Simple algorithm of wavelet-compression allowing realizing inexpensive hardware and software platforms for closed circuits and processing systems of high definition television in real time has been develope

Osokin, A. N.

Sidorov, D. V.

Electronic archive of Tomsk Polytechnic University

IntroductionStandard of shrinking digital images JPEG2000 onthe basis of wavelet transform is one of the most perspective ones. It consists of two extensive parts(JPEG2000 Part I, JPEG2000 Part II). In the range ofthe standard there is a majority of software implementations (ACDSee, LeadTools, Mjp2000, Jasper, Lurawaveetc.) differed significantly in degree and rate of shrinking images [1, 2]. Implementation of JPEG2000 fromACDSee according to [1] possesses the best characteristics: rate of shrinking, degree of shrinking and qualityof reconstructed image (further in the article at themention of standard JPEG2000 implementation fromACDSee is considered). JPEG2000 gains obviously in20...25 % as a reconstructed image by the metric PSNR[2, 3] (at equal shrinking ratio) relative to commonJPEG. The main reason of restricted distribution ofJPEG2000 is a complicity of software implementationand as a result low shrinking rate. Shrinking process duration is caused to a large extent by complicity of sampling stages, block coding and output stream arrangement occupying about 80 % of the total compression time [2]. Therefore, on the basis of the first and he secondparts of the standard JPEG2000 (Part I, Part II) a simpler and quicker algorithm of wavelet compressionQWC (Quick Wavelet Compress, authors name) possessing practically equal JPEG2000 quality of recoveredimage (at equal degree of shrinking) was developed.Shrinking algorithm QWCAlgorithm QWC has shrinking diagram similar tothe standard JPEG2000 (Fig. 1).Stages of preprocessing and wavelet transform of thestandard JPEG2000 and algorithm QWC coincide. According to the standard JPEG2000 input image preprocessing consists in adjustment of values of color component of RGB pixels relative to zero. Image preprocessing occurs only in the case if values of color componentof RGB pixels are in the range of [0,2N–1] (where N isthe capacity of color components bit of RGB or brightness of image pixels) then value of each color component of RGB pixels decreases by 2N–1:where Pi,j is the value of one of color components ofRGB or image pixel brightness, i,j are the coordinates ofpixel in image matrix.The next stage of algorithm is wavelet transform [2].Standard establishes two possible variants of transformation according to the part I:• integer transformation 5/3 used for lossless compression;• irrational transformation 9/7 used for lossy compression.In comparison with standard JPEG2000 where evenscalar quantizer is used, in algorithm QWC subbandscalar quantizer with different sampling coefficients forsubbands (levels) of highfrequency wavelet coefficients is used. Application of uneven quantizer increasesquality of reconstructed image at permanent degree ofshrinking regardless of applied wavelet transform [4]./ 2 / 2, ,, 1, ,, [ 2 , 2 1],2 , [0, 2 1]N Ni j i ji j N Ni j i jP PPP P−⎧ ⎫⊂ − −⎪ ⎪= ⎨ ⎬− ⊂ −⎪ ⎪⎩ ⎭Control, computer engineering and information science79UDC 004.93SIMPLE ALGORITHM OF WAVELETCOMPRESSION OF HALFTONE AND COLOUR IMAGESD.V. Sidorov, A.N. OsokinTomsk Polytechnic UniversityEmail: rauco@mail.ruSimple algorithm of waveletcompression allowing realizing inexpensive hardware and software platforms for closed circuits and processing systems of high definition television in real time has been developed.Fig. 1. Diagram of image compression of the standard JPEG2000 and algorithm QWCDiagram of compression according to the standard JPEG2000Diagram of compression according to the algorithm QWCPreprocessingOriginal image Shrunk image Shrunk imageDiscrete wavelettransformSampling Block codingOutput stream arrangementOn the basis of