Bayesian Compressive Sensing Based on Importance Models

Transcription

1 Senor & Tranducer 03 y IFSA Bayean Compreve Senng Baed on Importance odel * Qcong Wang, Shuang Wang, Wenxao Jang, Yunq Le Department of Computer Scence, Xamen Unverty, Xamen, Fujan, 36005, Chna Tel.: E-mal: qcwang@xmu.edu.cn Receved: 5 Aprl 03 /Accepted: 0 June 03 /Pulhed: 8 June 03 Atract: To olve the prolem that all row gnal ue the ame recontructon algorthm, a type of Bayean compreve enng aed on mportance model propoed, whch recontruct more mportant gnal frtly even f long ome unmportant gnal. Compared to Bayean compreve enng whoe performance not well when amplng rato lower, the propoed algorthm can mprove recontructon qualty effectvely. The mportance model nclude two procee, one judgng whether the gnal mportant and the other how to recontruct mportant gnal etter. In th paper, the mproved recontructon algorthm aed on pare mportant gnal and agnng meaure y mportant weght. The two algorthm gve prorty to the more mportant column coeffcent gnal n the recontructon proce. The expermental reult how that the propoed algorthm have etter recontructon effect than the tradtonal Bayean compreve enng, and epecally, the performance of recontructon algorthm aed on agnng meaure y mportant weght mproved ovouly when the amplng rate relatvely low. Copyrght 03 IFSA. Keyword: Bayean compreve enng, Importance model, Wavelet.. Introducton Compreve Senng [-5] ha an extenve applcaton perpectve n the feld of gnal proceng ecaue t reak through Nyqut theory whch the amplng rate mut e more than two tme the hghet frequency. Spare repreentaton of mage a reearch hotpot n the compreve enng. Currently, there are many mprovement of pare repreentaton of mage, whch are aed on dfferent tranformaton, uch a wavelet [, 6], curvelet [7], andlet [8], contourlet [9], dual-tree complex wavelet etc [0]. Thee algorthm are all only concerned on mprovng the pare repreentaton of mage and ue the unfed recontructon algorthm to all coeffcent after pare tranformaton, wthout conderng the coeffcent feature. Wavelet coeffcent have many feature [], uch a patal frequency and drecton electon, energy concentraton of frequency doman and energy attenuaton, patal cluterng of hgh frequency, the mlarty etween uand coeffcent, the relatve etween ampltude etc. The cale feature of wavelet coeffcent [3] and modelng pare pror y ung the mlarty etween uand coeffcent [4, 5] are ued to mprove the recontructon effect of Bayean compreve enng uccefully. In th paper, we frtly try to tudy the feature of wavelet energy dtruton and wavelet coeffcent tattc. Then we take full advantage of wavelet coeffcent feature to propoe two mportance model. They are aed on pare mportant gnal and meaure agned y mportant weght repectvely. The propoed algorthm gve prorty to the more Artcle numer P_SI_390 39

