The 2D wavelet transform on. a SIMD torus of scanline processors. R. Lang A. Spray H. Schroder. Application Specic Computer Design (ASCOD)

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1 The D wavelet transform on a SIMD torus of scanline processors R. Lang A. Spray H. Schroder Application Specic Computer Design (ASCOD) Dept. of Electrical & Computer Engineering University of Newcastle Callaghan, NSW Australia rlang/aspray/heiko@faceng.newcastle.edu.au Abstract In this paper, the implementation of the two-dimensional Wavelet Transform (WT) is considered on a torus of SIMD scanline processors. The most obvious method for implementing the WT is presented as well as a reformulation of the decomposition to make the implementation on such a machine faster. The newer method produces a dierent tiling of the frequency domain of the image, representing a new transform with potential for image coding. The implementation of these transforms is considered in detail in terms of VLSI complexity and future work investigating the benets of the modied transform is mentioned. 1 Introduction The two-dimensional Discrete Wavelet Transform (WT)[1, ] is a popular technique for image coding and compression[1, 3, 4, 5]. The WT transforms the image into the frequency domain where the image is compressed by eliminating various frequency coecients based on one or more of a number of possible algorithms. Finally, the result is coded, again with a choice of possible algorithms. Transform coding has been made popular by the Discrete Cosine Transform (DCT) and its use in the popular JPEG (Joint Photographic Experts Group) image compression format. Wavelet Transform coding has been considered for use in High Denition Television (HDTV) standards and has been shown to achieve high compression ratios with good image quality after decoding. We present a simple, non-mathematical description of the D-WT and show a standard implementation on the torus. We then look at reducing the number of transpose operations in this implementation to improve the speed and demonstrate that for a simple image coding scheme this method is as good as the original transform. The wavelet Transform.1 One Dimensional Since the D-WT is a separable transform constructed from the 1D-WT, it is natural to explain the 1D transform rst as a foundation for the D transform. The WT is a repeated ltering processing, using related QMF (Quadrature Mirror Filter) lters, one a \low-pass" lter, the other a \high-pass" lter. The output of each lter is decimated, that is, every second value is removed, halving the length of the output. The output of each lter stage is made up of \transform coecients" and each lter stage represents a \level" of the transform. The low-pass result is then transformed by the same process and this is repeated until either the output of a stage is of length 1 or the desired depth of processing has been achieved. This whole process yields the same number of data in the output as in the input. The 1D-WT process of ltering is illustrated in Figure 1.. Two Dimensional The separable D-WT is simply the application of the 1D-WT repeatedly to rst the horizontal data of the image, then the vertical data of the image. At the end of each stage, four regions are produced in the result, a \High-High" region, \High- Low" region, \Low-High" region, and a \Low-Low" region, which is then continually processed as for the 1D-WT. This produces a tiling of the frequency domain of the image as shown in Figure..3 Image Coding To achieve lossy image coding with the D-WT, we rst transform the image into the frequency domain using the WT and then eliminate some of the highfrequency coecients.

2 {x} LPF {a} 1 {a} {a} 3 {a} 4 LPF LPF LPF LPF {a} 5 {d} 5 {d} 4 {d} 3 {d} {d} 1 LPF Low-pass Filtering High-pass Filtering Decimation Figure 1: The 1D-WT ltering process. This elimination process can vary from a simple thresholding step to more complex entropy or edgebased decisions. As an example, the demonstrations shown at the end of this paper use a simple method which keeps the 5% most frequent coecients in the high-frequency results, zeroing the rest and leaving the lower-frequency bands alone. Once coecients are eliminated (through whatever means), the result must be coded. Since a lot of zeros are introduced it is wise to select a coding scheme that exploits this fact. The examples at the end of this paper use a scheme similar to JPEG in that each non-zero datum is coded using three pieces of information: 1. a count of the zero values preceding the datum. the number of bits needed to code the datum 3. the actual datum This technique also allows a variable number of bits to be used in the coding of each datum in order to reduce rounding errors. 3 Torus of SIMD Scanline Processors The torus of SIMD (Single Instruction stream, Multiple Data stream) Scanline Processors is a p n p n torus of n processors which operates on an n n image[6, 7]. An example layout of a 4 4 torus is shown in Figure 3, this machine is designed to process images. We have chosen this architecture because the scanline processing approach is popular [6] due to the ability to exploit the spatial locality of information (that is, processing of information in nearby scanlines). The Torus topology is capable of fast Figure 3: A 4 4 torus transpose operations [8], a fundamental part of the D-WT algorithm. Each processor in the torus is responsible for a scanline in the image. Hence scanline-based algorithms are easily implemented on the SIMD machine since each processor can process a scanline independently of the other processors. However, column operations are usually much slower on such a machine due to the amount of communication involved, hence it is best rst to transpose the image, then perform the operation on the resulting scanlines and nally transpose the image again. A toroidal machine can transpose an nn image in O(n p n) time since the network conguration reduces the communication distance between any two rows from O(n) to O( p n)[8]. 4 Traditional Implementation Since the D-WT involves operations rst on rows and then on columns of the image, the most obvious way to implement this on the torus would be to put a transpose between each half-stage of the algorithm. We call this the traditional implementation because it involves simply implementing the algorithm exactly as it is described, repeated ltering followed by transposes to get to the column data.

