FOR EFFICIENT IMAGE PROCESSING. Hong Tang, Bingbing Zhou, Iain Macleod, Richard Brent and Wei Sun

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1 A CLASS OF PARALLEL ITERATIVE -TYPE ALGORITHMS FOR EFFICIENT IMAGE PROCESSING Hong Tang, Bingbing Zhou, Iain Macleod, Richard Brent and Wei Sun Computer Sciences Laboratory Research School of Information Sciences and Engineering Australian National University Canberra, ACT 0200, Australia Abstract This paper proposes a class of parallel iterative median-type lters to restore signals degraded by Gaussian noise. An ecient implementation of these lters on distributed memory MIMD machines is outlined. Iterative median-type algorithms have been implemented with high eciency (> 90%) on the Fujitsu AP1000. We conclude that algorithms of this type are very suitable for parallel implementation on distributed memory machines. 1. Introduction Signal restoration, especially image restoration, requires large-volume and high-speed computations. Certain classes of image restoration operations such as spatial frequency ltering and convolution have relatively straightforward parallel implementations based on 2-D FFTs [1, 4, 5, 7, 11]. This paper discusses the more dicult case of non-linear ltering operations, with reference to application of a class of iterative median-type lters [2, 3, 4, 6, 8, 9, 10, 15] for enhancement of images degraded by additive Gaussian noise. The conventional median lter has the advantages that each point in the output image depends only on a limited neighbourhood of points in the input image (simplifying parallel implementation) and that it tends to preserve the shape of details in the original undegraded image. However, a conventional median lter usually needs to be applied iteratively (i.e. sequentially) to achieve the desired level of noise reduction. p-recursive median lters have better noise performance, but can lead to distortion of image detail. They also have a strong sequential aspect in that when computing a new output point they rely on values computed in the preceding p steps. One approach to parallel implementation of p-recursive median ltering divides the original image into sub-image blocks which are processed concurrently, but this is at the expense of introducing artefacts in the vicinity of the block boundaries. The following sections describe a hybrid parallel image lter, which combines p-recursive and conventional median lters in a way which gives superior image restoration (in terms of noise reduction, preservation of image detail and elimination of artefacts) and reduces overall computational and communication demands. 2. Median-Type Filters We discuss application of the standard median and the p-recursive median lters to restoration of noisy images. In a 2-D standard median lter with a square window of size (2k + 1) (2k + 1), the median replaces the central pixel: ^f(i; j) = MED f g(m; n) j i? k m i + k; j? k n j + kg (1) where ^f is the ltered image, g is the noisy image, and MED is the median operator. When the median lter is modied as follows, it becomes a \p-recursive median lter." The p-recursive median lter incorporates previous output values into the median decision process so that, at each step, it replaces the p points preceding the central point in the moving window with the medians from the previous p steps. The median operation can then be applied to obtain the new central value ^f(i; j). If the points in the window are numbered row by row in sequence, so that the central point is numbered l, where l = ((2k + 1) 2 + 1)=2, then the new method gives ^f l = MEDfg 1 ; :::; g l?p?1 ; ^f l?p ; :::; ^f l?1 ; g l ; :::; g (2k+1) 2g (2) where, for l? p r l? 1, ^f r is the median value of the window centred on the r th point. The parameter p can be changed to yield dierent lter types. The window moves from left to right and from top to bottom. Both standard median and p recursive median lters have good noise attenuation characteristics but at the expense of a possibly unacceptable amount of blurring. If p = 0, then Eq. (2) is the standard median lter. If p = 1, then there is only one output-feedback median point in the window. This has the eect of some noise reduction together with a very small amount of smoothing compared to the standard median lter. If p = l? 1, then Eq. (2) describes the recursive median lter [3, 8, 9] which has the greatest number of feedback median points in the window.

