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CONTINUOUS IMAGE MATHEMATICAL CHARACTERIZATION In the design and analysis of image processing systems, it is convenient and often necessary mathematically to characterize the image to be processed. There are two basic mathematical characterizations of interest: deterministic and statistical. In deterministic image representation, a mathematical image function is defined and point properties of the image are considered. For a statistical image representation, the image is specified by average properties. The following sections develop the deterministic and statistical characterization of continuous images. Although the analysis is presented in the context of visual images, many of the results can be extended to general two-dimensional. | Digital Image Processing PIKS Inside Third Edition. William K. Pratt Copyright 2001 John Wiley Sons Inc. ISBNs 0-471-37407-5 Hardback 0-471-22132-5 Electronic 1 CONTINUOUS IMAGE MATHEMATICAL CHARACTERIZATION In the design and analysis of image processing systems it is convenient and often necessary mathematically to characterize the image to be processed. There are two basic mathematical characterizations of interest deterministic and statistical. In deterministic image representation a mathematical image function is defined and point properties of the image are considered. For a statistical image representation the image is specified by average properties. The following sections develop the deterministic and statistical characterization of continuous images. Although the analysis is presented in the context of visual images many of the results can be extended to general two-dimensional time-varying signals and fields. . IMAGE REPRESENTATION Let C x y t X represent the spatial energy distribution of an image source of radiant energy at spatial coordinates x y at time t and wavelength X. Because light intensity is a real positive quantity that is because intensity is proportional to the modulus squared of the electric field the image light function is real and nonnegative. Furthermore in all practical imaging systems a small amount of background light is always present. The physical imaging system also imposes some restriction on the maximum intensity of an image for example film saturation and cathode ray tube CRT phosphor heating. Hence it is assumed that 0 C x y t X A 3 4 CONTINUOUS IMAGE MATHEMATICAL CHARACTERIZATION where A is the maximum image intensity. A physical image is necessarily limited in extent by the imaging system and image recording media. For mathematical simplicity all images are assumed to be nonzero only over a rectangular region for which -Lv x Lr x x -Ly y Ly The physical image is of course observable only over some .

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