TRANSFORM DOMAIN SUPERPOSITION 223 Also, the matrix form of the output for finite-area superposition is related to the extended image matrix K E by (9.2-6a) For sampled image superposition, (9.2-6b) The number of computational operations required to obtain k E by transform domain processing is given by the previous analysis for M = N = J. Direct transformation Fast transformation: If C is sparse, many of the filter multiplication operations can be avoided. From the discussion above, it can be seen that the secret to computationally effi- cient superposition is to select a transformation that possesses a fast computational algorithm that results in a relatively sparse transform domain superposition filter matrix. As an example, consider finite-area convolution performed by Fourier domain processing (2,3). Referring to Figure 9.2-1, let (9.2-7) where with for x, y = 1, 2, , K. Also, let denote the vector representation of the extended spatially invariant impulse response array of Eq. 7.3-2 for J = K. The Fou- rier transform of is denoted as (9.2-8) These transform components are then inserted as the diagonal elements of a matrix (9.2-9) QS1 J M() []K E S1 J M() [] T = GS2 J M() []K E S2 J M() [] T = 3J 4 J 2 4J 2 2 Jlog+ J 2 A K 2 A K A K ⊗= A K 1 K W x 1–()y 1–() = W 2πi– K ⎩⎭ ⎨⎬ ⎧⎫ exp≡ h E K() K 2 1× h E K() h E K() A K 2 []h E K() = K 2 K 2 × H K() diag h E K() 1()…h E K() K 2 (),,[]= 224 LINEAR PROCESSING TECHNIQUES Then, it can be shown, after considerable manipulation, that the Fourier transform domain superposition matrices for finite area and sampled image convolution can be written as (4) (9.2-10) for N = M – L + 1 and (9.2-11) where N = M + L + 1 and (9.2-12a) (9.2-12b) Thus the transform domain convolution operators each consist of a scalar weighting matrix and an interpolation matrix that performs the dimensionality con version between the - element input vector and the - element output vector. Generally, the interpolation matrix is relatively sparse, and therefore, transform domain superposition is quite efficient. Now, consider circulant area convolution in the transform domain. Following the previous analysis it is found (4) that the circulant area convolution filter matrix reduces to a scalar operator (9.2-13) Thus, as indicated in Eqs. 9.2-10 to 9.2-13, the Fourier domain convolution filter matrices can be expressed in a compact closed form for analysis or operational stor- age. No closed-form expressions have been found for other unitary transforms. Fourier domain convolution is computationally efficient because the convolution operator C is a circulant matrix, and the corresponding filter matrix C is of diagonal form. Actually, as can be seen from Eq. 9.1-6, the Fourier transform basis vectors are eigenvectors of C (5). This result does not hold true for superposition in general, nor for convolution using other unitary transforms. However, in many instances, the filter matrices D, B and C are relatively sparse, and computational savings can often be achieved by transform domain processing. DH M() P D P D ⊗[]= B P B P B ⊗[]H N() = P D uv,() 1 M 1 W M u 1–()– L 1–() – 1 W M u 1–()– – W N v 1–()– – = P B uv,() 1 N 1 W N v 1–()– L 1–() – 1 W M u 1–()– – W N v 1–()– – = H K() PP⊗() N 2 M 2 C JH J() = FAST FOURIER TRANSFORM CONVOLUTION 225 Figure 9.2-2 shows the Fourier and Hadamard domain filter matrices for the three forms of convolution for a one-dimensional input vector and a Gaussian-shaped impulse response (6). As expected, the transform domain representations are much more sparse than the data domain representations. Also, the Fourier domain circulant convolution filter is seen to be of diagonal form. Figure 9.2-3 illustrates the structure of the three convolution matrices for two-dimensional convolution (4). 9.3. FAST FOURIER TRANSFORM CONVOLUTION As noted previously, the equivalent output vector for either finite-area or sampled image convolution can be obtained by an element selection operation on the extended output vector k E for circulant convolution or its matrix counterpart K E . FIGURE 9.2-2. One-dimensional Fourier and Hadamard domain convolution matrices. ( b ) Sampled data convolution Signal Fourier Hadamard ( a ) Finite length convolution ( c ) Circulant convolution 226 LINEAR PROCESSING TECHNIQUES This result, combined with Eq. 9.2-13, leads to a particularly efficient means of con- volution computation indicated by the following steps: 1. Embed the impulse response matrix in the upper left corner of an all-zero matrix, for finite-area convolution or for sampled FIGURE 9.2-3. Two-dimensional Fourier domain convolution matrices. Spatial domain Fourier domain ( a ) Finite-area convolution ( b ) Sampled image convolution (c) Circulant convolution JJ× JM≥ JN≥ FAST FOURIER TRANSFORM CONVOLUTION 227 infinite-area convolution, and take the two-dimensional Fourier transform of the extended impulse response matrix, giving (9.3-1) 2. Embed the input data array in the upper left corner of an all-zero matrix, and take the two-dimensional Fourier transform of the extended input data matrix to obtain (9.3-2) 3. Perform the scalar multiplication (9.3-3) where . 