wavelet transform specification it isconsidered that energy of highfrequency (HF) subbands HL1, HH1, LH1 (Fig. 2) describes the smallest details of image which are badly perceived by a man, therefore, the coefficient of scalar sampling for these subbandsmay be rather high, for example specified by a user [4, 5].Fig. 2. Shift of signal energy presented by the image per threesteps of wavelet transformSubbands LH2, HH2, HL2 describe larger details andpossess higher energy value, therefore, sampling coefficient for these bands should be lower, otherwise, lossesat image reconstruction are more significant than atprevious levels HL1, HH1, LH1. For subbands of higherlevel (HL3, HH3, LH3 etc.) the process is similar:where Q is the sampling coefficient, n is the step of wavelet transform.It should be noted that for the rangesHH1,HH2...HHn coefficient of scalar sampling may beconsiderably higher than for subbands LH1...LHn andHL1...HLn due to weak pixel correlation bias and coefficients of subband HH1 may be zeroed without significant losses in quality [2, 4, 6].The above mentioned interaction of energy distribution among HF subbands allows constructing subbandscalar quantizer with irregular pitch of sampling coefficient and supporting image reconstruction more detailedthan at application of quantizer with one sampling coefficient. Unfortunately, a question of dependence of subband energy or a way of obtaining sampling coefficientsfor subbands is rather complicated and not well coveredat the moment in modern scientific literature. The onlyindication of dependence between subband coefficientsin scientific literature is that energy growth from each(n–1) band to n band depends only on the band (n–1)for pyramidal expansion, Fig. 2. Then it is appropriate tosuppose that dependence of sampling coefficients is similar that is coefficient for n subband is calculated ofthe coefficient of (n–1) subband [7]. It is obvious thatsampling coefficient Q depends for each band not onlyon sampling coefficient of the previous subband but ona concrete shrunk image, therefore, for simplicity of implementation and high rate of sampling only the dependence between sampling coefficients was kept and correction coefficients L was introduced:where i is the level of coefficient expansion or wavelettransform.In algorithm of shrinking QWC as well as in standardJPEG2000 the main parameter at shrinking which a user indicates is either compression ratio or a size of compressed file that is a user may indicate only one parameter for image shrinking – compression ratio Kс. But thecompression ratio is not the only parameter participating in sampling and arithmetical shrinking the sampling coefficient Q and correcting coefficient L shouldbe also specified. As the user specifies only the compression ratio (file size) then the dependence of samplingcoefficient Q and correction coefficient L on compression coefficient Kс should be determined. Theoreticaldefining of coefficients Q and L is rather complicatedprocess, therefore, to simplify obtaining coefficients Kс,Q and L the experiment was carried out.Experiment The aim of carrying out the experiment is to obtainfour experimental dependences (value arrays) of coefficients к Q and L depending on Kс and giving the bestquality of reconstructed image, that is, it is necessary toobtain two arrays of values of coefficients Q and L forhalftone and color images.Thus, for each value of compression ratio Kс it is necessary to determine experimentally an optimal pair ofvalues of coefficients Q and L giving the best quality ofreconstructed image.The initial data for the experiment:• five master images of a set of color images CalgaryCorpus with color depth 24 bits per a pixel and fivehalftone images with depth of 8 bits per a pixel obtained by lossless conversion from original color ones8 (images are selected subject to their reality);• compression ratio interval for color imagesKc∈[8, 200], for halftone images Kc∈[4, 200], step 1;• interval of sampling coefficient Q for halftone andcolor images Q∈[4, 60], step 1;• interval of correction coefficient L for halftone andcolor images L∈[10, 99], step 1;• downsampling coefficient for color components1:8:8 (only for color images) [4].Extreme values of coefficients Kс, Q and L intervalswere preliminary obtained experimentally.Quality of reconstructed image was determined bymetric PSNR* [3]:1,i iQ L Q −=1 2 3... ,nQ Q Q Q> >Bulletin of the Томsк Pоlytеchnic University. 