Xử lý hình ảnh kỹ thuật số P7

SUPERPOSITION AND CONVOLUTIONIn 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.
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Xử lý hình ảnh kỹ thuật số P7 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)7SUPERPOSITION AND CONVOLUTIONIn Chapter 1, superposition and convolution operations were derived for continuoustwo-dimensional image fields. This chapter provides a derivation of these operationsfor discrete two-dimensional images. Three types of superposition and convolutionoperators are defined: finite area, sampled image, and circulant area. The finite-areaoperator is a linear filtering process performed on a discrete image data array. Thesampled image operator is a discrete model of a continuous two-dimensional imagefiltering process. The circulant area operator provides a basis for a computationallyefficient means of performing either finite-area or sampled image superposition andconvolution.7.1. FINITE-AREA SUPERPOSITION AND CONVOLUTIONMathematical expressions for finite-area superposition and convolution are devel-oped below for both series and vector-space formulations.7.1.1. Finite-Area Superposition and Convolution: Series FormulationLet F ( n1, n 2 ) denote an image array for n1, n2 = 1, 2,..., N. For notational simplicity,all arrays in this chapter are assumed square. In correspondence with Eq. 1.2-6, theimage array can be represented at some point ( m 1 , m 2 ) as a sum of amplitudeweighted Dirac delta functions by the discrete sifting summation F ( m 1, m 2 ) = ∑∑ F ( n 1, n 2 )δ ( m 1 – n 1 + 1, m 2 – n 2 + 1 ) (7.1-1) n1 n2 161162 SUPERPOSITION AND CONVOLUTIONThe term 1 if m1 = n 1 and m 2 = n 2 (7.1-2a)  δ ( m 1 – n 1 + 1, m 2 – n 2 + 1 ) =   0 otherwise (7.1-2b)is a discrete delta function. Now consider a spatial linear operator O { · } that pro-duces an output image array Q ( m 1, m 2 ) = O { F ( m 1, m 2 ) } (7.1-3)by a linear spatial combination of pixels within a neighborhood of ( m 1, m 2 ) . Fromthe sifting summation of Eq. 7.1-1,   Q ( m 1, m 2 ) = O  ∑ ∑ F ( n 1, n 2 )δ ( m1 – n 1 + 1, m 2 – n 2 + 1 ) (7.1-4a)  n1 n2 or Q ( m 1, m 2 ) = ∑∑ F ( n 1, n 2 )O { δ ( m 1 – n 1 + 1, m 2 – n 2 + 1 ) } (7.1-4b) n1 n2recognizing that O { · } is a linear operator and that F ( n 1, n 2 ) in the summation ofEq. 7.1-4a is a constant in the sense that it does not depend on ( m 1, m 2 ) . The termO { δ ( t 1, t 2 ) } for ti = m i – n i + 1 is the response at output coordinate ( m 1, m 2 ) to aunit amplitude input at coordinate ( n 1, n 2 ) . It is called the impulse response functionarray of the linear operator and is written as δ ( m 1 – n 1 + 1, m 2 – n 2 + 1 ; m 1, m 2 ) = O { δ ( t1, t2 ) } for 1 ≤ t1, t2 ≤ L (7.1-5)and is zero otherwise. For notational simplicity, the impulse response array is con-sidered to be square. In Eq. 7.1-5 it is assumed that the impulse response array is of limited spatialextent. This means that an output image pixel is influenced by input image pixelsonly within some finite area L × L neighborhood of the corresponding output imagepixel. The output coordinates ( m 1, m 2 ) in Eq. 7.1-5 following the semicolon indicatethat in the general case, called finite area superposition, the impulse response arraycan change form for each point ( m 1, m 2 ) in the processed array Q ( m 1, m 2 ). Follow-ing this nomenclature, the finite area superposition operation is defined as FINITE-AREA SUPERPOSITION AND CONVOLUTION 163FIGURE 7.1-1. Relationships between input data, output data, and impulse response arraysfor finite-area superposition; upper left corner justified array definition. Q ( m 1, m 2 ) = ∑∑ F ( n 1, n 2 )H ( m 1 – n 1 + 1, m 2 – n 2 + 1 ; m 1, m 2 ) (7.1-6) n 1 n2The limits of the summation are MAX { 1, m i – L + 1 } ≤ n i ≤ MIN { N, m i } (7.1-7)where MAX { a, b } and MIN { a, b } denote the maximum and minimum of the argu-ments, respectively. Examination of the indices of the impulse response array at itsextreme positions indicates that M = N + L - 1, and hence the processed output arrayQ is of larger dimension than the input array F. Figure 7.1-1 illustrates the geometryof finite-area superposition. If the impulse response array H is spatially invariant,the superposition operation reduces to the convolution operation. Q ( m 1, m 2 ) = ∑∑ F ( n 1, n 2 )H ( m 1 – n 1 + 1, m2 – n 2 + 1 ) (7.1-8) n1 n2Figure 7.1-2 presents a graphical example of convolution with a 3 × 3 impulseresponse array. Equation 7.1-6 expresses the finite-area superposition operation in left-justifiedform in which the input and output arrays are aligned at their upper left corners. It isoften notationally convenient to utilize a definition in which the output array is cen-tered with respect to the input array. This definition of centered su ...