3 Lh Hh Lh Hh HhHv HhHv HhHv H - High Pass Result, L - Low Pass Result h - horizontal filtering, v - vertical filtering Figure : The D-WT frequency domain tiling. Hence the image is rst horizontally ltered, then transposed and horizontally ltered again. To properly orientate the image after this stage another transpose operation is needed as the nal operation. 5 \Two-pass" Implementation We can perform a similar transform faster by reducing the number of transposes required. This produces a slightly dierent tiling of the frequency image, with some regions transformed to further depth in a particular orientation (horizontal/vertical). This method simply transforms the image completely horizontally before considering the vertical data, which is then transformed completely. Hence on this machine there is only one transpose operation between these two complete transform steps and one transpose at the end to re-orientate the result. We call this method a \two-pass" method because of the count of Transpose operations, hence the traditional method could be called the \log(n)- pass" method. This is shown in Figure 4. The nal result of this can be compared to the nal result of the traditional WT shown in Figure. Since the lters being used in the operations are linear lters, the diagonals of the two tilings are the same, and in fact the non-diagonal regions of the new transform are similar to regions in the traditional transform but have been transformed Step 1. Step. Figure 4: The two steps of the faster transform, the second step yields the result. to greater depth in a particular direction (either horizontally or vertically). Since some regions of this new result have been transformed further, we hope to develop a better coding scheme designed to take advantage of the further transformation to achieve better compression and/or quality. At this stage comparisons have been made only for the same coding schemes. These results are presented at the end of the paper. 6 Analysis We present here an analysis of the algorithms presented. This analysis is done in terms of counting the number of Communication (COM) steps and the number of Multiply-Accumulate (MAC) steps for the algorithm, since these operations are the most dominant in complexity issues for VLSI implementation. Due to the simple connectivity of the torus, each COM operation will in general be much simpler (and hence faster) than any MAC operation. Hence