2007. V. 311. № 580* Use of estimation method of PSNR quality is justified by the fact that application of more exact methods of estimation, for example, VQM [3] or SSIM [3] is complicated as the initial files or binary libraries of techniques are implemented for OS Windows(r) family and their use at TPU cluster is not possible, the cluster is controlled by OS Linux(r) SuSe(r) in console mode.where n, m is the amount of signal samples, x are thesamples of original, initial signal for compression, y arethe samples of unpacked, regenerated signal.Compression ratio Кс was determined by the formula:where So and Sc are the volumes of initial and shrunkimage.To carry out the experiment and determine four arrays of values of coefficients Q and L the program defining the most optimal ratios of coefficients Kс, Q and Lat a set of master images was developed for TPU «SKIFPolitekh». Use of TPU cluster is justified by a large volume of calculations and simplicity of computation parallelization, for example, for the experiment on halftone images the initial data are as follows:• amount of master images 5 (halftone);• interval of compression ratio Kc∈[4, 200], step 1;• interval of sampling coefficient Q∈[4, 60], step 1;• interval of correction coefficient L∈[10, 99], step 1.For the experiment on color images the initial data are:• amount of master images 5 (color);• interval of compression ratio Kc∈[8, 200], step 1;• interval of sampling coefficient Q∈[4, 60], step 1;• interval of correction coefficient L∈[10, 99], step 1.Thus, for each value Kс all combinations of values ofcoefficients Q and L are selected by the program thatmay give without optimization of result selection6(200–4+1)(60–4+1)(99–10+1)=6063660 for halftone and 6(200–8+1)(60–4+1)(99–10+1)=5940540for color shrunk and reconstructed images.Maximal amount of parallel computing may be equal to a number of variants of values Kс, Q and L. Therefore, to decrease the amount of computing and furtherselection of optimal results the method of concessionswas applied [8]. The method allows specifying a rate ofconcessions λ by quality increment (criterion) of eachreconstructed image ensuring, thereby, that optimalvector of image quality increment and searched values Qand L is selected at maximal loss between values insidevector not more than λ, %. Therefore, maximal error atfaulty selection of quality increment vector (and valuesQ and L for the given vector) does not exceed λ, %. Application of other methods, for example, additive ormultiplicative ones of criteria check may result, in general case, in cancellation of special criteria and faultyselection of optimal vector.Experiment results The result of experimental program performance is4 arrays of values of Q and L for color and halftone images specifying values of sampling coefficient Q and correction coefficient L for compression ratio Kс selectedby a user. Because of low quality of image estimation bymetric PSNR a part of values of arrays Q and L turnedout to be inadequate; there are «gaps» (zero or start values) in coefficient values in value array. To correct thesituation the arrays of values of coefficients Q and L forcolor and halftone images were sectionally approximated and smoothed (the example is in Fig. 3, 4).Fig. 3. Кс – L initial curve for halftone imagesFig. 