2 mportant column coeffcent gnal n the recontructon proce. The experment reult how that compared to the tradtonal Bayean compreve enng [], the propoed algorthm have mprovement n ome extent. Epecally, the performance of recontructon algorthm aed on meaure agned y mportant weght mproved ovouly when the amplng rate relatvely low. The paper organzed a follow. In ecton two, we ntroduce the energy dtruton feature of wavelet coeffcent and coeffcent tattc n each wavelet level. In ecton three, we ntroduce the mportance model. In ecton four and fve, we propoe the mproved algorthm aed on pare mportant gnal and meaure agned y mportant weght, repectvely. They are two dfferent perpectve to recontruct manly mportant row gnal. Smulaton reult whch are preented n ecton x tetfy the mproved effect of Bayean compreve enng aed on mportance model. Concluon and dcuon of future work are provded n ecton even.. The Feature of Wavelet Coeffcent.. Wavelet Energy Dtruton After wavelet tranformaton lke Fg., the energy dtruton of mage changed. The energy formula can e gven y: N E N j A(, j), () level, the uand except LL 3 uand are howed n the rght of Fg.. We can ee that wavelet energy decreae level y level and concentrate on hgh level uand. LL 3 uand hghet. If uand coeffcent of the hgher level can e recontructed etter and LL uand coeffcent can e recontructed etter, the qualty of recontructed mage wll e mproved certanly... Wavelet Coeffcent Stattc Analyzng the wavelet coeffcent n dfferent uand, t eay concluded that the coeffcent value n HL uand and LH uand are larger than thoe n HH uand n the ame level. A the level hgher, the wavelet coeffcent are hgher. Large coeffcent are manly n the LL uand, few n the HL uand and LH uand and very rare n the HH uand. Fg. 3 how the aolute utracton value etween HL coeffcent and HH coeffcent of each level for a Lena mage. Aove the dotted lne, the coeffcent aolute value of HL uand are larger than HH uand, otherwe the oppote. We can conclude that mot HL uand coeffcent are larger than HH uand coeffcent n the ame level. 3. Importance Recontructon odel ot of energy gathered n the hgh level uand and many large coeffcent are n hgh level uand. If we can mprove the recontructon qualty of column coeffcent gnal whch contan much energy or large coeffcent, even long qualty column coeffcent gnal contanng few energy or mall coeffcent, the overall recontructon qualty wll e mproved. Importance recontructon model can e expreed lke Fg. 4. The man core of mportance recontructon model how to judge the mportance of row coeffcent gnal and how to focu on the recontructon of thee mportant gnal. The followng ecton, we conder from two drecton and propoe two recontructon algorthm aed on pare mportant gnal and more meaure agned to mportant gnal, repectvely. Fg.. Three level of wavelet tranformaton. Fg. how energy dtruton of wavelet coeffcent n the three level uand. X-coordnate correpondng uand and Y-coordnate energy rato. The left fgure how all uand. In order to etter oerve the energy dtruton n the three 4. Improved Algorthm Baed on Spare Important Sgnal The recontructon of Bayean compreve enng [] can e expreed a prolem that uppoe the pror proalty dtruton for a knd of pare dtruton lke Laplace dtruton and then maxmze poteror proalty to olve l norm prolem. 40

3 Fg.. Energy dtruton of wavelet coeffcent. Fg. 3. HL coeffcent v. HH coeffcent n every level. Importance odel Row y row Fg. 4. Importance recontructon model. Bayean compreve enng motly ue pare pror. Therefore, f parer, the modelng effect of pror dtruton etter. Suppoe mage ze N pxel. Recontruct wavelet coeffcent row y row, whch can decreae quantty of ojecton matrx [6]. In fact, th recontructon method a knd of lock compreve enng [7]. Wavelet coeffcent can e repreented,,..., and the wavelet coeffcent n the thrd level are,,..., /4. Accordng to the aove ecton, f we reaonaly ue wavelet energy feature and wavelet coeffcent tattc to make,,..., /4 parer, the whole recontructon effect wll e mproved. Exchange HL uand and HH uand n the ame level. Fg. 5 how the kurto dfference etween efore and after wavelet coeffcent exchange of 5656 Lena mage. From the Fg. 5 we can ee that kurto change of the former thrty two lne exchanged wavelet coeffcent w, w,..., w /4 not ovou. Becaue after wavelet tranlaton, the LL uand coeffcent n the three level are enough large. Even f after tranlaton, the kurto not affected. But we can ee 4

