TELK OMNIKA , V ol. 15, No . 1, March 2017, pp . 540 548 ISSN: 1693-6930, accredited A b y DIKTI, Decree No: 58/DIKTI/K ep/2013 DOI: 10.12928/telk omnika.v15.i1.3164 540 Ima g e De-noising on Strip Steel Surface Def ect Using Impr o ved Compressive Sensing Algorithm Dongy an Cui 1,2 , K e wen Xia* 1 , Jingzhong Hou 1 , and Ahmad Ali 1 1 School of Electronics Inf or mation Engineer ing, Hebei Univ ersity of T echnolo gy Tianjin/China/Hebei Univ ersity of T echnology 2 School of Inf or mation Engineer ing, Nor th China Univ ersity of Science and T echnology T angshan/China/Nor th China Univ ersity of Science and T echnology Corresponding author , e-mail: kwxia@heb ut.edu.cn Abstract De-noising f or the str ip steel surf ace def ect image is conductiv e to the accur ate det ection of the str ip steel surf ace def ects . In order to filter the Gaussian noise and salt and pepper noise of str ip steel surf ace def ect images , an impro v ed compressiv e sensing algor ithm w as applied to def ect image de-noising in this paper . First, the impro v ed Regular iz ed Or thogonal Matching Pursuit algor ithm w as descr ibed. Then, three typical surf ace def ects (scr atch , scar , surf ace upw ar ping) images w ere selected as the e xper imental samples . Last, detailed e xper imental tests w ere carr ied out to the str ip steel surf ace def ect image de-noising. Through compar ison and analysis of the test results , the P eak Signal to Noise Ratio v alue of the proposed algor ithm is higher compared with other tr aditional de-noising algor ithm, and the r unning time of the proposed algor ithm is only26.6% of that of tr aditional Or thogonal Matching Pursuit algor ithms . Theref ore , it has better de-noising eff ect and can meet the requirements of real-time image processing. K e yw or d: compressiv e sensing, surf ace def ects , image de-noising Cop yright c 2017 Univer sitas Ahmad Dahlan. All rights reser ved. 1. Intr oduction Image noise reduction is a classical prob lem in image processing which has o v er 50 y ears of research histor y[1, 2], and stil l is a hot topic. The str ip steel surf ace def ect images in the process of collection, acquisition and tr ansmission will be polluted to some e xtent b y visib le or in visib le noise , also due to the unstab le light, camer a vibr ation and other f actors etc. Theref ore , it is necessar y to carr y out the noise processing of the collected images . A large n umber of studies ha v e been carr ied out on the surf ace def ect image de-noising at home . In 2008, Liu W eiw ei, Y un Hui Y an et al of Nor theaster n Univ ersity put f orw ard an image de-noising method based on local similar ity analysis and neighborhood noise e v aluation [3]; In 2010, Bo T ang et al studied the r ules of str ip steel surf ace def ects image de-noising based on w a v elet t hreshold[4]; In 2012, Hao Xu of W uhan Univ ersity of Science and T echnology proposed the method of surf ace def ect of str ip steel based on mathematical mor phology , which could detect small def ect edge under strong noise and o wn strong noise imm unity[5]. F rom the abo v e , the e xisting str ip steel surf ace def ect image de-noising methods mainly f ocused on the tr aditional filter ing method. In the e xisting theor y , the or iginal signal is mostly pro- jected to a cer tain tr ansf or mation space , and the sparsity of the coefficient in the projection do- main is as a fundamental basis . While the e xistence of noise aff ected the sparsity of signals in the tr ansf or m space . So the optimization method is used to restore the signal, if only a single sparse constr aint pr inciple is used, it is difficult t o accur ately reconstr uct the or iginal signal. In this case , compressiv e sensing theor y still ma y tak e other eff ectiv e method of reconstr ucting. Numerous studies sho w that the reconstr uction algor ithm based on compression perception theor y is applied in signal de-noising can achie v e good eff ect[6]-[8]. Donoho[9]-[1 0],Candes[11]-[13],Romberg[11]- [13] and T ao[12]-[13] and other scientists initially put f orw ard the concept of compressed sensing from sparse signal decomposition and appro ximation theor y in 2004, f ollo w ed b y a large n umber Receiv ed September 9, 2016; Re vised December 27, 2016; Accepted J an uar y 13, 2017 Evaluation Warning : The document was created with Spire.PDF for Python.
