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Giới thiệu: Phương pháp nén ảnh Fractal (FIC: Fractal Image Compression) cung cấp 1 giải pháp nén ảnh mầu với tỉ lệ nén rất cao ; như vậy nó yêu cầu số lượng lớn tính toán, nghĩa là thời gian nén của FIC rất lâu. Trong bài báo này chúng ta phát triển 1 hệ số FIC để áp dụng đối với các ảnh mầu, nó được khởi tạo 1 mã Fractal biến đổi mầu trên RGB thành YUV, thích hợp mẫu hình thức và phương pháp đặc biệt để giảm thơi gian nén ảnh. Chúng ta cũng đề xuất 1 kỹ thuật song song trong trình tự giai đoạn tốc độ này. Trong đố vẫn vẫn lưu trữ được đặc tính thuân lợicủa đầu vào. Kết quả thực nghiệm phân tích bởi thuật toán Fisher đối với ảnh mầu và không có hỗ trợ xử lý song song đã được xác định khả năng thiết kết mã Fractal nhanh đối với các ảnh mầu. 1. GIỚI THIỆU Nén ảnh Fractal (FIC) là dựa vào phân khu hệ thống hàm lặp ( PIFS) nó được dùng chính đặc trưng tương tự ( giống nhau ) của ảnh để nén ảnh 1. Trong nén ảnh Fractal, 1 ảnh dược phân khu thành 1 tập hợp các khối không trùng ( đè) nhau được gọi là khoảng. 1 tập hợp khác của các khối lớn gọi là các dải được sử dụng để định nghĩa vùng tốt nhất trong ảnh đối với mỗi dải. nó giống nhất đối vởi dải. Tất cả, mã Fractal luôn luôn áp dụng với các ảnh xám. Phần lớn là hướng tới phương pháp để nén 1 ảnh mầu RGB bưởi ảnh Fractal xám, thuật toán mã hóa được chia nnhỏ vào trong 3 kênh đỏ , xanh da trời , xanh lá cây, và chúng được nén riêng biệt bởi việc nghiên cứu mỗi thành phần mầu như 1 ảnh xám đơn, bởi vậy được gọi là 3 thành phần riêng mã Fractal ( SFC)9. Các khác là dữ liệu mầu của 3 kênh thành phần , red, green, blu được biến đổi sang thành phần YUV, để thuận lợi cho quang phổ hiện nay có lợi hơn trong việc nén. Để đạt được kết quả thương mại đối với ảnh lena với kích thước 512 x 512 là PSNR đạt 31,4dB với tỉ lệ nén 25,55 và thời gian mã là 3.608 giây Trong công việc trước của chúng ta, chúng ta thực hiện nén ảnh mầu Fractal (FCIC) trên PC Linux với mã C++ và áp dụng chấm động trong xử lý FIC. Việc biến đổi thành phần RGB thành YUV có thể mang đến lợi thế của dấu chấm động đối với dải ảnh động cao và thiết lập RMS và lặp của FIC để tăng thêm tỉ lệ nén cho mức cao trong khi vẫn giữ lại 1 chấp nhận PSNR11. Trong bài báo này, mục đích của chúng ta là tối ưu hóa thời gian mã bằng cách biến đổi thành phần RGB thành YUV, sử dụng phương pháp hiệu quả để xử lý những thành phần này và áp dụng xử lý song song đối với thuật toán FIC. Kết quả thực nghiệm đã được chỉ ra rằng thuật toán FIC có thể thực hiện đối với các ảnh mầu và cũng áp dụng được đối với nén video Low bit rate 2. LÝ THUYẾT NÉN ẢNH MẦU 2.1. thuật toán nén ảnh Fractal Trong nén ảnh Fractal, 1 ảnh được phân khu vào trong 1 tập hợp R của n khối dải hình vuông không trùng nhau. Tập hợp khác là D kích thước 2n x 2n các miền khối vuông lớn được mẫu con bởi pixel trung bình để có được kích thước tương tự như các dải. Đối với mỗi Ri ϵ R, phương pháp nén này sẽ tìm kiếm thông qua tất cả các khối của D để tìm 1 Di ϵ D phần lớn các khối giống như dải Ri và có thể được ánh xạ bởi 1 hệ số biến đổi affine ở đây. Nó cũng tìm độ tương phản và độ sáng tốt nhất thiết lập với si và oi đối với việc biến đổi wi ánh xạ từ Di sang Ri ở đó : s được xác định thay đổi độ tương phản và o được các định thay đổi độ sáng của sự biến đổi, z là mức xám của 1 pixel tại vị trí (x,y) 1. Bởi vậy đối với mỗi Di ϵ D, oi và si sử dụng quy hồi bình phương nhỏ nhất được tính toán và Di có ít nhất rms khác biệt là đỉnh 1. 1 tập hợp tất cả wi được gọi là W là biến đổi ảnh mã hóa. ảnh f đó thỏa mãn f = W(f) được điền vào điểm W. Nếu W được co lại, f là duy nhất và là 1 ảnh gốc sấp xỉ. bởi vậy thủ tục giải mã là dựa trên đặc tính này. Nếu số lượng R rất lớn, thì số so sánh cũng rất lớn. kể từ đây hình thức phân khu ảnh rất quan trọng trong nén ảnh Fractal để giảm khối lượng ( giảm khối lượng thời gian) nhưng vẫn giữ được chất lượng giải mã ảnh. Trong bài báo này phân khu HV được sử dụng. Phân khu HV là mềm dẻo bởi vì nó có thể cho phép cả 2 dải vuông và hình chữ nhật. một sự thuận lợi của phân khu HV cũng như là dải được giảm đi bởi sự phân tích không rõ ràng như