Compressed Sensing Image Processing Based on Stagewise Orthogonal Matching Pursuit

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1 Senor & randucer, Vol. 8, Iue 0, October 204, pp Senor & randucer 204 by IFSA Publihing, S. L. Compreed Sening Image Proceing Baed on Stagewie Orthogonal Matching Puruit Kuangfeng Ning School of Information Science and Engineering, Hunan International Economic Univerity, Changha, 40205, China el.: Received: 2 July 204 /Accepted: 30 September 204 /Publihed: 3 October 204 Abtract: hi paper tudie the compreed ening image proceing baed on Stagewie Orthogonal Matching Puruit, the relative error, matching degree, and the running time of StOMP algorithm have been reearched from the perpective of two-dimenional compreible ignal. A imulation platform i built for StOMP algorithm imulation. Further, the performance and complexity are compared for everal typical greedy algorithm uch a SP, OMP, CoaMP, and StOMP. StOMP i ignificantly better than SP, OMP and CoaMP algorithm. Image parity K mut be known by OMP, SP, and CoaMP algorithm, and the original ignal can be gradually approached by StOMP, and it tep K i unknown. StOMP algorithm can take into account the recontruction time and recontruction quality. Copyright 204 IFSA Publihing, S. L. Keyword: Image proceing, Compreed ening, Spare tranform, StOMP, Recontruction algorithm.. Introduction Compreive Sening theory i a new ignal proceing theory which i born in recent year, it i uggeted by D. Donoho (American Academy of Science) [], E. Cande (Ridgelet, Curvelet founder) and Chinee cientit. ao (2006 Field Medal winner) [2], the attention of relevant reearcher i greatly attracted from the date of it birth [3]. he core idea of compreed ening i compreed and ampling merger, and the meaured value i far le than the amount of data that traditional ampling method, it break the bottleneck of Shannon ampling theorem, to make a highreolution ignal acquiition i poible. Compreed ening theory include a pare repreentation of the ignal, random meaurement and recontruction algorithm. Spare repreentation i a priori condition to apply compreed ening, random meaurement i a key proce of compreed ening, recontruction algorithm i a neceary mean to obtain the final reult. In a repreentation of the ignal, the parity i the theoretical bai of the compreive Sening application, the claic method of thinning ha a dicrete coine tranform (DC), a Fourier tranform (FF), dicrete wavelet tranform (DW) and etc. Cande and ao [4, 5] and other evidence: independent and identically ditributed Gauian random meaurement matrix can become univeral compreed ening meaurement matrix. Cande [6, 7] and other reearcher to etablih a well-known contraint iometric reitance (RIP) theory, in order to fully recontruct the ignal, we mut enure that two different K pare ignal will not be mapped into a ingle ample collection by the obervation matrix, which require the contituting 34