TELK OMNIKA ISSN: 1693-6930 541 of rele v ant theoretical research. M.A.T . Figueiredo proposed the g r adient projection f or sparse reconstr uction(GPSR)algor ithm based on the L1 nor m. The method obtaine d the good eff ect of denoising [14]. The compressed perception theor y is introduced into the str ip steel surf ace def ect image preprocessing, which is r arely mentioned in the liter ature at home and abroad. Theref ore , In this paper , the impro v ed R OMP algor ithm w as applied to the str ip steel surf ace def ect image de-noising, which has better de-noising eff ect and shor ter r unning time compared with tr aditional median filter ing, w a v elet threshold method and tr aditional compressed sensing algor ithm. 2. Resear c h Method 2.1. Description of weighted R OMP algorithm Needell et al proposed the regular iz ed or thogonal matching pursuit algor ithm (R OMP) based on the or thogonal matching pursuit (OMP)[15]-[16]. All matr ices satisfied the restr icted isometr y condition and all sparse signals can be reconstr ucted. The algor ithm w as impro v ed based on R OMP algor ithm. The selection of atomic inde x set f or the first time w as using w eighted correlation coefficient, not only consider ing the correlation coefficient of the current iter ation, also consider ing the correlation coefficient of the last iter ation, and e xpanding the selection scope of inde x v alue . W eighted f or m ula is as sho wn in (1). g t = g + g t 1 s:t:g = A T r T 1 ; 0 < < 1 ; 0 < < 1 ; + = 1 (1) The pseudo code of algor ithm is as f ollo ws . Input:(1)measurement matr ix y , y 2 R ;(2) M N dimensional sensing matr ix A =  ;(3) sparsity le v el K of the signal(the n umber of nonz ero elements in x ). Output N dimensional reconstr ucted signal(sparse appro ximation signal) ^ x 2 R N . Initializ e r 0 = y ; 0 = ; A 0 = Iter ation step(1)Calculate: g = abs A T r t 1 (which is: h r t 1 ; j i ; 1 j N ); step (2)Obtain u = j g t j according to f or m ula (1),choose a set J of the K biggest or nonz ero v alues , which corresponds to the column n umber of A and constr uct a set J ; step (3)Regular iz e: j u i j 2 j u j j f or all i; j 2 J 0 ,Among all subset J 0 ,choose J 0 with the maximal energy P j j u ( j ) j 2 ; j 2 J 0 ; step (4) t = t 1 S J 0 , A t = A t 1 S j (f or all j 2 J 0 ); step(5) Calculate the least squares solution of y = A t t : ^ t = ar g min t k y A t t k = ( A t t ) 1 A T t y ; step (6) Update residual: r t = y A t ^ t = y A t A T t A 1 A T t y step (7) t = t + 1 , if t K , then retur n step (1), if t > K or k t k 0 2 K ( k t k 0 represents the n umber of elements in the set or residua r t = 0 ,then the iter ation stop , and enter step (7); step (8) ^ t has nonz ero entr ies at t the v alue is respectiv ely ^ t obtained from reconstr uc- tion step (9) reconstr ucted the signal ^ x = ^ . 2.2. De-noising Model Based On The W eighted Correlation R OMP Algorithm Assuming that the receiv ed image signal is g ( x; y ) , which is contaminated b y noise . The clean image is f ( x; y ) . The additiv e noise is n ( x; y ) .Then the additiv e noise model is g ( x; y ) = f ( x; y ) + n ( x; y ) . When the signal is disturbed b y m ultiplicativ e noise , the model is e xpressed as in (2). g ( x; y ) = f ( x; y ) (1 + n ( x; y )) = f ( x; y ) + f ( x; y ) n ( x; y ) (2) Where , the output signal of the second ter m is the result of m ultiplying the noise , which is aff ected b y f ( x; y ) . The bigger f ( x; y ) , the bigger the noise . According to the theor y of compres- siv e sensing, the f ollo wing results can be obtained,as in (3). g ( x; y ) = f ( x; y ) + n ( x; y ) =  (3) Image De-noising on Str ip Steel Surf ace Def ect Using Impro v ed Compressiv e ... (Dongy an Cui) Evaluation Warning : The document was created with Spire.PDF for Python.