trong phân khu vuông. Sau đó tất cả các khoản được bảo vệ, chúng ta không lưu trữ tất cả các hệ số trong (1). Hệ số tương phản si và độ sáng oi được lượng tử hóa và lưu trữ trong 1 số bít cố định 1. Giải mã ảnh được tạo dựng bởi lặp W từ 1 ảnh khởi tạo. đối với mỗi (Ri, Di) không được đóng gói từ file nén, miền Di là mẫu con bởi trung bình mỗi khối vuông con 2 x 2 không trùng nhau. Sau đối mỗi giá trị pixel trong miền mẫu con được nhân lên bởi si , được thêm vào oi và địa điểm trong vị trí trong dải tương ứng Ri dược xác định bởi sự định hướng. tiến trình này được lặp lại cho đến khi việc giải mã ảnh được điền ( hoàn tất)
Journal of Electronic Imaging 8(1), 98 – 103 (January 1999) Lean domain pools for fractal image compression Dietmar Saupe Universitaăt Leipzig Institut fuăr Informatik Augustusplatz 10/11 04109 Leipzig, Germany E-mail: saupe@acm.org Abstract In fractal image compression an image is partitioned into ranges for each of which a similar subimage, called domain, is selected from a pool of subimages However, only a fraction of this large pool is actually used in the fractal code This subset can be characterized in two related ways: (1) It contains domains with relatively large intensity variation (2) The collection of used domains is localized in those image regions that show a high degree of structure Both observations lead to improvements of fractal image compression First, we accelerate the encoding process by a priori discarding the low variance domains from the pool that are unlikely to be chosen for the fractal code Second, the localization of the domains may be exploited for an improved encoding of the domain indices, in effect raising the compression ratio When considering the performance of a variable rate fractal quadtree encoder we found that a speedup by a factor of – does not degrade the ratedistortion curve ranging from compression ratio up to 30 © 1999 SPIE and IS&T [S1017-9909(99)00301-3] Introduction Fractal image compression1–3 is capable of yielding competitive rate-distortion curves; however, it suffers from long encoding times Therefore, large efforts have been undertaken to speed up the encoding process Most of the proposed techniques attempt to accelerate the searching and are based on some kind of feature vector assigned to ranges and domains The features can be discrete ͑leading to classification and clustering methods͒ or continuous ͑yielding functional or nearest neighbor methods͒ For a survey see our papers see Refs and When applied, these methods provide greater speed which is traded in for a loss in image fidelity and compression ratio A different route to increased speed can be chosen by less searching as opposed to faster searching This means that if we can devise a technique to estimate a priori whether a given domain will be used for the fractal code or not, then we can exclude all unlikely domains In this way greater speed is achieved by restricting the search to a reduced domain pool Of course, the search in the reduced domain pool may then be supported by a method of the other kind, e.g., by classification Several possible ap- Paper 96018 received April 16, 1996; accepted for publication September 15, 1998 1017-9909/99/$10.00 © 1999 SPIE and IS&T proaches along these lines have been reported in the literature One may argue that those domains that are close to a given range in the image ͑e.g., domains that overlap the range͒ are especially well suited as a partner for the given range, thereby localizing the domain pool relative to the range ͑see, e.g., the work of Monro and Dudbridge6,7 and Barthel et al.8͒ This is an adaptive domain pool reduction; for each range a different domain pool is constructed Another adaptive variant has been suggested in a functional method by Bedford, Dekking, and Keane.9,4 Depending on the range the domain pool is shrunk by excluding a number of domains that not satisfy a condition which involves certain inner products which are independent of the range and can be calculated for all domains in a preprocessing step These excluded domains are guaranteed not to be optimal for the given range; thus, no