4. Take the inverse Fourier transform (9.3-4) 5. Extract the desired output matrix (9.3-5a) or (9.3-5b) It is important that the size of the extended arrays in steps 1 and 2 be chosen large enough to satisfy the inequalities indicated. If the computational steps are performed with J = N, the resulting output array, shown in Figure 9.3-1, will contain erroneous terms in a boundary region of width L – 1 elements, on the top and left-hand side of the output field. This is the wraparound error associated with incorrect use of the Fourier domain convolution method. In addition, for finite area (D-type) convolu- tion, the bottom and right-hand-side strip of output elements will be missing. If the computation is performed with J = M, the output array will be completely filled with the correct terms for D-type convolution. To force J = M for B-type convolution, it is necessary to truncate the bottom and right-hand side of the input array. As a conse- quence, the top and left-hand-side elements of the output array are erroneous. H E A J H E A J = JJ× F E A J F E A J = K E mn,()JH E mn,()F E mn,()= 1 mn, J≤≤ K E A J 2 [] 1– H E A J 2 [] 1– = QS1 J M() []K E S1 J M() [] T = GS2 J M() []K E S2 J M() [] T = 228 LINEAR PROCESSING TECHNIQUES Figure 9.3-2 illustrates the Fourier transform convolution process with proper zero padding. The example in Figure 9.3-3 shows the effect of no zero padding. In both examples, the image has been filtered using a uniform impulse response array. The source image of Figure 9.3-3 is pixels. The source image of Figure 9.3-2 is pixels. It has been obtained by truncating the bot- tom 10 rows and right 10 columns of the source image of Figure 9.3-3. Figure 9.3-4 shows computer printouts of the upper left corner of the processed images. Figure 9.3-4a is the result of finite-area convolution. The same output is realized in Figure 9.3-4b for proper zero padding. Figure 9.3-4c shows the wraparound error effect for no zero padding. In many signal processing applications, the same impulse response operator is used on different data, and hence step 1 of the computational algorithm need not be repeated. The filter matrix H E may be either stored functionally or indirectly as a computational algorithm. Using a fast Fourier transform algorithm, the forward and inverse transforms require on the order of operations each. The scalar multiplication requires operations, in general, for a total of oper- ations. For an input array, an output array and an impulse response array, finite-area convolution requires operations, and sampled image convolution requires operations. If the dimension of the impulse response L is sufficiently large with respect to the dimension of the input array N, Fourier domain convolution will be more efficient than direct convolution, perhaps by an order of magnitude or more. Figure 9.3-5 is a plot of versus for equality FIGURE 9.3-1. Wraparound error effects. 11 11× 512 512× 502 502× 2J 2 2 Jlog J 2 J 2 14 2 Jlog+() NN× MM× LL× N 2 L 2 M 2 L 2 LN FAST FOURIER TRANSFORM CONVOLUTION 229 FIGURE 9.3-2. Fourier transform convolution of the candy_502_luma image with proper zero padding, clipped magnitude displays of Fourier images. ( a ) H E ( c ) F E ( e ) K E ( b ) E ( d ) E ( f ) E 230 LINEAR PROCESSING TECHNIQUES FIGURE 9.3-3. Fourier transform convolution of the candy_512_luma image with improper zero padding, clipped magnitude displays of Fourier images. ( a ) H E ( c ) F E ( e ) k E ( b ) E ( d ) E ( f ) E FAST FOURIER TRANSFORM CONVOLUTION 231 between direct and Fourier domain finite area convolution. The jaggedness of the plot, in this example, arises from discrete changes in J (64, 128, 256, ) as N increases. Fourier domain processing is more computationally efficient than direct process- ing for image convolution if the impulse response is sufficiently large. However, if the image to be processed is large, the relative computational advantage of Fourier domain processing diminishes. Also, there are attendant problems of computational FIGURE 9.3-4. 