2 Senor & randucer, Vol. 8, Iue 0, October 204, pp matrix of each M column vector, which are extracted from the obervation matrix, i non-ingular [8]. Due to the mall amount of the greedy algorithm, it rebuild reult i better and eaier to achieve, o the greedy algorithm i the mot widely ued. Such algorithm are ued for the minimum L0 norm, which allow a certain degree of recontruction error exit, the olving model i a follow. min l0.t. y ΦΨ < ε, () where ε i the mall contant [9, 0]. he behalf of uch algorithm are the matching puruit (MP, Matching Puruit) algorithm, and orthogonal matching puruit (OMP, Orthogonal Matching Puruit) algorithm. It alo include a erie of improvement baed on OMP algorithm, uch a regularization algorithm matching puruit (ROMP, Regularize OMP), tagewie orthogonal matching puruit (StOMP, tagewie orthogonal matching puruit), pare adaptive matching puruit (SAMP, Spare Adaptive MP), compreive ampling matching puruit method (CoSaMP, Compreed Sampling MP) and o on. 2. StOMP Algorithm Stagewie Orthogonal Matching Puruit (StOMP) [] i that the ignal i converted into a negligible margin though a erie of Operation. Initial margin r 0 = y, a matched filter Φ r i contituted under -th tatu, all differential amplitude are larger than the coordinate of a particular elected threhold, thee coordinate are elected to do leat quare, the leat quare fit value are then ubtracted to get a new margin. After a certain number of tate tranition, the proce will end. Contrat OMP algorithm, StOMP able to join more coefficient in each tate, OMP algorithm can only join a coefficient, StOMP algorithm require only a certain number of tate tranition, while OMP algorithm require more tate converion. StOMP algorithm i fater than many other method (uch a L minimum norm and OMP algorithm) for olving pare olution, and therefore there are more attractive in olving large-cale problem. StOMP algorithm tep: StOMP aim i to get approximation of the original ignal through y =Φ x0. StOMP run in S-th tage, the remainder r, r 2,... i eliminated to obtain a erie of approximation a erie of x 0, x,..., and the original ignal i retored. StOMP algorithm i hown in Fig.. StOMP algorithm follow the atomic election criteria in OMP erie algorithm, compared with the other algorithm of the matching track erie, StOMP algorithm feature i adaptive, without knowing the parity K' cae, the original ignal can gradually approached by tep. In addition, part of atom are elected in StOMP and MP, OMP to update the recontruction, the upport et Λ i needed, StOMP introduced the idea of backtracking, which i to elect ome value, which i combined with the reulting upport et Λ of the previou iteration, the candidate et Λ i gotten. Since the parity K i unknown, o the algorithm need to deign the appropriate condition of the iterative top, which i ued intead of K, and a better recontruction reult are achieved. In StOMP algorithm, the iterative recontruction proce will be divided into everal tage, x upport et ize at each tage i uncertainty, the product C =Φ r =<Φ, r > between the calculating meaurement matrix Φ and reidual r are greater than the threhold value to form candidate et J, then the candidate et d i updated through increaing tage value and it re-iteration, when the tage value reache the et value, we determined to meet the iteration top condition. Fig.. Schematic Repreentation of the StOMP algorithm. he main tep of StOMP algorithm [] are decribed a follow. ) the initial olution x = 0, the initial reidual = y, counter = ; r0 2) Calculating C =Φ r =<Φ, r > 3) "large" coordinate et J = { j: c ( j) > g δ } i generated by the threhold value, δ i a noie 35

3 Senor & randucer, Vol. 8, Iue 0, October 204, pp σ = r / n, level 2 g i a threhold parameter [2], the typical range i 2 g t 3; 4) Update upport etimate Λ =Λt J, and calculate approximate value x, ( x ) = ( Φ Φ ) Φ y, where Λ Λ Λ Λ Φ Λ repreent only a new compoed matrix of column in Φ, which i correpond to the upport et J; r = y Φ x ; 5) Update reidual 6) Check the termination condition (e.g., = 0), if it i not to top time, et = +, go to tep (2), if x = x i retored; the top time, the ignal ˆ Becaue StOMP algorithm can accurately recontruct pare ignal, in order to achieve the StOMP imulation, the picture i firt need to convert to pare ignal, uch a FF, DC, wavelet tranform, wavelet tranform i adopted in thi paper, and then the reulting pare ignal i multiplied by the meaured value meaurement to obtain matrix Y, then in StOMP algorithm, the original ignal i preciely recontructed baed on Y, and finally the original image i obtained through the invere wavelet tranform. Specific flow chart i hown in Fig StOMP Algorithm Simulation and it Analyi Reult 3.. Simulation Reult Under Different Sampling Rate When image proceing, we need to tranform the image, uch a FF, DC, wavelet tranform, the image i tranformed into the pare coefficient correponding to the group, then the coefficient matrix i treated by column, and finally the Invere tranform i done back on the treated coefficient, the pare recontructed image can be gotten. In thi paper, Lena 256 * 256 image i given to do wavelet tranform then recontructed, the ampling rate (M / N) are repectively 0.7,0.6,0.5,0.4,0.3, the retored image i hown in Fig. 3. Fig. 3. Recontruction effect of StOMP for Lena 256. For a more intuitive comparion StOMP algorithm recovery effect at different ampling rate, the paper alo give StOMP algorithm PSNR value after the recontruction from 0.2 to 0.7 ampling rate, and the relative error in ampling rate poition, they are a hown in able he Simulation Reult under Different hrehold Fig. 2. he flow diagram baed on StOMP algorithm. In StOMP algorithm, elect atom are needed to meet J = { j: c( j) > gδ }, where σ noie level i σ = r 2 / n, g i a threhold parameter, typical range i g, g value i a variable, in order to ee more clearly the threhold effect impact on the recontruction, thi paper preent the recontructed image with the ame ample rate and different threhold, they are hown in Fig. 4, Fig. 5 and Fig. 6. For a more intuitive analyi of different ample rate, the effect of the two-dimenional image i recontructed by StOMP algorithm under different threhold circumtance, the paper give the PSNR value at the M / N = 0.3,0.5,0.7, and the threhold parameter from of 2.2 to 2.7. A are hown in able 2. 36