4 that kurto change of the lat thrty two lne exchanged wavelet coeffcent w, w,..., w /4 ovou whch llutrate that the exchange w, w,..., w /4 parer than,,..., /4. We judge the row coeffcent gnal ncluded hgh level uand a mportant gnal and make them parer. The man procedure of algorthm can e decred a: Step : Apply pare tranformaton to the orgnal mage x and get wavelet coeffcent. In h paper we ue DWT tranformaton. Wavelet a ym 8. Step : Exchange HL uand coeffcent and HH coeffcent n each level of wavelet coeffcent and get modfed wavelet coeffcent w. Step 3: Recontruct ung BCS row y row and get recontructed modfed wavelet coeffcent ŵ. Step 4: Exchange HL uand coeffcent and HH coeffcent n each level of wavelet coeffcent. ŵ and otan wavelet coeffcent ˆ Step 5: Apply DWT nvere tranformaton to wavelet coeffcent ˆ and otan recontructed mage ˆx. etter recontructon of mportant gnal to mprove the whole recontructon effect. But ome energy cattered n hgh uand after wavelet decompoton of multlevel, we ort energy of all row gnal to judge each row mportance and then allocate more meaure to row gnal whoe mportance value hgher. The man procedure of algorthm can e decred a: Step : Apply pare tranform to the orgnal mage x and get wavelet coeffcent. Here, we ue DWT tranformaton. Wavelet a ym 8. Step : Otan energy value of each row coeffcent gnal, ort t and get a ortng ndex I. Step 3: Agn meaure accordng to the ndex I. Step 4: Recontruct ung BCS row y row and get recontructed wavelet coeffcent ˆ. Step 5: Apply DWT nvere tranform to wavelet coeffcent ˆ and otan recontructed mage ˆx. If the meaure numer of each row efore reagnng, et a threhold T,..., N to expre the numer of mportant gnal. If the gnal ndex atfe to I T, we judge t a a mportant gnal, then the meaure et. If the ndex I an unmportant gnal, then the meaure et, and atfy: T, t., T N T, () N Fg. 5. The kurto dfference etween efore and after exchange of the frt 64 lne wavelet coeffcent. 5. Improved Algorthm Baed on eaure Agned y Important Weght The prevou ecton we mprove the algorthm performance va pare mportant gnal. In th ecton, we wll recontruct etter the row gnal ncluded much energy and large coeffcent va agnng meaure quantty. Accordng to the econd ecton aout wavelet feature, hgh level uand contan mot of energy. We try to allocate more meaure to hgh level uand, and decreae correpondng meaure to low level uand. Therefore the total numer of meaure not change. We loe the recontructon qualty of unmportant row gnal n exchange for a where. Ung th agnment method, the whole meaure numer not change, jut reallocatng once and gve more meaure to mportant gnal to recontruct accurately. Th meaure allocaton method a mple approach, ut t no full ue of the energy order. Therefore, we gve another meaure allocaton method aed on energy weght. If the ndex of row gnal I T, meaure numer tll et. If the ndex of row gnal I T, et a weght factor: W T N I N I for thee row gnal. Accordng to the weght factor correpondng meaure: (3) W, get the Round WT, (4) 4

5 Round rounded value. The mean of where ummaton equal to, whch enure the total meaure unchanged. Weght factor W can e vewed a mportance factor determnng how mportant of a row gnal. It meet purpoe that more mportant gnal whch nclude more energy recontruct more uffcently. Therefore, t a good meaure agnment method. and, j the pxel pont of a A B mage. Larger PSNR ndcate more cloely to the orgnal R mage. 6. Experment Reult In th ecton we manly compare BCS algorthm and mproved algorthm aed on pare mportant gnal or meaure agned y mportant weght. To ntroduce ealy, the two mproved method can e expreed a BCS-Sparty and BCS-eaure. We recontruct ten 5656 RI mage, ung DWT tranformaton. Wavelet a ym6 and oervaton matrx Gauan random matrx. Set the threhold: N T, (5) Fg. 6. Compare recontructon effect ung three dfferent algorthm n the ame amplng rato. 3, (6) 4, (7) 4 Fg. 6 the recontructon reult when the amplng rate 0.5. Rerr expreed a the recontructon error, /. Rerr cloer to zero, the recontructon effect etter. Fg. 6 how that Rerr of the two algorthm oth decreae and the BCS-eaure algorthm mprove effectvely compared to the orgnal BCS and uperor to BCS-Sparty. Fg. 7 how the recontructon effect for a R mage n the dfferent amplng rato ung BCS, BCS-Sparty and BCS-eaure, repectvely. Stll et the threhold a equaton (5), (6), and (7). Ue DWT tranformaton, wavelet a ym6 and oervaton matrx Gauan random matrx. We ue PSNR to judge the three algorthm. The PSNR can e defned a: where PSNR 55 SE 0log0 SE x j x j A A B,, B 0 j0 (8), (9) Fg. 7. Recontructon effect ung three algorthm n the dfferent amplng rato. It how that no matter the amplng rato change, the PSNR ung BCS-Sparty and BCS-eaure are hgher than BCS method. When the amplng rato 0.88, BCS-Sparty mproved aove 5 db and BCS-eaure mproved aout 9 db. Although the mprovement ung BCS-Sparty method maller than BCS-eaure, the dtance etween the two algorthm grow maller wth the amplng rato ncreang. It llutrate that BCS-eaure method more uperor when the amplng rato low and BCS-Sparty method alo ha ome mprovement effect. Set the threhold a equaton (5), (6), (7) and N, (0) 43