2 This provides much greater noise reduction than the rst case (p = 1), but usually results in an unacceptable degree of low pass ltering. We can generally choose an intermediate value of p (1 < p < l? 1) which gives an acceptable trade-o between noise reduction and loss of ne detail. When p = k(k + 2) = (k + 1) 2? 1; (3) for window size (2k + 1) (2k + 1), k=f1,2,3,...g, then the lter becomes the \improved" median lter [13, 14]. For a 3 3 window (k = 1), a good value of p is 3; for a 5 5 window (k = 2), p = 8; and so on. 3. Iterative Enhancement A simple model of image degradation is g(i; j) = f(i; j) + n(i; j); (4) where g(i; j); f(i; j) and n(i; j) are the degraded and original image and the noise respectively. The aim of image enhancement is to reduce the noise as much as possible or to nd a method which, given g, derives an image as close as possible to the original, subject to a suitable optimality criterion. A number of approaches can be found in the literature for solving the image enhancement problem [12]. The basic solution to Eq. (4) is as follows: ^f = D[g] = D[f + n]; (5) where D is an image enhancement lter. When D is of higher order, it becomes an iterative lter: f k+1 = D[f k ] (6) f 1 = D[f 0 + n] = D[f + n]; (7) where Eq. (7) is equal to Eq. (5). Iterative median lters are applied repeatedly to suppress noise signals while preserving image edges, according to a suitable criterion. Although they tend to be computationally expensive, such lters are very popular in image processing because of their good performance. They would nd widespread use in interactive applications if they could be made to operate in real-time: the most promising approach here is through parallel implementation. 4. Parallel Algorithms This section describes parallel implementation of iterative median lters on distributed memory MIMD machines (these having better scalability than shared memory machines). The implementations given are also suitable for SIMD machines with appropriate conditional operators. Assume that P processing elements (PEs) are available in the parallel system. To process an N N image we rst divide the image into P x P y ; (8) blocks, where P x ; P y 1 and P x P y = P, letting each block hold N P x P y adjacent pixels. Each sub-image block is then allocated to the local memory of a separate PE, so that the image is evenly distributed across the entire system. As described above, the new value of the central pixel depends on all the pixels in the processing window of a median lter. If the window size is M M, K = (M?1)=2 rows or columns of pixels at each boundary of a block will aect and also be aected by the pixels close to them in the original image but forming part of the boundary region of other blocks. Since the image is distributed across the system, we need to access the local memories of other PEs by using explicit data communication when the new values of boundary region pixels in each PE are being calculated. 4.1 Parallel implementation of median lter With median ltering the output image values from the current iteration depend only on the values from the previous iteration. We may thus combine all the required pixels at each boundary of a block and transfer them as a single message between the corresponding PEs at the beginning of each iteration. The pixels received in each PE pad the corresponding boundaries of each block. We then work only with such locally augmented blocks and there is no further data communication required during each iteration. We give an example here of a two step procedure for communicating data between PEs when the parallel computing system is organized as a 2-D array. In the rst step, the K boundary rows of pixels from the top and bottom boundaries of a block are transferred vertically between neighbouring PEs; the received rows pad the top and bottom of the block. In the second step, the K boundary columns from the (padded) left and right boundaries are transferred horizontally between the neighbouring PEs. Note that transfer of these padded columns also transfers the required elements from diagonally neighbouring PEs. As discussed above, the original N N image is divided into P x P y blocks and evenly distributed across a system of P = P x P y PEs. Each block is of size N=P x N=P y and its shape varies with the values of P x and P y. It is easy to show that the total number of pixels to be transferred

3 from other PEs to a given PE is about N 2K + N : (9) P y P x In the system-wide sense, this number can be minimised by choosing similar values for P x and P y so that the block sub-images are approximately square. Minimising communication overheads is important with distributed memory architectures because transfer of data elements between PEs is typically much slower than within PEs. 4.2 Parallel hybrid median lter Although they reduce noise more eciently than conventional median lters, p-recursive median lters cannot be directly implemented in parallel because the image values to be determined rely on the feedback of median points computed from the previous windows in the same iteration. This section describes a hybrid median lter which combines p-recursive and conventional median lters in a way which gives superior image restoration, allows parallel implementation and reduces overall computational cost. Suppose that the image is divided into blocks in a similar manner to that described above for the conventional median lter. Our parallel algorithm has two stages. In the rst stage, K 0 ( K) boundary columns or rows of each block are transferred to adjacent blocks and are used to pad the corresponding boundaries. The p-recursive median ltering algorithm is then applied to each augmented block, using only pixels local to each PE. All blocks can be processed in parallel with this block expanding method. As a result of regarding each augmented block as an independent image, artefacts are introduced in the vicinity of the block boundaries. Two techniques which can be combined to reduce such boundary eects to acceptable levels are now considered. When the columns or rows are exchanged between adjacent blocks, the amount of information obtained from the neighbours depends on the extent of padding or augmentation. The boundary artefacts are reduced in magnitude as the width of overlap K 0 increases. However, if we make the overlap large the total commmunication and computational costs increase substantially. Fortunately, our experimental results (discussed in the next section) show that in practice the magnitude of boundary artefacts falls quite quickly with increasing K 0 (for K 0 K). The second technique for reducing boundary eects has a more subtle basis. Observe that there will be no boundary eects if each pixel has a signicant correlation only with those pixels which reside in the same block. For most classes of imagery, the degree of correlation between pairs of pixels depends on their physical separation. Thus for a rectangular block with a given number of pixels, the mutually relevant information regarding pixels within the block is maximised if the block is square. Thus when P x and P y are similar, the total number of pixels to be transferred between blocks and the magnitude of boundary artefacts are both reduced. Even with use of the above techniques, noticeable nonconstant gray level dierences may still exist in the vicinity of block boundaries. A conventional median lter can then be applied to smooth the entire image and to eliminate remaining visible artefacts, forming the second stage of our parallel hybrid algorithm. 5. Experimental Results We report the results of implementation experiments on a practical MIMD machine, comparing the restoration performance of various lters on representative degraded images. Parallel Machine: Parallel algorithms for various median-type lters have been implemented on the ANU's Fujitsu AP1000, a distributed memory MIMD machine with 128 (1024 max.) SPARC processors. Each processor has 16 MBytes of memory and a Weitek oatingpoint unit. The topology of the AP1000 is a 2-D torus, with hardware support for wormhole routing. The routing hardware (T-net) provides a theoretical bandwidth of 25 MByte/sec between any two cells. In practice, about 6 MByte/sec is attainable by user programs. The routing hardware supports row and column broadcasts, with broadcast times being comparable to those of PE to PE transfers. Parallel Parameters: The two most important criteria for evaluating the success of the parallel implementations of the algorithms used in our experiments are speedup and eciency, dened as follows: Speedup: This is dened as the ratio between T 1, the time to execute the fastest serial algorithm for a given problem on one processor and T P, the time to execute the parallel algorithm for the same problem on P processors, that is, Speedup = T 1 =T P : The Speedup measures the overall improvement through parallelisation in solving a given problem. Eciency: This is dened as the Speedup divided by P, the total number of processors used in executing the parallel algorithm, that is, Eciency = Speedup=P: If the eciency is close to unity, it is clear that the processors are doing useful work most of the time and that communication and synchronisation overheads associated with the parallel implementation are relatively small. This means that the hardware is being used eectively. Image Data: Our experiments used three dierent image sources of varying sizes and characteristics. These