542 ISSN: 1693-6930 Where , is a sparse representation of the tr ansf or med image . In this w a y , w e can reco v er the or iginal image b y estimating the sparse representation of the clean image so as to achie v e the pur pose of remo ving the noise . The de-noising model based on the compressiv e sensing model is as in (4). = ar g min k k 0 ; s:t: k g k 2 2 T (4) 3. Result and Anal ysis Three kinds of str ip steel def ect images (scr atch, scar , surf ace upw ar ping) are selected in this paper ,and the noise type is Gaussian noise and salt and pepper noise . 3.1. Sim ulation Experiment 1: Def ect ima g e de-noising polluted b y Gaussian noise Firstly , the eff ect of diff erent tr ansf or mation matr ices on the image de-noising processing is studied. Sampling r ate w as 0.4, 0.5 and 0.4, respectiv ely ,as sho wn in T ab le 1. T ab le 1. Compar ison of diff erent sampling r ate and diff erent tr ansf or mation matr ices FFT FFT DCT DCT D WT D WT Def ect type M/N PSNR time(s) PSNR time (s) PSNR time (s) scr atch 0.4 22.4548 3.451 20.6991 3.34 20.9255 0.98 0.5 23.1268 4.891 20.6288 4.703 19.9260 1.282 0.6 23.2428 6.493 20.6907 6.231 19.7568 1.843 scar 0.4 21.7367 3.541 18.7781 3.361 19.7284 0.94 0.5 21.5880 4.932 19.2251 4.671 21.1243 1.361 0.6 21.8828 6.392 19.9448 6.121 19.3047 1.841 surf ace upw ar ping 0.4 23.0123 3.641 19.9002 3.51 20.8668 1.21 0.5 23.4251 5.323 19.8785 5.044 19.3613 1.469 0.6 23.2042 6.924 20.1599 6.651 19.2570 1.9 F rom T ab le 1 ,w e can get the compar ison results of PSNR and r un time under the D WT and FFT , DCT tr ansf or m matr ix.The PSNR v alue using D WT tr ansf or mation matr ix to proce ss is higher than that of the DCT tr ansf or m matr ix, and is slightly lo w er than that of the FFT tr ansf or m matr ix. While the r unning time of the D WT tr ansf or m matr ix is m uch shor ten than that of the other tw o . Ob viously , the D WT tr ansf or mation matr ix can g reatly shor ten the r unning time , and it will not reduce the image quality . In this paper , the D WT tr ansf or m matr ix is applied in diff erent compression sensing algor ithms to process the str ip surf ace def ect image , so as to achie v e the pur pose of real-time processing. T ab le 2 sho ws the de-noising results of three types of def ects (scr atch, scar , f acial w ar p- ing) under Gaussian noise using OMP , Cosamp , stomp and the proposed algor ithm. The sampling r ate is 0.5, The tr ansf or mation matr ix is the D WT matr ix. The Gaussian noise mean is 0 and v ar i- ance is 0.1, 0.01, 0.001 respectiv ely , as sho wn in T ab le 2 belo w . F rom the e xper imental data and e xper imental results , the PSNR v alue is higher than the OMP algor ithm, Cosamp algor ithm and StOMP algor ithm using the proposed method to de-noise the str ip st eel surf ace def ect image . Although the r unning time is slightly higher than that of the StOMP algor ithm, b ut compared with the OMP algor ithm and the Cosamp algor ithm, the r unning time is g reatly reduced, about 26.6% of the OMP algor ithm and 41.7% of the Cosamp algor ithm. Exper iments sho w that, consider ing the de-noising eff ect and the r unning time , the perf or mance of this algor ithm to handle str ip surf ace def ect image Gaussian noise pollution is optimal. T ype of e xper imen t def ects respectiv ely are scr atch, scar and surf ace upw ar ping. Gauss noise means is 0, the v ar iance is 0.1, 0.02, 0.01, 0.005. 3 * 3 templates is selected in mean filter and median filter f or processing. Figure 1(a)-(g) to figure3(a)-(g) are respectiv ely the results of three kinds of def ects when the Gaussian noise mean is 0 and the v ar iance is 0.01. T ab le 4 sho ws the PSNR v alues of three kinds of def ects using v ar ious de- noising algor ithms (Gaussian noise with mean 0 TELK OMNIKA V ol. 15, No . 1, March 2017 : 540 548 Evaluation Warning : The document was created with Spire.PDF for Python.