4. Кс – L approximated curve for halftone imagesThe arrays of values of coefficients Q and L obtainedafter approximation and smoothing were included intoquantizer of algorithm QWC.The next stage of compression according to algorithm QWC is the stage of block coding (Fig. 1). Theprocess of block coding of algorithm QWC in comparison with the process of block coding in the standardJPEG2000 was significantly simplified and accelerated(Table 1).According to standard JPEG2000 HF wavelet coefficients are compressed by blocks. After compression ofall HF wavelet coefficients by blocks the blocks are rearranged (according to location, value of bit planes) forbeing recorded into output stream buffer. In the suggested algorithm blocks are compressed in the required or,occSK S=210 ,2, ,1, 1255PSNR( , ) 10log ,( )n mi j i ji jnmx yx y= ==−∑Control, computer engineering and information science81der therefore, the arrangement of output buffer is notrequired.Table 1. Stage of block coding of standard JPEG2000 and algorithm QWCThe originals color image (image size 512×512 pixels, color depth 24 bits) compressed by the standardJPEG2000 (Fig. 5, b, PSNR=33,77; SSIM=0,89179;Kс=100) and algorithm QWC (Fig. 5, c, PSNR=30,71;SSIM=0,86779; Kс=100) is given in Fig. 5.The results of comparative test of the standardJPEG2000 and algorithm QWC at mark images of a setCalgary Corpus (images were initial data in the experiment of obtaining arrays of values of coefficients Q andL) are given in Table. 2.Table 2. Comparison of quality of compression of standardJPEG2000 and algorithm QWCAccording to the carried out test (Table 2) algorithmQWC loses to JPEG2000 on the average 3 % by metricSSIM, by metric PSNR – 12,5 %. Expert judgmentshows that metric PSNR does not virtually give more orless adequate estimation of quality as images are weaklydistinguishable visually.Master image ofthe set CalgaryCorpusCompression ratioIndex PSNR Index SSIMDifference in% in indexfor QWC relative toJPEG2000JPEG2000 QWC JPEG2000 QWC PSNR SSIMlena.bmp20 40,25906 33,88858 0,95792 0,95333 18,79 0,4850 36,84676 32,79163 0,92970 0,91856 12,36 1,21100 33,77797 30,71334 0,89179 0,86779 9,97 2,79150 32,07620 29,42957 0,86843 0,83540 8,99 3,95Mean value of difference by PSNR and SSIM 12,53 2,11bat.bmp20 43,94340 34,93433 0,98256 0,97225 25,78 1,0650 36,73378 31,91089 0,96110 0,95133 15,11 1,02100 32,25620 29,47235 0,92720 0,91092 9,44 1,78150 30,15820 27,39804 0,89842 0,87320 10,07 2,88Mean value of difference by PSNR and SSIM 15,10 1,68papr.bmp20 41,20751 34,94453 0,96824 0,95629 17,92 1,2550 37,36990 31,30622 0,93541 0,92038 19,36 1,63100 33,74086 28,04916 0,89790 0,87349 20,29 2,79150 31,64930 26,45925 0,85934 0,83862 19,61 2,47Mean value of difference by PSNR and SSIM 19,30 2,04serf.bmp20 34,50090 29,93922 0,93553 0,88968 14,11 5,1550 29,66250 27,36494 0,82053 0,77156 7,07 6,34100 27,15076 25,78640 0,70902 0,65324 4,40 8,53150 26,05561 24,87761 0,63702 0,58829 3,70 8,28Mean value of difference by PSNR and SSIM 7,32 7,07fl.bmp20 39,26349 34,17872 0,96982 0,95998 14,87 1,0250 33,25353 31,44448 0,90210 0,90895 5,75 +0,75100 30,00079 28,28929 0,83054 0,80631 6,05 3,01150 28,33185 26,63861 0,77671 0,74357 6,36 4,45Mean value of difference by PSNR and SSIM 8,26 1,93Mean value of difference by PSNR and SSIM 12,5 2,97 Standard JPEG2000 Algorithm QWCHF wavelet coefficients are dividedinto square blocks of a free size, often 32×32 or 64×64 but not morethan 4096 coefficients.HF wavelet coefficients are divided into blocks according tothe levels of expansion of coefficient (conversion steps). State of arithmetic coder is resetted. ExcludedBlock is presented in the form of 11bit planes. ExcludedFor each bit plane the value of adjacent bits is forecasted; for each 8bits in the plane the value of 9 bit isforecasted. ExcludedPlanes are packed in the form of bitcuts that, for example, for a blockof 4096 coefficients can give from1 to 11 passages (respectively, 1 passage at maximal compression and 11passages at minimal one) or4096.11=45056 searches of coefficients at minimal compression andnot less than 4096 searches of coefficients at maximal one. Only the values of coefficients themselves are packedthat, for example, for blockof 4096 coefficients alwaysgives only 1 passage regardless of compression degree.On the average, the quantityof passages is 6...7 times lessthan for the standardJPEG2000.Bulletin of the Томsк Pоlytеchnic University. 