image or compression degradation can occur with this method Complementing these methods one can work with a fixed domain pool, which is initially scanned once in order to discard domains that are unlikely to be of any use In principle, this last mentioned approach has already been implemented in the early work of Jacquin.3 He used a classification scheme coming from a study of Ramamurthi and Gersho10 which classifies domain blocks according to their perceptual geometric features Three major types of blocks are differentiated: shade blocks, edge blocks, and midrange blocks In shade blocks the image intensity varies only very little Since ranges that would be classified as shade blocks can be approximated well by scaled constant fixed blocks, it is not necessary to search for corresponding domains Thus, in this scheme all domains classified as shade blocks are never used and effectively are excluded from the domain pool However, in Jacquin’s approach the class of shade blocks cannot be very large For example, only 11% of all blocks for the 256ϫ256, bit/pixel image Lenna have been classified as shade blocks in Jacquin’s 98 / Journal of Electronic Imaging / January 1999 / Vol 8(1) Downloaded From: https://www.spiedigitallibrary.org/journals/Journal-of-Electronic-Imaging on 11/17/2017 Terms of Use: https://www.spiedigitallibrary.org/terms-of-use Domain pools for image compression work.11 The reason for this is that otherwise too many range blocks would also be classified as shade blocks and thus be coded as constant fixed blocks yielding poor approximations Therefore, no variations of shade block definitions have been investigated in these studies In this article we consider a parametrized and nonadaptive version of domain pool reduction Here we allow an adjustable number of domains to be excluded ͑ranging from 0% to almost 100%͒ and investigate the effects on computation time, image fidelity and compression ratio We will see that there is no need for keeping domains with low intensity variance in the pool Thus, we propose to eliminate a fraction 1Ϫ ␣ , ␣ (0,1͔ , of the domain pool consisting of the domains with least variance In this way we remove the mostly useless domains from the pool, achieving a lean and more productive domain pool Using the adaptive quadtree method of Fisher ͑see Ref 2, Appendix A͒ we will show the following: Fig Histogram of variances in the domain pool of domain blocks of size 8ϫ8 vs that for domains actually used in an adaptive quadtree fractal code of Lenna The computation time scales linearly with ␣ Even for low values of ␣, e.g., ␣ ϭ0.15, there is no degradation in image quality On the contrary, the fidelity improves slightly For medium values of ␣, e.g., ␣ ϭ0.50, the compression ratio suffers a little, decreasing by about 2% The fractal code for an image essentially consists of the partitioning of the image into ranges and the data for one affine transformation per range These data are given by an offset o ͑typically bits͒, a scaling factor s ͑5 bits͒, a domain D k from the domain pool ͑ log2 ND bits where N D is the size of the domain pool͒ and the code for an isometry ͑3 bits͒ The intensity values in the coded range are then taken from the scaled, transformed copy of the domain plus the added offset The domains from the pool are indexed and referenced by that index Let us discuss the simple example of a gray scale image of resolution 512ϫ512 with a domain pool of nonoverlapping domains of size ϫ8 Thus, there are 642 ϭ4096ϭ2 12 domains in this pool altogether and the storage of one domain index costs 12 bits It turns out that only a certain fraction, e.g., say 1000, of these domains are used We propose to make use of this observation in the following way We use a standard ‘‘white block skipping’’ ͑WBS͒ quadtree storage scheme to identify the 1000 domains used out of the total of 4096 With the quadtree on hand we can now code indices of domains in the range from to 1000, costing only 10 bits each Thus, we save bits per transformation If M is the total number of ranges we have an overall file size reduction if the code for the quadtree