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Xử lý hình ảnh kỹ thuật số P7

SUPERPOSITION AND CONVOLUTION In Chapter 1, superposition and convolution operations were derived for continuous two-dimensional image fields. This chapter provides a derivation of these operations for discrete two-dimensional images. Three types of superposition and convolution operators are defined: finite area, sampled image, and circulant area. The finite-area operator is a linear filtering process performed on a discrete image data array. | 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 7 SUPERPOSITION AND CONVOLUTION In Chapter 1 superposition and convolution operations were derived for continuous two-dimensional image fields. This chapter provides a derivation of these operations for discrete two-dimensional images. Three types of superposition and convolution operators are defined finite area sampled image and circulant area. The finite-area operator is a linear filtering process performed on a discrete image data array. The sampled image operator is a discrete model of a continuous two-dimensional image filtering process. The circulant area operator provides a basis for a computationally efficient means of performing either finite-area or sampled image superposition and convolution. 7.1. FINITE-AREA SUPERPOSITION AND CONVOLUTION Mathematical expressions for finite-area superposition and convolution are developed below for both series and vector-space formulations. 7.1.1. Finite-Area Superposition and Convolution Series Formulation Let F n1 n2 denote an image array for nb 2 1 2 . N. For notational simplicity all arrays in this chapter are assumed square. In correspondence with Eq. 1.2-6 the image array can be represented at some point m1 m2 as a sum of amplitude weighted Dirac delta functions by the discrete sifting summation F m1 m2 F n-1 n2 8 m-1 - n-1 1 m2 -n2 1 7.1-1 n1 n2 161 162 SUPERPOSITION AND CONVOLUTION The term 1 8 m1 - 1 1 m2 - 2 1 0 if m1 1 and m2 2 otherwise 7.1-2a 7.1-2b is a discrete delta function. Now consider a spatial linear operator O that produces an output image array Q mp m2 O F mp m2 7.1-3 by a linear spatial combination of pixels within a neighborhood of m1 m2 . From the sifting summation of Eq. 7.1-1 Q m1 m2 OiZZ F 1 2 8 m1 - 1 1 m2 - 2 1 f 7.1-4a L 1 2 J or Q mx m2 zz F nx n2 O 8 mj - n 1 m2 -n2 1 7.1-4b n1 n2 recognizing that O is a linear operator and that F