4 g(i,j) STANDARD ^ f(i,j) TABLE I The performance parameters of median-type filters in minimum CPU time and optimal image quality. g(i,j) Fig. 1. The median lter. RECURSIVE STANDARD ^ f(i,j) Fig. 2. Combined ltering by recursive and median lters in one iteration step. were \Girl" of size (smoothly varying photographic image), \Lenna" of size (many ne details and sharp edges) and \Christine" of size (large but relatively smooth). Gaussian noise was added to each image to give a signal-to-noise ratio (SNR) of about 10dB. Images of dierent sizes were used to study the way in which the speedup and eciency varied with problem size. Experimental Design: We have tested three types of iterative median lters with their parallel implementation. The performance of each of the lters and its combined forms with two of the other median lters have also been realised and given in Figures 1, 2 and 3. Their optimal combination regarding CPU time and image quality is given in Table I. From the table, it is clearly seen that parallel hybrid median lters greatly reduce the CPU times while still achieving the same improvement in signal-tonoise ratios. 5.1 Speedup and eciency of parallel lter implementations The overall results here were excellent, with speedups ranging from better than 15.5 with 16 processors up to 117 or so with 128 processors. The eciencies ranged from a low of 92% with 128 processors to nearly 100% with 16 processors. The measured eciencies were slightly better for the larger images. Parallel execution times ranged g(i,j) IMPROVED STANDARD ^ f(i,j) Fig. 3. Combined ltering by one iteration of improved median plus two iterations of median lter. Filter Iterations Total CPU S/N Gain sequential median sequential improved sequential recursive parallel improved+median parallel recursive+median from tenths of a second through to about ten seconds, compared with sequential times of from one to twenty minutes or so. The good results are in part a consequence of our data communication strategy which greatly reduces the total communication cost, so all the processors involved in the computation are active most of the time. 5.2 Image enhancement performance The experimental results conrmed previous observations [13, 14] that the conventional median lter requires more iterations to achieve a given gain in S/N ratio than either the recursive or improved median lters. Parallel implementations of recursive and improved median lters (based on the block expanded method) gave useful gains in overall S/N ratio with one or two iterations. Important requirements of image enhancement processes are that during noise reduction operations they should as far as possible preserve the sharpness and shape of detail in the original image and minimise visible artefacts such as \ringing." Repeated application of a conventional median lter to achieve the desired noise reduction tended to result in excessive blurring. Recursive median ltering gave good noise reduction without excessive blurring, but tended to distort high contrast image detail. A further problem was that in the parallel implementation of improved and recursive median lters, artefacts associated with block boundaries were distinctly visible with K = 1 (and less visible as K was increased). Hybrid lters based on one or more iterations of either an improved or recursive median lter, followed by two iterations of a conventional median lter, achieved useful S/N gains and were eective in reducing the visibility of block boundary artefacts. The measured overall gains in S/N ratio do not clearly indicate the reductions in artefact amplitude with such second stage ltering. To get a better measure of the improvements here, the gain in S/N ratio in narrow strips of the image centred on the block boundaries was calculated. This localised measurement was more sensitive to artefact amplitude and gave an objective assessment of the eect of increasing K and using