does not exceed 2M bits This approach will become beneficial in combination with lean domain pools since the collection of domains used will have even more structure yielding a smaller code for the domain quadtree Parallel to this work researchers are currently investigating other methods for domain pool reduction for fractal image compression Kominek12,13 and Signes14 propose to remove domain blocks from the domain pool if they can be covered well by other blocks still in the pool Also high variance domain blocks are generally favored over low variance blocks; however, no analysis of the time/ performance trade-off is attempted for this approach The remainder of this article is organized as follows In Sec we present the details and results of the domain pool reduction by eliminating domains with low intensity variation In Sec the optimized domain storage scheme is presented with results Finally, in Sec we analyze the performance of the fractal encoder with lean domain pools and optimized storage scheme over a range of compression ratios Acceleration by Lean Domain Pools In a first experiment we check our hypothesis that there is no need for keeping domains with low intensity variance in the pool We carry out a fractal encoding of a test image using the adaptive quadtree method and record a histogram of intensity variances of blocks of size 8ϫ8 from the domain pool and also the corresponding histogram for the variances of those domains actually used in the code ͑see Fig 1͒ The result is very clear There is a very large subset of domains in the pool with small variances while there is no such trend in the histogram for the blocks used Thus, we may indeed expect that discarding a large fraction of low variance blocks will effect only a few ranges For these ranges a suboptimal domain with a larger variance may be found If, however, there is no longer a domain available in the pool which admits a collage error within the prescribed tolerance, then the range needs to be subdivided into four smaller ranges In the main study of this paper we scan each domain pool ͑i.e., the pools for block sizes 8ϫ8, 16ϫ16, 32 ϫ32, and 64ϫ64͒ and keep only a fraction ␣, ␣ (0,1͔ , of them in the pool, namely those domains that have the largest variances For differing choices of the parameter ␣ we compute the fractal code and record the computation time used, the peak-signal-to-noise ratio ͑PSNR͒, and the comJournal of Electronic Imaging / January 1999 / Vol 8(1) / 99 Downloaded From: https://www.spiedigitallibrary.org/journals/Journal-of-Electronic-Imaging on 11/17/2017 Terms of Use: https://www.spiedigitallibrary.org/terms-of-use Saupe Table Performance of the adaptive quadtree method with lean domain pools The parameter ␣ indicates the fraction of domains which are kept in the pool The time is measured on an Indy R4600SC of Silicon Graphics in seconds The compression ratio is in the fourth column When applying the optimized coding procedure for the domains, we obtain the ratios of the fifth column with the difference in file size measured in bytes indicated in the last column See also Fig Results for 512ϫ512 Lenna Compression ␣ CPU Time s PSNR dB Comp ratio New ratio Bytes saved 1.00 15.2 32.73 14.88 14.84 Ϫ39 0.90 14.0 32.71 14.86 14.84 Ϫ34 0.80 12.6 32.75 14.85 14.83 Ϫ21 0.70 11.3 32.76 14.83 14.82 0.60 0.50 10.1 8.7 32.80 32.87 14.75 14.57 14.77 14.62 Ϫ7 24 60 0.40 0.30 0.20 0.15 0.10 7.4 6.0 4.6 3.9 3.1 32.90 32.93 32.88 32.78 32.53 14.45 14.19 13.49 13.10 12.64 14.56 14.88 14.23 13.89 13.98 137 856 1009 1135 1982 0.08 0.06 0.04 0.02 2.8 2.4 2.1 1.7 32.40 32.03 31.80 31.03 12.36 12.21 11.86 11.39 13.69 14.13 13.79 13.86 2070 2921 3101 4103 pression ratio ͑see the four left columns in Tables and 2, and Fig 2͒ The results are as follows: Time Regarding the computation times there seems to be an overhead of about 1–2 s The remaining time scales linearly with the parameter ␣ This is as expected since the major computational effort in the encoding lies in the linear search through the domain pool Fidelity The quality of the encoding in terms of fidelity measured by the PSNR increases by 0.1–0.2 