6 wavelet a ym6. Fg. 8 compare the recontructon effect for 5656 ran R mage ung three dfferent algorthm. Fg. 8 (a) the orgnal mage, Fg. 8 (), Fg. 8 (c) and Fg. 8 (d) are recontructon mage ung BCS, BCS-Sparty or BCS-eaure repectvely. It how that the recontructon mage ung BCS-Sparty and BCS-eaure are clearer than BCS algorthm. three algorthm n hgh level. Fg. 0 (c) how the recontructon wavelet coeffcent n HH uand ung three algorthm. Fg. how recontructon error ung BCS, BCS-Sparty and BCS-eaure algorthm n dfferent uand. The recontructon error ung BCS-eaure algorthm all lower than BCS n the hgh level uand. The recontructon error ung BCS-Sparty algorthm lower than BCS n the hgh level uand except HH uand, ut the dfference not large. From the Fg. 0 and Fg., we can ee that the two mproved method recontruct etter n hgh level uand and wore n low level uand than BCS. (a) () (a) () (c) (d) Fg RI recontructed mage: (a) orgnal mage, () BCS algorthm, (c) BCS-Sparty algorthm, (d) BCS-eaure algorthm. To oerve the mage detal clearly, Fg. 9 how partal enlarged detal around the eye. Fg. 9 (a) the partal enlarged detal of orgnal mage, Fg. 9 (), Fg. 9 (c) and Fg. 9 (d) are partal enlarged detal of recontructon mage ung BCS, BCS-Sparty or BCS-eaure repectvely. It how that the detal le clear ued BCS algorthm, ut the recontructed mage ung the mproved algorthm are almot mlarly clear to orgnal mage. It vual llutraton of the mprovement effect va Bayean compreve enng aed on mportance model. To llutrate that the man dea of mproved algorthm aed on mportance model whch are empha on the recontructon of mportant row gnal and focu on recontructon of LL uand and hgh level uand, we how recontructon wavelet coeffcent ˆ of 88 ran mage n Fg. 0. Set the threhold a equaton (5), (6) and (7). Ue DWT tranformaton, wavelet a ym8 and oervaton matrx Gauan random matrx. Fg. 0 (a) how the whole recontructon wavelet coeffcent ung three algorthm. In Fg. 0 (), we compare the recontructon wavelet coeffcent ung (c) (d) Fg RI partal enlarged detal around eye of recontructed mage. 7. Concluon Wavelet coeffcent have many feature. In th paper, we tudy energy feature of wavelet coeffcent to ee that uand energy n hgh level the hghet and the nfluence to recontructed qualty the larget. Accordng to the wavelet coeffcent tattc, we can ee that large coeffcent concentrate motly n hgh uand of hgh energy. Takng full advantage of wavelet coeffcent feature, we propoe two recontructon algorthm aed on mportance model. The man dea utlze the wavelet feature to judge whch row coeffcent gnal are mportant. And then accordng the judgment, we propoe two recontructon algorthm from two drecton whch are more parng or gve more meaure to mportant gnal. 44

7 (a) The whole wavelet coeffcent. ()The wavelet coeffcent n hgh level. (c) The wavelet coeffcent n HH uand. Fg. 0. Compare the recontructed wavelet coeffcent ung BCS, BCS-Sparty or BCS-eaure. 45