TELK OMNIKA ISSN: 1693-6930 543 T ab le 2. Gaussian noise de-noising eff ect when the sampling r ate is 0.5 Def ect Gaussian OMP OMP Cosamp Cosamp StOMP StOMP Propo sed Proposed type Noise PSNR Time(s) PSNR Time(s) PSNR Time(s) PS NR Time(s) scr atch 0.1 15.0269 11.373 11.3264 8.1589 9.6994 1.995 26. 7728 2.9952 0.01 24.1047 11.546 17.9604 7.1916 18.7309 2.139 29.9563 3.0888 0.001 33.4488 11.345 21.1458 7.3788 27.6014 2.124 33.6600 3.0576 scar 0.1 18.3338 11.152 11.4972 7.2696 9.7740 1.58 8 26. 7509 2.9952 0.01 24.0836 11.500 17.4030 7.1760 18.5289 1.907 29.2152 3.0420 0.001 33.1111 11.729 19.9023 7.1760 27.2120 1.484 32.3062 3.1356 surf ace 0.1 15.1427 11.431 11.3537 7.2384 9.7414 0.856 9 26. 5816 3.1512 upw ar ping 0.01 23.7802 11.607 17.0315 7.1448 18.2017 0.3639 29. 3068 2.9952 0.001 32.5775 11.400 19.6132 7.0824 24.7782 0.1734 32.5256 3.0108 and v ar iance 0.01). Figure 4 (a)-(c) are respectiv ely the PSNR compar ison cur v e of th ree kinds of def ects noisy images obtained b y using v ar ious de-noising algor ithms . Figure 1. scr atch(Gaussian noise with mean 0 and v ar iance 0.01)(a) the or iginal image of scr atch (b) the image with Gaussian noise (c) Median filter ing image (d) Mean filter ing image (e) W a v elet de-noising image (f)CS de-noising image (g) de-noising image of the proposed algor ithm As sho wn in T ab le 3 and Figure 4, the PSNR v alues of the v ar ious algor ithms are all decreased with the increasing of the noise intensity . Compared with other tr aditional de-noising methods , the proposed method in this paper has higher PSNR v alue , that is , the eff ect is better . 3.2. Sim ulation Experiment 2: Def ect ima g e de-noising polluted b y salt and pepper noise T ype of def ects are respectiv ely scr atch, scar and f acial w ar ping in the e xper iment. Salt and pepper noise intensity are 0.1, 0.01,0.005,0.001. 3*3 templates is selected in mean filter and median filter f or processing. Figure 5(a)-(g) to figure7(a)-(g) are respectiv ely the results of three kinds of def ects de-noising images . T ab le 4 sho ws the PSNR v alues of three kinds of def ects using v ar ious de-noising algor ithms (Salt and pepper noise intensity is 0.1). Figure 8 (a)-(c) are respectiv ely the PSNR compar ison cur v e of three kinds of def ects noisy images obtained b y using v ar ious de-noising algor ithms . As sho wn in T ab le 4 and Figure 8, the PSNR v alues of the v ar ious algor ithms are all decreased with the increasing of the noise intensity . The median filter ing method is v er y eff ectiv e f or the de-noising of salt and pepper noise . Compared with other tr aditional de-noising methods , Image De-noising on Str ip Steel Surf ace Def ect Using Impro v ed Compressiv e ... (Dongy an Cui) Evaluation Warning : The document was created with Spire.PDF for Python.
544 ISSN: 1693-6930 Figure 2. scar(Gaussian noise with mean 0 and v ar iance 0.01)(a) the or iginal image of scar (b) the image with Gaussian noise (c) Med ian filter ing image (d) Mean filter ing image(e) W a v elet de-noising image (f)CS de-noising image (g) de-noising image of the proposed algor ithm Figure 3. surf ace upw ar ping(Gaussian noise with mean 0 and v ar iance 0.01)(a) the or iginal image of surf ace upw ar ping (b) the image with Gaussian noise (c) Median filter ing image (d) Mean filter ing image(e) W a v elet de-noising image (f)CS de-noising image (g) de-noising image of the proposed algor ithm the proposed method in this paper has higher PSNR v alue , that is , the eff ect is better . 4. Conc lusion F or cold-rolling comple x en vironment, and its image in the acquisition, acquisition, tr ansf er process will be polluted b y visib le or in visib le noise , w e f ocus on de-noising meth od based on the impro v ed compressiv e sensing algor ithm. The conclusions are as f ollo ws: (1) In the compressiv e sensing algor ithm, the image quality and r unning time are aff ected b y diff erent tr ansf or mation matr ix. Consider ing the tw o , the D WT tr ansf or m matr ix is the best. (2) Under the same noise intensity , the proposed algor ithm has a little diff erence on the TELK OMNIKA V ol. 15, No . 1, March 2017 : 540 548 Evaluation Warning : The document was created with Spire.PDF for Python.