5 Distortion 10 (db) (a) Fig. 4. Signal-to-noise ratio versus K K second stage ltering. Figure 4 shows how the artefacts reduce with increasing K after parallel hybrid median ltering. Figure 5 gives an example of the ltered results using our lters on degraded 10dB Christine. 6. Conclusions The overall approach adopted here is to take an image processing procedure which has sequential aspects and to apply it in parallel to a number of sub-images, at the same time recognising that this may introduce artefacts in the vicinity of the sub-image boundaries. The magnitude of the artefacts can be reduced by analysing their causes and making suitable modications to the processing techniques. A nal ltering stage can then reduce any residual artefacts to insignicant levels. The parallel iterative median-type noise ltering algorithms presented in this paper are quite suitable for implementation on distributed memory MIMD and SIMD machines. The block expanded method allows all the required data to be transferred between adjacent PEs as single messages. This greatly reduces the overall communication overheads resulting from message latency. For a given number of processors we also show how to partition the original image so that the total amount of data to be transferred between PEs can be minimised without affecting the nal result, further reducing communication costs. High eciencies were achieved with the ANU's AP1000, indicating that the algorithms implemented are well matched to a mesh-connected processor array architecture. References [1] R. P. Brent, \Parallel algorithms for digital signal processing", Numerical Linear Algebra, Digital Signal Processing and Parallel Algorithms (edited by G. H. Golub and P. Van Dooren), Springer-Verlag, 1991, pp.93{110. [2] H. U. Dohler: \Generation of root signals of two dimensional median lters," Signal Processing, Vol.18, 1989, pp.269{276. [3] N. C. Gallagher and G. L. Wise: \A theoretical analysis of the properties of median lters," IEEE Trans. Acoust. Speech & Signal Process., Vol-29, 1981, pp.1136{1141. [4] K. S. Fu, VLSI for Pattern Recognition and Image Processing, Springer-Verlag, New York, (b) (c) (d) Fig. 5. The ltered results on degraded 10dB 1024X1024 Christine. (a) the degraded image; (b) after parallel median ltering; (c) after parallel hybrid improved median ltering (k = 1); (d) after parallel hybrid recursive median ltering (k = 1).

6 [5] L. Kronsjo and D. Shumsheruddin, Advances in Parallel Algorithms, Blackwell Scientic Publications, London, [6] T. S. Huang (Ed.): Two-Dimensional Digital Signal Processing II Spring-Verlag, Berlin Heidelberg, 1981 [7] A. Nakamura, M. Nivat, A. Saoudi, P. S. P. Wang and K. Inoue, Lecture Notes in Computer Science { Parallel Image Analysis, Springer-Verlag, 1992 [8] T. A. Nodes and N. C. Gallagher: \Median lters: Some modications and their properties," IEEE Trans. Acoust. Speech & Signal Process., Vol-30, 1982, pp.739{746. [9] I. Pitas and A. N. Venetsanopoulos: \Nonlinear mean lters in image processing", IEEE Trans. Acoust. Speech & Signal Process., Vol-34, 1986, pp.537{584 [10] A. Rosenfeld and A. C. Kak, Digital Picture Processing, Vols. 1 and 2, Academic Press [11] D. Rover, V. Tsai, Y. S. Chow and J. Gustafson: \Signalprocessing algorithms on parallel architectures: A performance update," Journal of Parallel and Distributed Computing, Vol- 13, 1991, pp.237{245. [12] M.Sarrafzadeh, A. K. Katsaggelos and S. P. R. Kumar, \Parallel architectures for iterative image restoration," in Parallel Algorithms and Architectures for DSP Applications, Academic Publishers, [13] Tang H., Zhou B.B., Macleod I.D., Brent R.P. and Sun W., \Comparisions of Parallel Iterative Noise Filters for Real-time Image Processing", to appear in the Proceedings of International Conference on Systems, Control, Information, Wuhan, China, October [14] Tang H., Zhou B.B., Macleod I.D., Brent R.P. and Sun W., \Parallel implementation of an adaptive dierential iterative RMS lter for ecient signal restoration", to appear in the Proceedings of IEEE Region 10 International Conference on Parallel Computation and Applications, Singapore, August [15] Tang H., \A human-machine interactive system for ecient image restoration", Proceedings of International Conference on Acoustics, Speech and Signal Processing 1994, Adelaide, Australia, April, Vol.5, 1994, pp [16] Tang H., \A combining approach for the enhancement of unknown blurred and noisy image", Proceedings of 1994 International Conference on Speech, Image and Neural Networks, Hong Kong, April, Vol.2, 1994, pp

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