4 the MAC operations will dominate the complexity of either transform on this machine D-WT The one-dimensional Wavelet Transform involves ltering a signal with two (related) lters. The ltering operation involves Multiply and Accumulate (MAC) operations between the coecients of the lter and the current values windowed by the lter. Hence for a lter width of k and an input signal of length n, one step of the WT involves k n MAC operations for each of the lters, so for both the low and high-pass lters the total is kn MAC operations. The following levels of the transform operate on data of length half that of the previous level, hence the total work of these levels is: = k( n ) + k( n 4 ) + k( n 8 ) + : : : + 1 This simple sum yields k(n? 1), hence the total 1D-WT involves k(n? 1) MAC operations including the rst level kn MAC operations. When implementing the D-WT, 1D-WT operations are performed on scanlines by each processor. Since the processors each control a scanline and can access that scanline as a memory block, no Communication (COM) steps are required to implement this part of the algorithm. 6. Transpose The transpose operation involves no MAC operations, only COM operations. It has an asymptotic computational complexity of O(n p n), but we are interested in actuals steps required. The algorithm for the Transpose on a SIMD torus of processors is shown in Figure 5. This is described clearly in [8], but we are interested only in the communication steps and a count of them, ie. those lines indicated in the gure. The key to understanding the transpose operation is in understanding how a value is routed from its current scanline to its new scanline, taking advantage of the p n scanline jumps made possible by a write to the east or west in the torus. We can work out the total number of communications by separately working out the count for each of the four communication lines in the algorithm and adding the result. comm1 is executed 1++3+: : :+ p n = p n( p n+1) times. comm is executed p n( : : :+ p n) = p n p n( p n+1) times. comm3 is executed the same number of times as comm. inaddr = scanline number of pe ; inaddr increments after each read outaddr = inaddr+64 ; outaddr decrements after each write repeat k = 0 to n-1 read pixel repeat j = 0 to k communicate pixel west ; comm1 end j write pixel repeat d = 1 to n-1 read pixel repeat j = 1 to d communicate pixel north ; comm end j repeat j = 0 to k communicate pixel west ; comm3 end j disable all PEs with row index > d communicate pixel west ; comm4 enable all PEs write pixel end d end k Figure 5: The Transpose Algorithm. comm4 is executed p n p n times. Simplifying all these results and adding them gives the total count for the transpose operation as 5n+(n+1) p n COM operations. 6.3 Traditional WT Transform The traditional method involves transforming at each level horizontally and vertically. Each level is only one level of the 1D-WT, however, so at the rst level, this is equal to kn MAC operations (for the ltering operations) and 5n+(n+1)p n COM operations (for the transposes). As in the 1D-WT, the further levels of the D- WT are performed by transposing the lowest lter result, half the work of the previous level. Hence the total number of MAC operations is 4kn+4k(n? 1) = 4k(n? 1) and we can approximate the total count of COM operations p p to 5n+(n+1) n+5(n? 1)+((n?1)+1) n? p 1 = 10n?5+(n+1) n+ (n? p 1) n? 1 COM operations. This is only an approximation because the mathematics involved in the summing of subsequent levels requires a bit more work for this case due to the square root component. In fact, this approximation falls short of the actual count for a given level, biasing the comparison in favour of the traditional implementation. The results graphed in Figure 6 show actual COM count comparisons for the two methods. We see

5 10000 Comparison of COM operations for traditional and new implementations trad new COM operations Image width Figure 6: Comparison of COM operations for traditional and faster implementations that the traditional implementation is increasing at a much higher rate with increasing image size than the faster implementation. This method doesn't use all the processors all of the time. The number of processors used in a particular level is half that used in the previous level. This totals up to approximately 50% utilisation, hence it is desirable to look into pipelining more than one image into the array for processing. This can be done with more complex control by processing the rst level of two images initially and then the remaining levels of each of the images in parallel on the array. This would require some complex algorithmic work, particularly in scheduling the two simultaneous transpose operations. The very nature of the WT algorithm with its alternating row and column operations makes it an undesirable algorithm to implement on such a torus of Scanline Processors, hence we intend investigating custom architectures more suited to the data- ow of the WT. 6.4 \Two-pass" Transform The newer method requires fewer transpose operations. The ltering is rst done completely in the horizontal direction, giving a total of k(n? 1) MAC operations and no COM operations. This intermediate result is transposed and the ltering is performed again (now on the vertical data), for another k(n?1) MAC operations and 5n+(n+1)p n COM operations (for the transpose). Finally the image is transposed again. The total for this method is 4k(n? 1) MAC operations and 5n + (n + 1) p n COM operations, hence we see that this method has at least halved the COM operations. All of the processors are active all of the time in this method, ie. 100% utilisation. The improvement over the traditional WT is gained by reducing the amount of row and column switching required and hence reducing the number of transpose operations. While the speed benet is benecial, it remains to be seen if the extra processing is useful or not to decide whether the increased utilisation of the processors is of any advantage or disadvantage. 6.5 Reducing the MAC Operations Both algorithms here suer the same number of MAC operations, hence it is desirable to increase the speed of these operations. Since the complexity is already O(n), no asymptotic improvement is available hence we must look at constant factors, aiming to reduce the number of MACs 8kn to (optimally) n. To improve this complexity, we could replace each processor in the torus with a systolic array as we have previously designed [9, 10], where each array has k simpler processors and improves the speed of the result by k. Alternatively each processor could include two MAC logic circuits to double the speed of the processing by performing the low and high pass ltering simultaneously. 7 Compression Comparisons Table 1 shows a comparison of compression rates and quality for the two methods. Both of these methods use a simple, crude quantization and coding scheme that yields around 60% compression,

