Gaussian Blurring-Deblurring for Improved Image Compression

Transcription

1 aussian Blurring-Deblurring for mproved mage Compression Moi Hoon Yap 1 Michel Biser Hong Ta Ewe 1 1 Mulimedia Universi (MMU) Jalan Mulimedia 100 Cberjaa Selangor Darul Ehsan Malasia {mhap hewe}@mmu.edu.m hp:// Used o be wih Mulimedia Universi; joined Universi of Noingham on 01 June 00 michel.biser@noingham.edu.m Absrac. The deblurring of aussian blur b invering he acion of he diffusion equaion has long been known. This echnique is ineresing bu wihou much pracical applicaion since he images have o be blurred b convoluion wih a aussian o be "de-blur-able" wih his echnique. n his paper we invesigae he possibili o use his blurring-deblurring process as a pre-pos-processing sep in classical image compression. is known indeed ha he compressibili of an image increases wih he blurring wih a relaion beween compression raio () and he blurring scale sigma (σ ) which we show o be roughl linear. So b preprocessing and image wih aussian blurring before compression he will increase. The echnique of deblurring aussian blur is hen used as a pos-processing sep afer decompression. Of course quanisaion of he blurred image prior o deblurring decreases he quali of he deblurring inroducing new errors. Hence he quanisaion sep inroduced b he compression algorihm also affecs he deblurring performed in he posprocessing sep resuling in a smaller Peak Signal o Noise Raio (). n his aricle he complemenar effecs of increased and decreased on he /-curve of various compression algorihms are sudied in funcion of sigma of he order of he deblurring and of he compression echnique. 1 nroducion A large varie of image compression algorihms have been developed over he ears. n his paper we ake a special look a one possible approach in image compression based on he following hree premises: 1) smooh images can be compressed much more efficienl han images wih a lo of (high-frequenc) deails; ) high-frequenc deails can be removed wih appropriae low-pass filering bu of course his inroduces blurring errors; ) aussian blurring can efficienl be de-blurred using he approach in 15

2 [1] and []. The whole quesion is: how do he loss of image quali inroduced b blurring he gain in compressibili inroduced b blurring and he gain in image quali inroduced b deblurring muuall relae? To analze each of hese facors we will refer o he hree differen processes menioned in Figure 1. A number of parameers and choices are o be made: 1) choice of he image; ) choice of he compression algorihm; ) compression raio () bi rae (bbp) or quali facor (Q) used during compression; ) amoun of blurring (σ b ); 5) order of deblurring (R); ) sigma value used for calculaing he derivaives o do he deblurring (σ d ). n his paper we will firs eamine he clues as o he success of he idea. Ne we will presen eperimenal resuls highlighing he imporance of each of he parameers. Following his we will discuss he resuls and presen conclusions and clues for fuure work. Fig. 1. Saring from an inpu image A he classical approach o compression is given in he firs branch: compression followed b decompression resuling in an image C which is an approimaion of A. The echniques inroduced in [1] are given in he second branch: aussian blurring wih variance σ b resuling in an image B followed b deblurring resuling in an image D. The approach proposed here is presened in he hird branch: pre-process he image o be compressed wih a blurring sep and pos-process wih a deblurring sep. E is he decompressed blurred image and F is he deblurred version of E. Moivaion n his secion we will analze each of he hree premises more in-deph. We will base our discussion on Figure 1 and presen some graphs supporing he ideas presened in he premises. n he presen paper we are no necessaril ineresed in he absolue performance of he compression algorihm bu in he improvemen ha he proposed pre- and pos-processing sep (blurring and de-blurring) could bring abou. Hence we chose for wo ver basic compression schemes. The firs one consiss of calculaing he Discree Cosine Transform (DCT) of he image and hresholding he coefficiens. The raio of image size o number of non-zero coefficiens can be aken as an approimaion o he. The second one is he plain JPE compression algorihm from ndependen 1