TELK OMNIKA ISSN: 1693-6930 545 T ab le 3. The de-noising eff ect of v ar ious algor ithms with diff erent intensity Gaussian noise Def ect Noise PSNR0 Median filter Mean filter w a v elet CS-OMP proposed scr atch 0.1 12.7296 17.5589 19.0174 24.1869 15.0130 26.7728 0.02 18.6791 24.0706 23.2626 24.0905 20.9537 26.8191 0.01 21.5005 26.7408 24.3423 24.0711 23.8867 29.9563 0.005 24.2483 29.2443 25.2858 24.0755 26.7766 32.6602 scar 0.1 12.9701 17.4094 18.8318 23.8928 15.1302 26.7509 0.02 18.7187 23.8616 22.7633 23.8432 20.9413 26.9985 0.01 21.5438 26.3853 23.7873 23.8198 23.6766 29.2152 0.005 24.1825 28.8335 24.7421 23.8063 26.5264 31.2532 surf ace 0.1 12.8528 17.1803 18.6847 23.6883 15.0231 26.5816 upw ar ping 0.02 18.3413 23.0985 22.4577 23.6199 20.8064 26.8462 0.01 20.8621 25.2768 23.5022 23.6304 23.4988 29.3068 0.005 22.9852 26.9534 24.0940 23.5888 26.1000 32.6708 Figure 4. the PSNR compar ison cur v e obtained b y using v ar ious de-noising algor ithms(a)The PSNR cur v e of scr atch image (b) The PSNR cur v e of scar image (c) The PSNR cur v e of surf ace upw ar ping image de-noising eff ect f or diff erent kinds of def ects . (3) Compared with the tr aditional algor ithm such as median filter ing, mean filter ing, w a v elet de-noising and con v entional compression sensing method ,the proposed method has better de- noising eff ect. Ac kno wledg ement This w or k w as suppor ted b y the National Natur al Science F oundation of China (No . 51208168), and Hebei Pro vince Natur al Science F oundation (No . E2016202341). Ref erences [1] Zhang Y e , Jia Meng, ”Underg round Image Denoising” TELK OMNIKA Indonesian Jour nal of Electr ical Engineer ing , v ol. 12(6), pp .4438-4443, 2014. [2] W ANG Jianw ei, ”A Noise Remo v al Algor ithm of Color Image” TELK OMNIKA Indonesian Jour nal of Electr ical Engineer ing , v ol. 12(1), pp .565-574, 2014. [3] W eiw ei Liu, Y an Y un-hui, Sun Hong-w ei et al., ”Impulse noise reduction in surf ace def ect of steel str ip images based on neighborhood e v aluation, Chinese Jour nal of Science Instr u- ment , V ols 29, pp . 1846-1850, 2008. [4] Bo T ang, K ong Jian-yi, W ang Xing-dong et al., ”W a v elet threshold denoising f or steel str ip surf ace def ect image , Jour nal of W uhan Univ ersity of Science and T echnology , V ols 33,pp . Image De-noising on Str ip Steel Surf ace Def ect Using Impro v ed Compressiv e ... (Dongy an Cui) Evaluation Warning : The document was created with Spire.PDF for Python.
546 ISSN: 1693-6930 Figure 5. scr atch(Salt and pepper noise intensity is 0.1)(a) the or iginal image of scr atch (b) the image with Gaussian noise (c) Median filter ing image (d) Mean filter ing image (e) W a v elet de- noising image (f)CS de-noising image (g) de-noising image of the proposed algor ithm Figure 6. scar(Salt and p epper noise intensity is 0.1)(a) t he or iginal image of scar (b) the image with Gaussian noise (c) Median filter ing image (d) Mean filter ing image(e) W a v elet de-noising image (f)CS de-noising image (g) de-noising image of the proposed algor ithm 38-42,2010. [5] Hao Xu, ”Image processing and identification of str ip steel surf ace def ects based on machine vision, Thesis f or master’ s deg ree of W u Han Univ ersity of science and technology ,2012. [6] Amin T a v ak oli, Ali P our mohammad, ”Image Denoising Based on Compressed Sensing, In- ter national Jour nal of Computer Theor y and Engineer ing ,V ols 4,pp . 266-269,2012. [7] Shunli Zhang, ”Compressed Sensing Method Applicatio n in Image Denoising, Inter national Jour nal of Signal Processing, Image Processing and P atter n Recognition ,V ols 8,pp . 203- 212,2015. [8] M. T . Alonso , P . L. Dekk er and J . J . Mallorqui, ”A No v el Str a tegy f or Radar Imaging Based TELK OMNIKA V ol. 15, No . 1, March 2017 : 540 548 Evaluation Warning : The document was created with Spire.PDF for Python.