2007. V. 311. № 582Fig. 5. Image: а) original; b) compressed by JPEG2000;c) compressed by QWCFig. 6. Image area: а) original; b) shrunk by standard JPEG2000; c) shrunk by algorithm QWC? ? ?? ? ?The developed and implemented algorithm QWCshrinks efficiently the images not used in the experimentof forming arrays of values of coefficients Q and L. Thearea of free image (it did not take part in determiningcoefficients, image size 1600?1200 pixels, color depth is24 bits) shrunk by the standard JPEG2000 (Fig. 6, b,SSIM=0,70929; Kс=60) and algorithm QWC (Fig. 6, c,SSIM=0,64978; Kс=60) is given as the example in Fig 6,a. Expert judgment prefers the image shrunk by QWC(Fig. 6, c) to JPEG2000 (Fig. 6, b) as algorithm QWCkept the structure of small details of image (grass stems)at shrinking.Conclusion A more rapid and simple algorithm of wavelet compression QWC with pyramidal diagram of sampling andcoefficient keeping was developed. It possesses quality ofreconstructed images (at equal degree of compression)almost equal to standard JPEG2000 and in some casesexcels it. Algorithm QWC may be rather easily transferred on inexpensive hardware platform capable of processing HDTV videoflow in real time operation mode.Development of shrinking algorithm is supported by the program«Uchastnik molodezhnogo nauchnoinnovatsionnogo konkursa»(«U.M.N.I.K.») by the Fund of assistance of development of minorform of enterprises in scientific and technical field. Control, computer engineering and information science83REFERENCES1. Vatoiln D., Moskvin A., Petrov O. Comparison of image codec ofstandard JPEG2000 and Windows Media Photo (new name: Microsoft HD photo) [Electronic resource]. – access mode: http://compression.ru/video/codec_comparison/wmp_codecs_comparison.html (20.06.2007).2. Vatolin D., Ratushnyak A., Smirnov M., Yukin V. Techniques of data compression. Archiver arrangement, image and video compression. – Moscow: DIALOGMIFI, 2002. – 384 p.3. Vatolin D. Software product for estimating quality of reconstructedimage MSU Quality Measure [Electronic resource]. – access mode:http://compression.ru/video/quality_measure/video_measurement_tool.html (20.06.2007).4. Sidodrov D.V., Osokin A.N. Hardwareoriented subband quantizerfor wavelet compression of halftone images // Molodezh i Sovremennnye Informatcionnye Tehnologii: Collected papers of the VAllRussian theoretical and practical conference of students, graduates and young scientists. Tomsk, February 27 – March 1, 2007. –Tomsk: TPU Press, 2007. – P. 446–448.5. Wellsteed S. Fractals and wavelets for compression of images in action. – Moscow: Triumf, 2003. – 320 p.6. Rabbani M., Santa Cruz D. The JPEG2000 StillImage Compression Standart. – Lausanne: Swiss Federal Institute of Technology(EPFL), 2002. – 246 p.7. Sagetong P., Ortega A. Analytical modelbased bit allocation for wavelet coding with applications to multiple description coding and region of interest coding // Proc. IEEE Int. Conf. on Multimedia andExpo (ICME). Tokyo, Japan August 2001. – Los Angeles: SIPI Department of EES University of Southern California, 2001. –P. 300–304.8. Shtoyer R. Multiobjective optimization: theory, computations andapplications: translation from English / R. Shtoyer; ed. by A.V. Lotov. – Moscow: Radio i Svyaz, 1992. – 504 p.Received on 12.10.2007

Simple algorithm of wavelet-compression of halftone and colour images

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