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161 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 devel- oped below for both series and vector-space formulations. 7.1.1. Finite-Area Superposition and Convolution: Series Formulation Let denote an image array for n 1 , n 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 as a sum of amplitude weighted Dirac delta functions by the discrete sifting summation (7.1-1) Fn 1 n 2 ,() m 1 m 2 ,() Fm 1 m 2 ,() Fn 1 n 2 ,()δm 1 n 1 1+– m 2 n 2 1+–,() n 2 ∑ n 1 ∑ = 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) 162 SUPERPOSITION AND CONVOLUTION The term if and (7.1-2a) otherwise (7.1-2b) is a discrete delta function. Now consider a spatial linear operator that pro- duces an output image array (7.1-3) by a linear spatial combination of pixels within a neighborhood of . From the sifting summation of Eq. 7.1-1, (7.1-4a) or (7.1-4b) recognizing that is a linear operator and that in the summation of Eq. 7.1-4a is a constant in the sense that it does not depend on . The term for is the response at output coordinate to a unit amplitude input at coordinate . It is called the impulse response function array of the linear operator and is written as for (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 spatial extent. This means that an output image pixel is influenced by input image pixels only within some finite area neighborhood of the corresponding output image pixel. The output coordinates in Eq. 7.1-5 following the semicolon indicate that in the general case, called finite area superposition, the impulse response array can change form for each point in the processed array . Follow- ing this nomenclature, the finite area superposition operation is defined as δ m 1 n 1 1+– m 2 n 2 1+–,() 1 0      = m 1 n 1 = m 2 n 2 = O · {} Qm 1 m 2 ,()OFm 1 m 2 ,(){}= m 1 m 2 ,() Qm 1 m 2 ,()OFn 1 n 2 ,()δm 1 n 1 1+– m 2 n 2 1+–,() n 2 ∑ n 1 ∑    = Qm 1 m 2 ,() Fn 1 n 2 ,()O δ m 1 n 1 1+– m 2 n 2 1+–,(){} n 2 ∑ n 1 ∑ = O · {} Fn 1 n 2 ,() m 1 m 2 ,() O δ t 1 t 2 ,(){}t i m i n i 1+–= m 1 m 2 ,() n 1 n 2 ,() δ m 1 n 1 1+– m 2 n 2 1 m 1 m 2 ,;+–,()O δ t 1 t 2 ,(){}= 1 t 1 t 2 , L≤≤ LL× m 1 m 2 ,() m 1 m 2 ,() Qm 1 m 2 ,() FINITE-AREA SUPERPOSITION AND CONVOLUTION 163 (7.1-6) The limits of the summation are (7.1-7) where and denote the maximum and minimum of the argu- ments, respectively. Examination of the indices of the impulse response array at its extreme positions indicates that M = N + L - 1, and hence the processed output array Q is of larger dimension than the input array F. Figure 7.1-1 illustrates the geometry of finite-area superposition. If the impulse response array H is spatially invariant, the superposition operation reduces to the convolution operation. (7.1-8) Figure 7.1-2 presents a graphical example of convolution with a impulse response array. Equation 7.1-6 expresses the finite-area superposition operation in left-justified form in which the input and output arrays are aligned at their upper left corners. It is often notationally convenient to utilize a definition in which the output array is cen- tered with respect to the input array. This definition of centered superposition is given by FIGURE 7.1-1. Relationships between input data, output data, and impulse response arrays for finite-area superposition; upper left corner justified array definition. Qm 1 m 2 ,() Fn 1 n 2 ,()Hm 1 n 1 1+– m 2 n 2 1 m 1 m 2 ,;+–,() n 2 ∑ n 1 ∑ = MAX 1 m i L 1+–,{}n i MIN Nm i ,{}≤≤ MAX ab,{} MIN ab,{} Qm 1 m 2 ,() Fn 1 n 2 ,()Hm 1 n 1 1+– m 2 n 2 1+–,() n 2 ∑ n 1 ∑ = 33× 164 SUPERPOSITION AND CONVOLUTION (7.1-9) where and . The limits of the summa- tion are (7.1-10) Figure 7.1-3 shows the spatial relationships