TELK OMNIKA ISSN: 1693-6930 547 Figure 7. surf ace upw ar ping(Salt and pepper noise intensity is 0.1)(a) the or iginal image of surf ace upw ar ping (b) the image with Gaussian noise (c) Median filter ing image (d) Mean filter ing image(e) W a v elet de-noising image (f)CS de-noising image (g) de-noising image of the proposed algor ithm T ab le 4. The de-noising eff ect of v ar ious algor ithms with diff erent intensity Salt and pepper noise Def ect Noise PSNR0 Median Mean w a v elet CS-OMP Proposed scr atch 0.1 17.5990 31.5663 22.3715 23.4050 20.1083 31.5687 0.01 26.7632 36.4494 26.3376 30.6689 32.4924 37.3122 0.005 28.6169 36.4548 26.7347 33.5300 35.9337 42.4640 0.001 32.2959 36.4613 26.9791 37.1267 41.6660 48.3173 scar 0.1 17.3623 29.7687 21.9811 22.9747 19.7776 30.1205 0.01 26.5068 34.8033 25.2883 29.3508 30.9695 36.5890 0.005 28.4849 34.8155 25.6405 31.6003 34.4295 40.7194 0.001 32.0367 34.8203 25.8779 35.0911 38.5237 45.1762 surf ace 0.1 17.0 655 28.1653 21.9634 23.1624 19.8053 29.2302 upw ar ping 0.01 24.3441 29.7292 24.5819 28.2800 30.6486 34.7397 0.005 25.7489 29.7268 24.8582 30.2394 32.7035 38.1562 0.001 26.9719 29.7334 25.0317 32.4196 36.0263 42.2687 Figure 8. the PSNR compar ison cur v e obtained b y using v ar ious de-noising algor ithms(a)The PSNR cur v e of scr atch image (b) The PSNR cur v e of scar image (c) The PSNR cur v e of f acial w ar ping image on Compressiv e Sensing, IEEE T r ans . Geoscience and Remote Sensing ,V ols 48,pp . 4285- 4295,2010. [9] D Donoho , ”Compressed sensing, IEEE T r ans , on Inf or mation Theor y ,V ols 52,pp . 1289- Image De-noising on Str ip Steel Surf ace Def ect Using Impro v ed Compressiv e ... (Dongy an Cui) Evaluation Warning : The document was created with Spire.PDF for Python.
548 ISSN: 1693-6930 1306,2006. [10] D Donoho , Y Tsaig, ”Extensions of compressed sensing, Signal Processing ,V ols 86,pp . 533- 548,2006. [11] E. Candes , J . Romberg, ”Sparsity and incoherence in compressiv e sampling, In v erse Prob ,V ols 23,pp . 969-985,2007. [12] E. Candes ,J . Romberg, T . T ao , ”Rob ust uncer tainty pr inciples: Exact signal reconstr uction from highly incomplete frequency inf or mation, IEEE T r ans . Inf or m. Theor y ,V ols 52,pp . 489- 509,2006. [13] E. Candes ,J . Romberg, T . T ao , ”Stab le signal reco v er y from incomplete and inaccur ate mea- surements , Comm. Pure Appl. Math ,V ols 58,pp . 1207-1223,2006. [14] Figuueiredo M A T , No w ak R D , Wr ight S J , ”Gr adient projection f or sparse reconstr uction: Application to compressed sensing and other in v erse prob lems , Jour nal of Select ed T opics in Signal Processing: Special Issue on Con v e x Optimization Methods f or Signal Processing ,V ols 194,pp . 586-598,2007. [15] Needell D V ersh ynin R, ”Signal reco v er y from incomplete and inaccur ate measurements via regular iz ed or thogonal matching pursuit, IEEE Jour nal on Sele cted T opics in Signal Process- ing ,V ols 4,pp . 310-316,2010. [16] Zhenzhen Y ang, Y ang Zhen, Sun Linhui, ”A Sur v e y on Or thogonal Matching pursuit T ype Algor ithms f or Signal Compression and Reconstr uction, Jour nal of Signal Processing ,V ols 29,pp . 486-496,2013. TELK OMNIKA V ol. 15, No . 1, March 2017 : 540 548 Evaluation Warning : The document was created with Spire.PDF for Python.