8 Fg.. Recontructon error n dfferent uand ung BCS, BCS-Sparty and BCS-eaure algorthm. The experment reult analyze the performance of the two mproved algorthm. When the amplng rato low, Bayean compreve enng aed on agnng meaure y mportant weght how the relatvely good performance compared to BCS or the other mproved algorthm. But wth the amplng rato ncreae, Bayean compreve enng aed on pare mportant gnal gradually how t performance. When the amplng rato reache ome value, the two method have good performance for dfferent mage. But n general, the two algorthm are etter than BCS algorthm, whch ndcate that ntroducng the mportance model effectve. In the future reearch, we can ntroduce wavelet coeffcent feature deeply nto compreve enng to mprove recontructon qualty. Acknowledgement Th paper wa upported y the Natonal Natural Scence Foundaton of Chna (No ) and y the Key Project Fund from the Provnce Offce of Sc. & Tech. of Fujan n Chna (Grant No. 0H604). Reference []. Donoho, D. Compreed enng, IEEE Tran. Inf. Theory, Vol. 5, Iue, 006, pp []. E. J. Candè and. B. Wakn, An ntroducton to compreve amplng, IEEE Sgnal Proce. ag., Vol. 5, 008, pp. 30. [3]. Candè, E., Romerg, J., and Tao, T. Rout uncertanty prncple: exact gnal recontructon from hghly ncomplete frequency nformaton, IEEE Tran. Inf. Theory, Vol. 5, Iue, 006, pp [4]. Lutg,., Donoho, D., and Pauly, J.., Spare RI: the applcaton of compreed enng for rapd R magng, agn. Reon. ed., agn. Reon. ed., Vol. 58, Iue 6, 007, pp [5]. E. Candè and J. Romerg, Sparty and ncoherence n compreve amplng, Invere Prolem, Vol. 3, 007, pp [6]. Baranuk R. Compreve enng, IEEE Sgnal Proceng agazne, Vol. 4, Iue 4, 007, pp. 8-. [7]. a J. W., Compreed enng y nvere cale pace and curvelet threholdng, Appled athematc and Computaton, Vol. 06, Iue, 008, pp [8]. E. Le Pennec and S. allat, Spare geometrc mage repreentaton wth andlet, IEEE Tran. Image Proce, Vol. 6, Iue 9, 008, pp [9]. L. L, L. F. Kong and Q. S. Lan, Image compreve enng aed on contourlet and vna flterng, Intrument Journal, Vol. 30, Iue 0, 009, pp [0]. S. Z. Chen, P. P. Hao and Q. S. Lan, Image compreve enng aed on dual-tree complex wavelet and wave atom, Sgnal Proceng, Vol. 6, Iue, 00. []. Roett, F., Katto, J. and Ohta,., Improved cannng method for wavelet coeffcent of vdeo gnal, Sgnal Proceng, Vol. 8, Iue 4, 996, pp []. J S, Xue Y, and Carn L., Bayean compreve enng, IEEE Tran. Sgnal Proceng, Vol. 56, Iue 6, 008, pp [3]. Vera E., ancera, L., Baacan, S. D., olna, R. and Kataggelo, A. K., Bayean compreve enng of wavelet coeffcent ung multcale Laplacan pror, n Proceedng of the IEEE/SP 5 th Stattcal Sgnal Proceng Workhop, Cardff, UK, 009, pp [4]. Lhan He., Haojun Chen., and Carn, L., Tree-tructured compreve enng wth varatonal Bayean analy, IEEE Sgnal Proceng Letter, Vol. 7, Iue 3, 00, pp [5]. Lhan He and Carn, L., Explotng tructure n wavelet-aed Bayean compreve enng, IEEE Tran. Sgnal Proceng, Vol. 57, Iue 9, 009, pp [6]. Can Feng., Zhhu We and Lang Xao, A novel mage compreve enng approach wth column pare pror, n Proceedng of the t Internatonal Conference on Informaton Scence and Engneerng (ICISE), 009, pp [7]. Gan L., Block compreed enng of natural mage, n Proceedng of the Int. Conf. on Dgtal Sgnal Proceng (DSP), Cardff, UK, Copyrght, Internatonal Frequency Senor Aocaton (IFSA). All rght reerved. ( 46