3 JPE roup (J) [] wih a quali facor Q o une he. Taking ino accoun ha mos processing was done in Malab all he images were convered o floaing-poin represenaion and rescaled o values beween 0 and 1. Firs premise: smooh images can be compressed much more efficienl han images wih a lo of (high-frequenc) deails. This makes a lo of sense since he sharp deails in he image produce man high-frequenc componens in he ransformed domain (be i DCT or DWT(Discree Waveles Transform)) hence man significan coefficiens which all have o be encoded. Second premise: high-frequenc deails can be removed wih appropriae lowpass filering bu of course his inroduces blurring errors. Simpl pu low-pass filering removes eacl he high frequencies ha ake up a lo of cos o encode. To illusrae how srong he effec is we refer o Figure (a) for DCT compression and Figure (b) for JPE compression where he classical ena image [5] was blurred wih successivel larger values of sigma and hen compressed - his in comparison wih he compression of he original ena image. Referring o Figure 1 we compare he compressed blurred image E wih he blurred image B and for he classical approach he compressed image C wih he original image A. The gain in compressibili is of course dependen on he image and he algorihm used and on he. n his case we noice a gain of o 15 db for σ 1 and 1 o 5 db for σ depending of he. vs vs (σ1) (σ) (σ) s1 s s (a) (b) Fig.. mproved compressibili of he ena image as i is being blurred wih aussian blurring wih: (a) DCT image compression algorihm (b) JPE image compression algorihm. Of course blurring also degrades he image quali. The severi of his degradaion is illusraed in Figure where were calculaed beween he original and blurred image in funcion of σ b. Referring o Figure 1 his means comparing he blurred image B wih he original image A. We noice a loss of 11 o 1dB for σ beween 8 and 1. 17

4 Fig.. oss of quali as a resul of aussian blurring in funcion of he amoun of blurring. Third premise: aussian blurring can efficienl be de-blurred. The approach proposed in [1] and [] is based on a Talor epansion of he image along he scale ais and following he epansion in for negaive scales i.e. de-blurring. This Talor epansion has he following shape: ( ) ( ) ( ) ( ) ( ) ( ) ; O whereb ( ; ) is he image wih and he spaial coordinaes and he scaling parameer. The derivaives o he scale can easil be calculaed using he diffusion equaion: Hence he scale derivaives can easil be calculaed as combinaions of spaial derivaives which in urn can be calculaed as convoluions of he inpu image wih he derivaive of he aussian: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ω iω i ( ) ( ) { } ( ) ( ) i 1 1 ω The deblurring process depends on wo parameers: he amoun of blurring used o calculae he derivaive operaors (which should logicall be as small as possible) and he order o which he Talor epansion is carried. n he case of rd order epansion (R ) we come o following epression wih derivaives up ill he h order (from he diffusion equaion we noice ha an n h order derivaive along he scale corresponds o a combinaion of spaial derivaives of order n): ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ; vs sigma sigma (1) () () () 18

5 Original image Blurred image wih σ Deblurred image wih R1 Deblurred image wih R Deblurred image wih R Deblurred image wih R8 Fig. 5. Comparison of he original ena image blurred ena image (sigma) deblurred ena 19

6 We now wan o evaluae he quali improvemen brough abou b his pe of deblurring depending on he amoun of blurring originall inroduced (σ b ) and on he order of he deblurring (R). Referring o Figure 1 his means comparing images D and A. We see ha as he order increases he quali of he deblurring improves wih roughl 1dB for R1 1.5dB for R db for R and.5db for R8. 1 vs sigma (sigma deblur) B-A R1 R R R8 Noe: R1 means D-A wih order 1 R means D-A wih order and so on. Fig.. mproved quali as a resul of deblurring aussian blur in funcion of he amoun of blurring for differen values of he order of he deblurring epressed in RMSE (lef) and (righ) sigma Puing i all ogeher i seems like we lose 1dB o 1dB b blurring bu improved compressibili gives us a gain of db o 5 db and deblurring can added 1dB o.5db resuling in a global gain of up o 1dB for σ and R as illusraed in Table 1. σ 1 σ σ Blurring -1dB -1dB -1dB Compression gain 5dB db 15dB 5dB 5dB 1dB deblurring 1dB o.5db Epeced gain -10dB o -8.5dB -1dB o -11.5dB db o.5db -8dB o -.5dB 1 o 1.5dB 1dB o.5db Table 1. Epeced lose and gain in in funcion of σ and R image (wih order R1 8). However i is o be noiced in his cone ha he reconsrucion procedure is ver sensiive o noise. n paricular since pracicall all digial images are saved in ineger forma he saving of he blurred image before deblurring would resul in a rounded-off image B which we will designae as ˆ. The deblurring of such a roundedoff image will be of much poorer quali - even becoming unsable for higher values of σ b and/or R as can be seen in Figure. n [] he auhor wisel remained wihin he low range of iniial blurring where no insabili is noiced unil a much higher order (according o his resuls unil a nd order). n his case referring o Figure 1 images ˆ and A are being compared. 170