6 Traditional Faster E rms Comp. E rms Comp. Lenna 15:97 41:79% 14:75 41:79% F16 13:07 41:8% 13:6 4:01% eye 9:67 4:9% 9:68 4:87% airwolf 18:11 40:37% 18:10 40:43% simpsons 0:85 41:93% 0:55 4:15% cat1 19:5 41:4% 19:30 41:5% Table 1: Simple compression comparisons. which is far from the results of other Wavelet techniques or JPEG compression. The purpose of this table, however, is not to demonstrate impressive compression, but to show that the new, faster method is comparable in use to the older method. In that respect, it is expected that the new method will perform similarly to the older method under better schemes with obvious speed benets. It is our intention to try to develop coding schemes suited to the slightly dierent decomposition achieved by the new method. Although the results of each transform are hard to distinguish between visually, the E rms (rootmean-square error per pixel over the whole image) value of the new method tends to be slightly lower than that of the traditional method. An example image and reconstruction results from the two methods are shown in Figures 7, 9, and 8. Although the faster method exhibits a slightly worse rms error for this image, both transforms are visually comparable. 8 Conclusions and Future work The new transform presented is slightly faster than the traditional WT transform at about half the communication complexity and at the expense of higher processor utilisation. The extra utilisation is either a benet or a penalty dependent on future work which will look at whether or not the idle processors in the traditional method can be utilised for other operations and/or whether or not the further transformation provided by the new transform is of any use for better compression and/or reconstruction quality. The new transform has been shown to be comparable to the old transform in terms of simple objective quality measures and compression ratios. Future work involves trying the new transform with better elimination and coding methods and comparing with the traditional method, as well as developing schemes to take advantage of the further depth of decomposition in some areas of the image resulting from the new transform. Our aim is to design WT architectures achieving optimal speedup for real-time applications, in the Figure 7: Original greyscale image of F16. Figure 8: Reconstruction after compression with the traditional method. Figure 9: Reconstruction after compression with the faster method.

7 sense that the implementation on O(n) processors introduces an O(n) speedup. References [1] Stephane G. Mallat. A theory for multiresolution signal decomposition: The wavelet representation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 11(7):674{ 693, Jul [] Olivier Rioul and Martin Vetterli. Wavelets and signal processing. IEEE Signal Processing Magazine, October [3] A.S. Lewis and G. Knowles. Image compression using the -d wavelet transform. IEEE Transactions on Image Processing, 1():44{ 50, Apr 199. [4] Jerome M. Shapiro. Embedded image coding using zerotrees of wavelet coecients. IEEE Transactions on Signal Processing, 41(1):3445{346, Dec [5] Marc Antonini, Pierre Mathieu, and Ingrid Daubechies. Image coding using wavelet transform. IEEE Transactions on Image Processing, 1():05{0, Apr 199. [6] Allan L. Fisher and Peter T. Highnam. The SLAP image computer. In Parallel Architectures and Algorithms for Image Understanding, pages 307{337. Academic Press, [7] A. Spray, H. Schroder, K.T. Lie, E.Plesner, and P.D.Bray. A low-cost real-time visualisation tool. In Proceedings of Supercomputing 93, pages 398{407, Portland, Nov [8] Meiyun Zheng. Algorithms for parallel scan{ line image processing. Master's thesis, Department of Electrical and Computer Engineering, University of Newcastle, 199. [9] R. Lang, E.Plesner, H.Schroder, and A.Spray. An ecient systolic architecture for the onedimensional wavelet transform. In Proceedings of SPIE conference on Wavelet Applications, pages 95{935, Orlando, Apr [10] R. Lang, A. Spray, and H.Schroder. 1- d wavelet transform architecture. Technical Report EE9457, Department of Electrical and Computer Engineering, The University of Newcastle, Australia, Nov 1994.

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