dB when lowering ␣ ͑except for the Baboon image͒ This is caused by the fact that some larger ranges can be covered well for ␣ ϭ1.0 by some domains which are removed from the pool at smaller values of ␣ As a consequence, some of these larger ranges are subdivided and their quadrants can be covered better by smaller domains than the large range previously This mechanism works for values of ␣ down to about 0.15 Compression The range splitting mentioned above also increases the number of ranges, thus causing the compression rate to decrease slightly For example, this drop is 1%–2% at ␣ ϭ0.5 and 2%–9% at ␣ ϭ0.2 It is remarkable that only relatively little loss in overall quality of the encoding is encountered for speed up factors of 10 and higher Improved Bitrate by Exploiting Spatial Domain Entropy Figure shows the domains of size 8ϫ8 that are used in the fractal code of Lenna ͑from the first row in Table 1͒ As Table Results as in Table for a few more test images Compression ␣ CPU Time s PSNR dB 1.00 16.6 Results for 512ϫ512 peppers 32.43 15.20 15.06 0.50 0.20 10.0 5.1 32.49 32.55 14.93 14.00 14.95 14.73 Ϫ161 17 921 0.10 0.05 3.3 2.4 32.35 32.11 13.22 12.31 14.56 14.22 1828 2859 1.00 33.2 25.15 5.68 5.59 0.50 17.3 25.13 5.61 5.75 Ϫ727 1148 0.20 0.10 8.0 4.7 24.81 24.37 5.55 5.52 5.93 6.16 3028 4915 0.05 3.3 23.87 5.50 6.42 6766 1.00 0.50 0.20 0.10 25.6 14.6 7.3 4.4 boats 10.18 10.19 10.51 10.85 185 335 1476 2531 0.05 2.8 11.29 3562 1.00 0.50 0.20 0.10 0.05 21.3 12.6 6.2 3.8 2.4 F16 12.60 12.55 12.75 13.23 13.65 75 211 1186 2079 2967 Comp ratio New ratio Bytes saved Results for 512ϫ512 baboon Results for 512ϫ512 32.03 10.11 32.07 10.06 31.89 9.92 31.53 9.82 30.95 9.79 Results for 512ϫ512 32.86 12.55 32.97 12.42 32.94 12.05 32.63 11.98 32.13 11.82 expected the indicated domains are located mostly along edges and in regions of high contrast of the image These black squares can be interpreted as a bitmap ͑of resolution 64ϫ64 in this case͒ and the goal of the procedure outlined in the introduction is to store this bitmap efficiently Then the number of bits required to identify a particular used domain is reduced If the structure of the bitmap is strong, then these savings are greater than the overhead necessary for coding the bitmap and an overall reduced file size for the code can be achieved We use the WBS quadtree storage scheme described in Gonzales and Woods ͑see Ref 15, p 354͒ For a bitmap of size k ϫ2 k we proceed recursively starting with the block given by the entire bitmap A solid white block is coded as 0; all other blocks are coded with a prefix and followed by the four codes of their four subquadrants, which are generated in the same way until a subblock of size is reached which is coded as ͑white͒ or ͑black͒ For an example, see Fig The last two columns in Table report the results of this procedure for the Lenna test image For the case without domain pool decimation ( ␣ ϭ1.00), there are no savings The costs for storing the WBS quadtree outweigh the savings from shorter domain codes However, as we decrease the value of ␣ below 0.7, we are obtaining some gain in 100 / Journal of Electronic Imaging / January 1999 / Vol 8(1) Downloaded From: https://www.spiedigitallibrary.org/journals/Journal-of-Electronic-Imaging on 11/17/2017 Terms of Use: https://www.spiedigitallibrary.org/terms-of-use Domain pools for image compression Fig Domains of size 8ϫ8 that are used for a fractal code of 512ϫ512 Lenna are shown in black We have carried out the same experiment with domain pools enlarged by factors of and 16 In these cases the break-even point of ␣ ϭ0.70 is reduced to 0.25 and 0.15, respectively Thus, the proposed method for domain address encoding seems to be applicable in situations where either the domain pool is not very large or when extremely fast encodings are desired ͑e.g., by choosing ␣ р0.05͒ and quality can be compromised In a production implementation one could carry out the WBS quadtree coding, check whether it yields any savings or not, and then use the better storage scheme The WBS storage scheme for bitmaps is simple and effective However, there are more