4 Senor & randucer, Vol. 8, Iue 0, October 204, pp able. Comparion of StOMP with different ampling. Sampling Rate M/N Image PSNR ime PSNR ime PSNR ime PSNR ime PSNR ime Lena Fig. 4. Recontruction effect of StOMP when M/N=0.3. Fig. 5. Recontruction effect of StOMP when M/N=0.5. Fig. 6. Recontruction effect of StOMP when M/N=

5 Senor & randucer, Vol. 8, Iue 0, October 204, pp able 2. Comparion of StOMP with different threholding. hrehold Sample PSNR PSNR PSNR PSNR PSNR PSNR M=0.3*N M=0.5*N M=0.7*N From the table, under the lower ampling rate, if the threhold parameter i lower, the elected candidate et i more, overetimation likely occur, it affect recontruction reult. If the threhold parameter i too high, candidate et may be le, which i the number of elected column, it will reduce the quality of recontruction, and therefore at different ample cae, the threhold elect would affect the recontruction reult, but Donoho did not give a pecific determine threhold parameter g method, how to chooe one threhold alo be one of the need tudy direction. In StOMP algorithm, the image i a matrix, each column of the matrix i proceed eparately, then the invere tranform i done, the vertical crack may occur under certain circumtance, it influence the remodeling effect to a certain extent Simulation Reult in Adding Noie Cae Since the data tranmiion channel will add a lot of noie, it i neceary to tudy the performance of the StOMP algorithm in adding noie. In thi paper, different SNR value with the added white Gauian noie, a comparative analyi i carried on the effect of image retoration. hey are hown in Fig. 7 and Fig. 8. Comparion of StOMP with different ampling i in able 3. Fig. 7. Recontruction of StOMP with different ampling when SNR=5, 0, 5. Fig. 8. Recontruction of StOMP with different ampling when SNR=20, 25, 30. able 3. Comparion of StOMP with different ampling when SNR=5, 0, 5, 20, 25, 30. SNR Sampling PSNR ime PSNR PSNR ime PSNR ime PSNR ime ime PSNR ime Rate M/N

6 Senor & randucer, Vol. 8, Iue 0, October 204, pp he figure how that, in the cae of adding noie, StOMP algorithm ha a good effect for twodimenional image recontruction, but at a lower ampling rate, the recontruction quality will be lower Performance Analyi of Algorithm In order to analyze the variou performance of StOMP algorithm, thi article will analyi the effect between image recontruction of OMP, SP, CoSaMP' algorithm and one of StOMP algorithm (Fig.9). Firt, we analyzed the rebuilding effect of the each algorithm in the abence of noie. parity K mut be known by OMP, SP, and CoaMP algorithm, and the original ignal can be gradually approached by StOMP, and it tep K i unknown. StOMP algorithm can take into account the recontruction time and recontruction quality (to ee Fig. 9), StOMP i a more practical recontruction algorithm. A are hown in able 4. Secondly, in the cae of added noie, image recontruction quality of each algorithm i in the following Fig. 0. Fig. 9. Comparion of Recontruction reult of SP, OMP, CoaMP, StOMP when M/N=0.5 (a) Original image, (b) Recontructed image uing SP, (c) Recontructed image uing OMP, (d) Recontructed image CoSaMP, (e) Recontructed image uing StOMP. A can be een from the recontructed image quality, at lower ampling rate, the effect of OMP' recontruction i inferior to other algorithm, but at a higher ampling rate, the recontruction reult in four algorithm i little difference. But from rebuilding time perpective, StOMP i ignificantly better than SP, OMP and CoaMP algorithm. Image Fig. 0. Comparion of recontruction reult of SP, OMP, CoaMP, StOMP when M/N=0.5 and SNR=30 (a) Original image, (b) Recontructed image uing SP, (c) Recontructed image uing OMP, (d) Recontructed image CoSaMP, (e) Recontructed image uing StOMP. From Fig. 0, when SNR i 5 db in the added noie, the recontruction i ineffective with repect noie-free ituation, PSNR value i low, but when the SNR value increae, the recontruction effect become better, and rebuilding time i nor increaed ignificantly. hee four algorithm have certain antinoie performance. o viually ee the performance of each algorithm after adding noie, thi paper compare with different PSNR value and the time in the different algorithm and the different SNR. A are hown in able 5. able 4. Comparion of StOMP, SP, OMP, CoaMP in different ampling. Sampling Rate M/N Recontruction Algorithm PSNR ime PSNR ime PSNR ime PSNR ime PSNR ime StOMP SP OMP CoSaMP able 5. Comparion of the performance of four algorithm when M/N=0.5. SNR Recontruction Algorithm PSNR ime PSNR ime PSNR ime StOMP SP OMP CoSaMP