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Both difficulties can be alleviated by a block-mode filtering technique in which a large image is separately processed in adjacent overlapped blocks (2, 7–9). Figure 9.3-6a illustrates the extraction of a pixel block from the upper left corner of a large image array. After convolution with a impulse response, the resulting pixel block is placed in the upper left corner of an output FIGURE 9.3-5. Comparison of direct and Fourier domain processing for finite-area convolution.3 FIGURE 9.3-6. Geometric arrangement of blocks for block-mode filtering. N B N B × LL× M B M B × [...]... of SVD/SGK Convolution Filters for Image Processing, ” Report USCIPI 950 , University Southern California, Image Processing Institute, January 1980 PART 4 IMAGE IMPROVEMENT The use of digital processing techniques for image improvement has received much interest with the publicity given to applications in space imagery and medical research Other applications include image improvement for photographic... Conference, 1966, 56 3 57 8 244 LINEAR PROCESSING TECHNIQUES 4 W K Pratt, “Vector Formulation of Two-Dimensional Signal Processing Operations,” Computer Graphics and Image Processing, 4, 1, March 19 75, 1–24 5 B R Hunt, “A Matrix Theory Proof of the Discrete Convolution Theorem,” IEEE Trans Audio and Electroacoustics, AU-19, 4, December 1973, 2 85 288 6 W K Pratt, “Transform Domain Signal Processing Techniques,”... and industrial radiographic analysis Image improvement is a term coined to denote three types of image manipulation processes: image enhancement, image restoration and geometrical image modification Image enhancement entails operations that improve the appearance to a human viewer, or operations to convert an image to a format better suited to machine processing Image restoration has commonly been defined... highpass filters for a 51 2 × 51 2 pixel image SMALL GENERATING KERNEL CONVOLUTION (a) Zonal low-pass (b) Butterworth low-pass (c) Zonal high-pass 241 (d ) Butterworth high-pass FIGURE 9.4-4 Zonal and Butterworth low- and high-pass transfer functions; 51 2 × 51 2 images; cutoff frequency = 64 9 .5 SMALL GENERATING KERNEL CONVOLUTION It is possible to perform convolution on an N × N image array F( j, k) with... Proc IEEE Conference on Pattern Recognition and Image Processing, Chicago, May 1978 14 W K Pratt, J F Abramatic and O D Faugeras, “Method and Apparatus for Improved Digital Image Processing, ” U.S patent 4,330,833, May 18, 1982 15 J F Abramatic and O D Faugeras, “Sequential Convolution Techniques for Image Filtering,” IEEE Trans Acoustics, Speech and Signal Processing, ASSP-30, 1, February 1982, 1–10 16... simplify the processing task of a dataextraction machine There is no general unifying theory of image enhancement at present because there is no general standard of image quality that can serve as a design criterion for an image enhancement processor Consideration is given here to a variety of techniques that have proved useful for human observation improvement and image analysis Digital Image Processing: ... and present methods of point and spatial image restoration The final chapter of this part considers geometrical image modification 10 IMAGE ENHANCEMENT Image enhancement processes consist of a collection of techniques that seek to improve the visual appearance of an image or to convert the image to a form better suited for analysis by a human or a machine In an image enhancement system, there is no conscious... fidelity of a reproduced image with regard to some ideal form of the image, as is done in image restoration Actually, there is some evidence to indicate that often a distorted image, for example, an image with amplitude overshoot and undershoot about its object edges, is more subjectively pleasing than a perfectly reproduced original For image analysis purposes, the definition of image enhancement stops... IEEE Trans Audio and Electroacoustics, AU- 15, 2, June 1967, 85 90 8 M P Ekstrom and V R Algazi, “Optimum Design of Two-Dimensional Nonrecursive Digital Filters,” Proc 4th Asilomar Conference on Circuits and Systems, Pacific Grove, CA, November 1970 9 B R Hunt, “Computational Considerations in Digital Image Enhancement,” Proc Conference on Two-Dimensional Signal Processing, University of Missouri, Columbia,... visualizing an image with negatively valued pixels This is a useful transformation for systems that utilize the two's complement numbering (a) Linear image scaling (b) Linear image scaling with clipping (c) Absolute value scaling FIGURE 10.1-2 Image scaling methods 250 IMAGE ENHANCEMENT (a) Linear, full range, − 0.147 to 0.169 (b) Clipping, 0.000 to 0.169 (c) Absolute value, 0.000 to 0.169 FIGURE 10.1-3 Image . image . 0.001 0.002 0.003 0.0 05 0.006 0.007 0.008 0.009 0.010 0.011 0.012 0.012 0.012 0.012 0.012 0.002 0.0 05 0.007 0.009 0.011 0.014 0.016 0.018 0.020 0.023 0.0 25 0.0 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