between the arrays F, H, and Q c for cen- tered superposition with a impulse response array. In digital computers and digital image processors, it is often convenient to restrict the input and output arrays to be of the same dimension. For such systems, Eq. 7.1-9 needs only to be evaluated over the range . When the impulse response FIGURE 7.1-2. Graphical example of finite-area convolution with a 3 × 3 impulse response array; upper left corner justified array definition. Q c j 1 j 2 ,() Fn 1 n 2 ,()Hj 1 n 1 L c +– j 2 n 2 L c j 1 j 2 ,;+–,() n 2 ∑ n 1 ∑ = L 3–()2⁄– j i NL1–()2⁄+≤≤ L c L 1+()2⁄= MAX 1 j i L 1–()2⁄–,{}n i MIN Nj i L 1–()2⁄+,{}≤≤ 55× 1 j i N≤≤ FINITE-AREA SUPERPOSITION AND CONVOLUTION 165 array is located on the border of the input array, the product computation of Eq. 7.1-9 does not involve all of the elements of the impulse response array. This situa- tion is illustrated in Figure 7.1-3, where the impulse response array is in the upper left corner of the input array. The input array pixels “missing” from the computation are shown crosshatched in Figure 7.1-3. Several methods have been proposed to deal with this border effect. One method is to perform the computation of all of the impulse response elements as if the missing pixels are of some constant value. If the constant value is zero, the result is called centered, zero padded superposition. A variant of this method is to regard the missing pixels to be mirror images of the input array pixels, as indicated in the lower left corner of Figure 7.1-3. In this case the centered, reflected boundary superposition definition becomes (7.1-11) where the summation limits are (7.1-12) FIGURE 7.1-3. Relationships between input data, output data, and impulse response arrays for finite-area superposition; centered array definition. Q c j 1 j 2 ,() Fn′ 1 n′ 2 ,()Hj 1 n 1 L c +– j 2 n 2 L c j 1 j 2 ,;+–,() n 2 ∑ n 1 ∑ = j i L 1–()2⁄– n i j i L 1–()2⁄+≤≤ 166 SUPERPOSITION AND CONVOLUTION and for (7.1-13a) for (7.1-13b) for (7.1-13c) In many implementations, the superposition computation is limited to the range , and the border elements of the array Q c are set to zero. In effect, the superposition operation is computed only when the impulse response array is fully embedded within the confines of the input array. This region is described by the dashed lines in Figure 7.1-3. This form of superposition is called centered, zero boundary superposition. If the impulse response array H is spatially invariant, the centered definition for convolution becomes (7.1-14) The impulse response array, which is called a small generating kernel (SGK), is fundamental to many image processing algorithms (1). When the SGK is totally embedded within the input data array, the general term of the centered convolution operation can be expressed explicitly as (7.1-15) for . In Chapter 9 it will be shown that convolution with arbitrary-size impulse response arrays can be achieved by sequential convolutions with SGKs. The four different forms of superposition and convolution are each useful in var- ious image processing applications. The upper left corner–justified definition is appropriate for computing the correlation function between two images. The cen- tered, zero padded and centered, reflected boundary definitions are generally employed for image enhancement filtering. Finally, the centered, zero boundary def- inition is used for the computation of spatial derivatives in edge detection. In this application, the derivatives are not meaningful in the border region. n' i 2 n i – n i 2Nn i –          = n i 0≤ 1 n i N≤≤ n i N> L 1+()2⁄ j i NL1–()2⁄–≤≤ NN× Q c j 1 j 2 ,() Fn 1 n 2 ,()Hj 1 n 1 L c +– j 2 n 2 L c +–,() n 2 ∑ n 1 ∑ = 33× Q c j 1 j 2 ,()H 33,()Fj 1 1 j 2 1–,–()H 32,()Fj 1 1 j 2 ,–()H 31,()Fj 1 1 j 2 1+,–()++= H 23,()Fj 1 j 2 1–,()H 22,()Fj 1 j 2 ,()H 21,()Fj 1 j 2 1+,()+++ H 13,()Fj 1 1 j 2 1–,+()H 12,()Fj 1 1 j 2 ,+()H 11,()Fj 1 1 j 2 1+,+()+++ 2 j i N 1–≤≤ FINITE-AREA SUPERPOSITION AND CONVOLUTION 167 Figure 7.1-4 shows computer printouts of pixels in the upper left corner of a convolved image for the four types of convolution boundary conditions. In this example, the source image is constant of maximum value 1.0. The convolution impulse response array is a uniform array. 7.1.2. Finite-Area Superposition and Convolution: Vector-Space Formulation If the arrays F and Q of Eq. 7.1-6 are represented in vector form by the vec- tor f and the vector q, respectively, the finite-area superposition operation can be written as (2) (7.1-16) where D is a matrix containing the elements of the impulse response. It is convenient to partition the superposition operator matrix D into submatrices of dimension . Observing the summation limits of Eq. 7.1-7, it is seen that (7.1-17) FIGURE 7.1-4 Finite-area convolution boundary conditions, upper left corner of convolved image. 0.040 0.080 0.120 0.160 0.200 0.200 0.200 0.080 0.160 0.240 0.320 0.400 0.400 0.400 0.120 0.240 0.360 0.480 0.600 0.600 0.600 0.160 0.320 0.480 0.640 0.800 0.800 0.800 0.200 0.400 0.600 0.800 1.000 1.000 1.000 0.200 0.400 0.600 0.800 1.000 1.000 1.000 0.200 0.400 0.600 0.800 1.000 1.000 1.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1.000 1.000 1.000 1.000 1.000 0.000 0.000 1.000 1.000 1.000 1.000 1.000 0.000 0.000 1.000 1.000 1.000 1.000 1.000 0.000 0.000 1.000 1.000 1.000 1.000 1.000 0.000 0.000 1.000 1.000 1.000 1.000 1.000 0.360 0.480 0.600 0.600 0.600 0.600 0.600 0.480 0.640 0.800 0.800 0.800 0.800 0.800 0.600 0.800 1.000 1.000 1.000 1.000 1.000 0.600 0.800 1.000 1.000 1.000 1.000 1.000 0.600 0.800 1.000 1.000 1.000 1.000 1.000 0.600 0.800 1.000 1.000 1.000 1.000 1.000 0.600 0.800 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 ( a ) Upper left corner justified ( b ) Centered, zero boundary ( c ) Centered, zero padded ( d ) Centered, reflected 55× N 2 1× M 2 1× qDf= M 2 N 2 × MN× D D 11, 0 … …… … 0 D 21, D 22, 0 D L 1, D L 2, D ML – 1 N, + 0D L 1 + 1, 0 … …… … 0D MN, = … … … … … … … 168 SUPERPOSITION AND CONVOLUTION The general nonzero term of D is then given by (7.1-18) Thus, it is observed that D is highly structured and quite sparse, with the center band of submatrices containing stripes of zero-valued elements. If the impulse response is position invariant, the structure of D does not depend explicitly on the output array coordinate . Also, (7.1-19) As a result, the columns of D are shifted versions of the first column. Under these conditions, the finite-area superposition operator is known as the finite-area convo- lution operator. Figure 7.1-5a contains a notational example of the finite-area con- volution operator for a (N = 2) input data array, a (M = 4) output data array, and a (L = 3) impulse response array. The integer pairs (i, j) at each ele- ment of D represent the element (i, j) of . The basic structure of D can be seen more clearly in the larger matrix depicted in Figure 7.l-5b. In this example, M = 16, FIGURE 7.1-5 Finite-area convolution operators: (a) general impulse array, M = 4, N = 2, L = 3; (b) Gaussian-shaped impulse array, M = 16, N = 8, L = 9. ( b ) 11 21 31 0 0 11 21 31 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 12 22 32 0 0 12 22 32 11 21 31 0 0 11 21 31 13 23 33 0 0 13 23 33 0 13 23 33 12 22 32 0 0 12 22 32 13 23 33 0 11 21 31 12 22 32 13 23 33 H = D = ( a ) D m 2 n 2 , m 1 n 1 ,()Hm 1 n 1 – 1+ m 2 n 2 1 m 1 m 2 ,;+–,()= m 1 m 2 ,() D m 2 n 2 , D