efficient methods, for example, using adaptive context based arithmetic coding Thus, there is room for further improvement Also we note that with strongly decimated domain pools, domains are used more often than once in the fractal code Thus, standard entropy coding of the domain indices will yield additional savings in storage Domains used in Fig Performance of the adaptive quadtree method with lean domain pools (from Table 1) The three plots show the computation times, the PSNR, and the compression ratio for varying parameters ␣ The horizontal lines indicate the performance without enhancement, i.e., for ␣ ϭ1.0 The vertical line is for ␣ ϭ0.3, for which the performance is improved in PSNR and CPU time and equal in bit rate compression An especially notable result is obtained for ␣ ϭ0.30 ͑see the vertical lines in Fig 2͒ The new enhanced storage scheme completely makes up for the loss of compression which occurred due to the domain pool decimation Thus, in effect, when compared with the original method ͑no domain pool decimation, no enhanced domain storage, line in Table 1͒, we arrive at a fractal encoding with exactly the same compression ratio of 14.88, an improved PSNR ͑by 0.2 dB͒ and a computation time reduced from 15.2 s down to only 6.0 s! Fig Example of the white block skip coding of a bitmap of size 8ϫ8 The code is ͓ ͑ ͒ ͑ 1001͒ ͑ 0010͒ ͑ 0101͔͒ ͓0͔ ͓ ͑ 0011͒ ͑ ͒ ͑ 0011͒ ͑ ͔͒ ͓0͔ obtained by recursively processing subblocks counterclockwise each time starting from the corresponding upper right quadrant For better readability we have written prefix codes in bold face, and bracketed the codes for the main and sub quadrants Journal of Electronic Imaging / January 1999 / Vol 8(1) / 101 Downloaded From: https://www.spiedigitallibrary.org/journals/Journal-of-Electronic-Imaging on 11/17/2017 Terms of Use: https://www.spiedigitallibrary.org/terms-of-use Saupe the pool carry different costs with respect to ͑w.r.t.͒ the bitrate of the code Frequently used domains reduce the entropy, thus, are cheaper to encode than domains used only once Moreover, domains corresponding to outliers in the bitmap require extra bits in the white block skip code and are therefore more expensive than domains corresponding to bits in clusters One can design a postprocessing for a fractal code with the goal of eliminating such outliers: It is possible that ranges covered by domains that correspond to outliers in the bitmap may also be covered well by other domains that are cheaper to encode We have not implemented these options Variable Rate Encodings with Lean Domain Pools While the experiments in the previous sections considered only a particular compression ratio for each image tested, we discuss in this section the performance of the fractal quadtree encoder over a range of compression ratios with and without lean domain pools and optimized storage scheme for the domain addresses An adaptive partitioning of an image may hold strong advantages over encoding range blocks of fixed size There may be homogeneous image regions in which a sufficient collage can be attained using large blocks, while in high contrast regions smaller block sizes may be required to arrive at the desired quality The first approach ͑already taken by Jacquin͒ was to consider square blocks of varying sizes, e.g., being 4, 8, and 16 pixels wide This idea leads to the concept of using a quadtree partition, first explored in the context of fractal coding in Refs and 16 In contrast to fixed block size encodings, the output file must also contain the specification of the quadtree underlying the encoding Adaptive partitionings naturally lend themselves to the design of a variable rate encoder The user may specify targets either for the image quality or the compression ratio The encoder recursively breaks up the image into suitable portions until the target quality or rate is reached In more detail the algorithm targeting fidelity might proceed as follows Define a minimal range size and a tolerance for the root-mean-square ͑rms͒ error of the collage obtained by the encoder Initialize a stack of ranges by pushing the entire image