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DISCRETE IMAGE MATHEMATICAL CHARACTERIZATION Chapter 1 presented a mathematical characterization of continuous image fields. This chapter develops a vector-space algebra formalism for representing discrete image fields from a deterministic and statistical viewpoint. Appendix 1 presents a summary of vector-space algebra concepts. | 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 5 DISCRETE IMAGE MATHEMATICAL CHARACTERIZATION Chapter 1 presented a mathematical characterization of continuous image fields. This chapter develops a vector-space algebra formalism for representing discrete image fields from a deterministic and statistical viewpoint. Appendix 1 presents a summary of vector-space algebra concepts. . VECTOR-SPACE IMAGE REPRESENTATION In Chapter 1 a generalized continuous image function F x y t was selected to represent the luminance tristimulus value or some other appropriate measure of a physical imaging system. Image sampling techniques discussed in Chapter 4 indicated means by which a discrete array F j k could be extracted from the continuous image field at some time instant over some rectangular area -J j J -K k K. It is often helpful to regard this sampled image array as a N1 x N2 element matrix F F np n2 for 1 nt Nt where the indices of the sampled array are reindexed for consistency with standard vector-space notation. Figure illustrates the geometric relationship between the Cartesian coordinate system of a continuous image and its array of samples. Each image sample is called a pixel. 121 122 DISCRETE IMAGE MATHEMATICAL CHARACTERIZATION FIGURE . Geometric relationship between a continuous image and its array of samples. For purposes of analysis it is often convenient to convert the image matrix to vector form by column or row scanning F and then stringing the elements together in a long vector 1 . An equivalent scanning operation can be expressed in quantitative form by the use of a N2 x 1 operational vector vn and a N1 N2 x N2 matrix Nn defined as 0 1 0 1 0 n - 1 0 n-1 vn 1 n Nn 1 n 0 n 1 0 n 1 0 N2 0 N2 Then the vector representation of the image matrix F is given by the stacking operation N f Z NnFvn n 1 In essence the vector vn .