m 2 1 + n 2 1 +, = 22× 44× 33× Hij,() FINITE-AREA SUPERPOSITION AND CONVOLUTION 169 N = 8, L = 9, and the impulse response has a symmetrical Gaussian shape. Note that D is a 256 × 64 matrix in this example. Following the same technique as that leading to Eq. 5.4-7, the matrix form of the superposition operator may be written as (7.1-20) If the impulse response is spatially invariant and is of separable form such that (7.1-21) where and are column vectors representing row and column impulse responses, respectively, then (7.1-22) The matrices and are matrices of the form (7.1-23) The two-dimensional convolution operation may then be computed by sequential row and column one-dimensional convolutions. Thus (7.1-24) In vector form, the general finite-area superposition or convolution operator requires operations if the zero-valued multiplications of D are avoided. The separable operator of Eq. 7.1-24 can be computed with only operations. QD mn , Fv n u m T n 1 = N ∑ m 1 = M ∑ = Hh C h R T = h R h C DD C D R ⊗= D R D C MN× D R h R 1() 0 … 0 h R 2() h R 1() h R 3() h R 2() … 0 h R 1() h R L() 0 0 … 0 h R L() = … … ……… … QD C FD R T = N 2 L 2 NL M N+() 170 SUPERPOSITION AND CONVOLUTION 7.2. SAMPLED IMAGE SUPERPOSITION AND CONVOLUTION Many applications in image processing require a discretization of the superposition integral relating the input and output continuous fields of a linear system. For exam- ple, image blurring by an optical system, sampling with a finite-area aperture or imaging through atmospheric turbulence, may be modeled by the superposition inte- gral equation (7.2-1a) where and denote the input and output fields of a linear system, respectively, and the kernel represents the impulse response of the linear system model. In this chapter, a tilde over a variable indicates that the spatial indices of the variable are bipolar; that is, they range from negative to positive spatial limits. In this formulation, the impulse response may change form as a function of its four indices: the input and output coordinates. If the linear system is space invariant, the output image field may be described by the convolution integral (7.2-1b) For discrete processing, physical image sampling will be performed on the output image field. Numerical representation of the integral must also be performed in order to relate the physical samples of the output field to points on the input field. Numerical representation of a superposition or convolution integral is an impor- tant topic because improper representations may lead to gross modeling errors or numerical instability in an image processing application. Also, selection of a numer- ical representation algorithm usually has a significant impact on digital processing computational requirements. As a first step in the discretization of the superposition integral, the output image field is physically sampled by a array of Dirac pulses at a resolu- tion to obtain an array whose general term is (7.2-2) where . Equal horizontal and vertical spacing of sample pulses is assumed for notational simplicity. The effect of finite area sample pulses can easily be incor- porated by replacing the impulse response with , where represents the pulse shape of the sampling pulse. The delta function may be brought under the integral sign of the superposition integral of Eq. 7.2-la to give (7.2-3) G ˜ xy,() F ˜ αβ,()J ˜ xyαβ,;,()αd βd ∞ – ∞ ∫ ∞ – ∞ ∫ = F ˜ xy,() G ˜ xy,() J ˜ xyα; β,,() G ˜ xy,() F ˜ αβ,()J ˜ x α– y β–,()αd βd ∞ – ∞ ∫ ∞ – ∞ ∫ = 2J 1+()2J 1+()× ∆S G ˜ j 1 ∆Sj 2 ∆S,()G ˜ xy,()δxj 1 ∆S– yj 2 ∆S–,()= J– j i J≤≤ J ˜ xyαβ,;,() ᭺ * Px– y–,() Px– y–,() G ˜ j 1 ∆Sj 2 ∆ S,()F ˜ αβ,()J ˜ j 1 ∆Sj 2 ∆S αβ,;,()αd βd ∞ – ∞ ∫ ∞ – ∞ ∫ =