as a range onto it While the stack is nonempty, carry out the following steps: Fig Rate-distortion curves for 512ϫ512 Lenna using lean domain pools of varying relative size ␣ and optimized domain address coding ings, i.e., by iteration of the collage image operator Typically, seven or eight iterations are required to get sufficiently close to the attractor Our test image is the 512ϫ512 gray scale image Lenna, for which we let the quadtree encoder produce fractal codes with compression ratios ranging from up to 30 The parameter ␣ is decreased from 1.00 to 0.50, 0.20, 0.10, 0.05, and for each value of ␣ a rate-distortion curve is obtained, shown in Fig The result clearly shows that with ␣ ϭ0.5 the rate-distortion curve practically is the same as that for the standard method ͑i.e., for ␣ ϭ1͒ When setting ␣ ϭ0.2 we also obtain comparable performance for compression ratios from about 15 and up For smaller compression ratios there is a slight penalty of up to 0.3 dB in PSNR Still lower values of ␣ degrade the rate-distortion curve more noticeably Figures and display the coding results at compression ratios 15 and 30 for ␣ ϭ0.2 Some artifacts are visible, especially for the high compression image Let us remark a Pop a range block R from the stack and search the corresponding domain pool yielding an optimal approximation with a corresponding least rms error for that range block b If the rms error is less than the tolerance or if the range size is less than or equal to the minimum range size, then save the code for the range c Otherwise, partition the range block R into four quadrants and push them onto the stack By using different fidelity tolerances for the collage one obtains a series of encodings of varying compression ratios and fidelities The decoder for the quadtree codes proceeds in the same way as for the case of fixed block size encod- Fig Decoded image at compression ratio 15.75, PSNRϭ32.4 dB, ␣ ϭ0.2 102 / Journal of Electronic Imaging / January 1999 / Vol 8(1) Downloaded From: https://www.spiedigitallibrary.org/journals/Journal-of-Electronic-Imaging on 11/17/2017 Terms of Use: https://www.spiedigitallibrary.org/terms-of-use Domain pools for image compression Acknowledgments The author appreciates the invaluable contribution of Matthias Ruhl, who organized the computer programs and ran the experiments References Fig Decoded image at compression ratio 30.17, PSNRϭ28.2 dB, ␣ ϭ0.2 that our rate-distortion curves in Fig are produced only for the study of how the choice of the parameter ␣ effects the performance of our method No postprocessing is carried out to remove blocking artifacts These curves should not be used for judging the quality of fractal image compression as such since the simple quadtree approach does not yield the best possible numbers In fact, with a state-ofthe-art fractal encoder, based on irregular adaptive partitionings, one can achieve PSNR values of 35.45 and 32.74 dB for the given test image at compression ratios of 15.3 and 29.9, respectively.17 Conclusion We have introduced the concept of lean domain pools in which a fraction 1Ϫ ␣ of low intensity variance domains are discarded from the domain pool This scales the time complexity of the encoding by a factor of roughly ␣ With this procedure, implemented in an adaptive quadtree fractal encoder, the trade-off between increased speed and quality in terms of fidelity and compression has been investigated Also we have introduced a new way for the specification of the domains used for the fractal code which improves efficiency when the collection of domains used shows a high degree of structure which often is the case when lean domain pools are used Our techniques are simple and can easily be incorporated into existing fractal coding programs, even in combination with other acceleration methods such as classification In summary, the lean domain pools introduced in this work cause only negligible or no loss with ␣ ϭ0.5 or even 0.2, thereby reducing the encoding time to about one half or 20% of the normally required time Moreover, with smaller values of ␣, the method also has a strong potential in