the mathematical approach of stating general theorems and then giving corollaries for special situations. Instead, I have often begun with special cases in which probabilistic insight is more readily achievable. Moreover, I have not taken a theorem-proof approach. When provided, proofs are in the main body of the text and clearly delineated; sometimes they are either not provided or outlines of conceptual arguments are given. The intent is to state theorems carefully and to draw clear distinctions between rigorous mathematical arguments and heuristic explanations. When a proof can be given at a mathematical level commensurate with the text and when it enhances conceptual understanding, it is usually provided; in other cases, the effort is to explain subtleties of the definitions and properties concerning random functions, and to state conditions under which a proposition applies. Attention is drawn to the differences between deterministic concepts and their random counterparts, for instance, in the mean-square calculus, orthonormal representation, and linear filtering. Such differences are sometimes glossed over in method books; however, lack of differentiation between random and deterministic analysis can lead to misinterpretation of experimental results and misuse of techniques. My motivation for the book comes from my experience in teaching graduate-level image processing and having to end up teaching random processes. Even students who have taken a course on random processes have often done so in the context of linear operators on Signals. This approach is inadequate for image processing. Nonlinear operators playa widening role in image processing, and the spatial nature of imaging makes it significantly different from one-dimensional signal processing. Moreover, students who have some background in stochastic processes often lack a unified view in terms of canonical representation and orthogonal projections in inner product spaces.Probability Theory. Random Processes. Canonical Representation. Optimal Filtering. Random Models.
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