7 Senor & randucer, Vol. 8, Iue 0, October 204, pp Concluion and Outlook hi paper tudie a threhold noiy image enhancement method, the relative error, matching degree, and the running time of StOMP algorithm have been reearched from the perpective of twodimenional compreible ignal. A imulation platform i built for StOMP algorithm imulation. Further, the performance and complexity are compared for everal typical greedy algorithm uch a SP, OMP, CoaMP, and StOMP. Although the calculation of greedy algorithm erie i imple, it achievement i convenient, recontruction effect are good, but it can not directly olve the original optimization problem, the recontruction quality i inferior to the mallet L norm baed algorithm. StOMP algorithm have following defect in term of elected atom: StOMP algorithm i mainly concentrated on dictionary Φ for uniform phere, for Φ, StoMP ha good performance, but it doe not eaily extended to a general dictionary. In addition to the propoed threhold g i taken from 2 to 3, it did not give a more accurate election method. herefore, you alo need to continue to tudy in the greedy algorithm. Compreed ening theory mean will be effective for compreible or pare ignal, but mot of natural ignal do not have the parity, from the orthogonal tranform nature, we firt need to do orthogonal tranformation to ignal, then the ignal i dialed with the next. herefore, orthogonal tranformation i alo an important apect in compreed ening. he elect appropriate orthogonal tranformation i directly related to how the ignal meet the requirement of parity, which will affect whether the ignal i the exact recontruction, thu the elect appropriate orthogonal tranform bae i an important part of conidering recontruction algorithm. Acknowledgement hi paper i ponored by Hunan Provincial Education Science "welfth Five-Year" plan project (No. XJK04BGD046). Reference []. D. Donoho, Compreed ening, IEEE ranaction on Information heory, Vol. 52, Iue 4, 2006, pp [2]. E. Cande,. ao, Near optimal ignal recovery from random projection: univeral encoding trategie, IEEE ranaction on Information heory, Vol. 52, Iue 2, 2006, pp [3]. Compreive Sening Reource ( [4]. E. Candè,. ao, Near optimal ignal recovery from random projection: univeral encoding trategie, IEEE ranaction on Information heory, Vol. 52, Iue 2, 2006, pp [5]. E. Candè,. ao, Decoding by linear programming, IEEE ranaction on Information heory, Vol. 5, Iue 2, 2005, pp [6]. Emmanuel Candè, Compreive ampling, in Proceeding of the International Congre of Mathematician, Madrid, Spain, 2006, Vol. 3, pp [7]. E. Candé, N. Braun, M. B. Wakin, Spare ignal and image recovery from compreive ample, in Proceeding of the 4 th IEEE International Sympoium on Biomedical Imaging: From Nano to Macro, Wahington, D. C., USA, 2007, pp [8]. D. Donoho, Y. aig, Extenion of compreed ening, Signal Proceing, Vol. 86, Iue 3, 2006, pp [9]. Yang Hairong, Zhang Cheng, et al, he theory of compreed ening and recontruction algorithm, Acta Electronica Sinica, Vol. 39, Iue, 20, pp [0].. Blumenath, M. E. Davie, Iterative threholding for pare approximation, Journal of Fourier Analyi and Application, Vol. 4, Iue 5-6, 2007, pp []. D. L. Donoho, Y. aig, I. Drori, and J.-L. Starck, Spare olution of underdetermined linear equation by tagewie orthogonal matching puruit, echnical Report , Department of Statitic, Stanford Univerity, Stanford, California, USA, Copyright, International Frequency Senor Aociation (IFSA) Publihing, S. L. All right reerved. ( 40