applications where extremely fast encodings are desired and some quality can be compromised M Barnsley and L Hurd, Fractal Image Compression, AK Peters, Wellesley ͑1993͒ Y Fisher, Fractal Image Compression—Theory and Application, Springer-Verlag, New York ͑1994͒ A E Jacquin, ‘‘Image coding based on a fractal theory of iterated contractive image transformations,’’ IEEE Trans Image Process 1, 18–30 ͑1992͒ D Saupe and R Hamzaoui, ‘‘Complexity reduction methods for fractal image compression,’’ in I.M.A Conf Proc on Image Processing; Mathematical Methods and Applications, Sept 1994, J M Blackledge, Ed., Oxford University Press, Oxford ͑1997͒ D Saupe, ‘‘Fractal image compression via nearest neighbor search,’’ in Fractal Image Encoding and Analysis, Y Fisher, Ed., Conf Proc NATO Advanced Study Institute, Trondheim, July 1995, SpringerVerlag, New York ͑1998͒ D M Monro, ‘‘A hybrid fractal transform,’’ Proc IEEE Int Conf Acoust Speech Signal Process 5, 169–172 ͑1993͒ D M Monro and F Dudbridge, ‘‘Fractal approximation of image blocks,’’ Proc IEEE Int Conf Acoust Speech Signal Process 3, 485–488 1992 K U Barthel, J Schuăttemeyer, T Voye, and P Noll, ‘‘A new image coding technique unifying fractal and transform coding,’’ IEEE Int Conf on Image Processing (ICIP’94), pp 112–116, Austin, Texas ͑1994͒ T Bedford, F M Dekking, and M S Keane, ‘‘Fractal image coding techniques and contraction operators,’’ Nieuw Arch Wisk 10͑3͒, 185–218 ͑1992͒ 10 B Ramamurthi and A Gersho, ‘‘Classified vector quantization of images,’’ IEEE Trans Commun COM-34, 1105–1195 ͑1986͒ 11 A E Jacquin, ‘‘Image coding based on a fractal theory of iterated contractive Markov operators, Part II: Construction of fractal codes for digital images,’’ Technical Report Math 91389-17, Georgia Institute of Technology ͑1989͒ 12 J Kominek, ‘‘Advances in fractal compression in multimedia applications,’’ manuscript ͑1995͒; available from ftp://links.uwaterloo.ca/ pub/Fractals/Papers/Waterloo/ 13 J Kominek, ‘‘Codebook reduction in fractal image compression,’’ Proc SPIE 2669, 33–41 ͑Jan 1996͒ 14 J Signes, ‘‘Geometrical interpretation of IFS based image coding,’’ Fractals Suppl 5, 133–143 ͑1997͒ 15 R C Gonzales and R E Woods, Digital Image Processing, Addison–Wesley, Reading, MA ͑1992͒ 16 E W Jacobs, Y Fisher, and R D Boss, ‘‘Image compression: A study of the iterated transform method,’’ Signal Process 29, 251–263 ͑1992͒ 17 H Hartenstein, ‘‘Topics in fractal image compression and nearlossless image coding,’’ dissertation, University of Freiburg, 1998 Dietmar Saupe received his Dr rer nat and Habilitation degrees, both from the University of Bremen, in 1982 and 1993, respectively He has served as visiting assistant professor of mathematics at the University of California at Santa Cruz (1985–1987), assistant professor at the University of Bremen (1987–1993), professor of computer science at the AlbertLudwigs-University of Freiburg (1993– 1998), and at the University of Leipzig (since 1998) His research has focused on image processing, computer graphics, visualization, experimental mathematics, and dynamical systems He is the co-author and editor of several books on fractals, e.g., Chaos and Fractals (Springer-Verlag, New York, 1992) He is a member of the IEEE Signal Processing Society, ACM SIGGRAPH, Eurographics, and others Journal of Electronic Imaging / January 1999 / Vol 8(1) / 103 Downloaded From: https://www.spiedigitallibrary.org/journals/Journal-of-Electronic-Imaging on 11/17/2017 Terms of Use: https://www.spiedigitallibrary.org/terms-of-use ... ‘‘Advances in fractal compression in multimedia applications,’’ manuscript ͑1995͒; available from ftp://links.uwaterloo.ca/ pub/Fractals/Papers/Waterloo/ 13 J Kominek, ‘‘Codebook reduction in fractal. .. extra bits in the white block skip code and are therefore more expensive than domains corresponding to bits in clusters One can design a postprocessing for a fractal code with the goal of eliminating... Our test image is the 512ϫ512 gray scale image Lenna, for which we let the quadtree encoder produce fractal codes with compression ratios ranging from up to 30 The parameter ␣ is decreased from