7 vs sigma (rounded sigma deblur) 0 8 B-A R1 R R R8 Noe: R1 means ˆ -A wih order 1 R means ˆ -A wih order and so on. Fig.. Quali of de-blurring aussian blur afer rounding off he blurred image in funcion of he amoun of blurring for differen values of he order of he deblurring sigma All in all i seems ha one would loose 0.5dB o db in image quali bu gain 15 db in compressibili b blurring he images as pre-processing sep o compression while gaining anoher 1dB o db b de-blurring he image as a pos-processing sep. e us now see how i all fis ogeher. Resuls Referring o Figure 1 we will now follow he hird branch which is he complee processing: blur he image as pre-processing sep compress-decompress i and hen de-blur i. Figures 7 and 8 show he pical evoluion of he : a firs plo shows he compression performance wihou pre- or pos-processing (image C from Figure 1 compared wih image A); a second plo wih onl he pre-processing sep (image E compared wih image A); and a hird plo wih boh pre- and pos-processing (image F compared wih image A). vs 0 8 R1 R R R8 Noe: R1 means F-A wih order 1 R means F-A wih order and so on. Fig. 7. nfluence of pre- and posprocessing on he quali of he hresholded DCT as image compression algorihm

8 vs (sigma blur) vs (sigma blur) R1 R R Noe: R1 means F-A wih order 1 R means F-A wih order and so on. Fig. 8. nfluence of pre- and pos-processing on he quali of he JPE image compression algorihm. To evaluae he influence of he differen parameers on he resuls Figures 9 give he resuls of he modified compression algorihms (wih pre- and pos-processing i.e. image F compared o A) for differen values of he iniial blur (σ b ) and of he deblurring order (R) for boh compression algorihms. R1 R R vs (R1) vs (R) 0 s1 s s 0 s1 s s vs (R) s1 s s Noe: s1 means F-A wih sigma1 s means F-A wih sigma and s means F-A wih sigma.. Fig. 9. nfluence of pre- and posprocessing on DCT image compression algorihm for differen values of sigma (σ 1 ) in he preprocessing and differen values of he order in deblurring (R1 ) in he posprocessing. Discussion 17

9 The resuls seem o be discouraging: far from improving he quali of he decompressed image he de-blurring onl degrades i furher. The eplanaion for his can be found in Figure and in he JPE compression algorihm from J []. The deblurring process was shown o be sensiive o rounding off of he floaing-poin values of he blurred image. n our processing chain however TWO such quanisaion seps can be observed. Firs here is he unavoidable quanisaion of he DCT coefficiens. However for low (high Q values) i would be epeced ha hese rounding-off errors should be under conrol. However he resuls remain discouragingl poor. Second he JPE compression algorihm from J [] like mos of he compression algorihms available work wih ineger inpu images since - as noed earlier - mos images are saved under ineger forma. Hence we are faced wih a DOUBE quanisaion sep: one ha rounds off image B o ˆ and a second one in he compression mechanism iself. This double quanisaion is oo much and makes he deblurring algorihm alogeher unsable. To illusrae his poin he eperimen of Figure was re-done wih coarser and finer quanisaions. Figure showed he resuls of rounding off he values of B o 5 possible values before deblurring. Figure 10(a) shows he same process wih 10 oupu levels and Figure 10(b) wih levels. is clear ha coarser quanisaions lead o more insabili in he reconsrucion. vs sigma (rounded sigma deblur) vs sigma (rounder sigma deblur) B-A 8 B-A R1 R R R8 R1 R R R sigma (a) (b) Noe: R1 means F-A wih order 1 R means F-A wih order and so on sigma Fig. 10. Quali of de-blurring aussian blur afer rounding off he blurred image o (a) 10 inpu values (b) inpu values in funcion of he amoun of blurring for differen values of he order of he deblurring. 17

10 5 Conclusions seems a promising idea o improve a compression algorihm b pre-processing i wih a aussian blurring sep so as o improve compression performance and posprocessing i wih a aussian de-blurring process o reduce he error in he image. However he double quanisaion occurring firs a he inpu of he compression algorihm ne as a necessar sep in an loss compression algorihm seem o defea he epeced resuls. Two possible avenues should sill be eplored. Firs he modificaion of he compression algorihm so as o ake floaing poin images as inpu. This would eliminae he spurious quanisaion a he enrance of he compression algorihm. Anoher possibili would be he use of a lossless compression algorihm. However an lossless compression algorihm could onl work on ineger images. Hence he quanisaion sep inside he compression algorihm would be avoided bu no he one which round of he blurred image. B pursuing boh avenues i would be possible o deermine which of he wo quanisaion seps is he mos derimenal o he proposed pre- and pos-processing sep for improved image compression. References 1. Romen BMT Florack MJ Salden Ah Viergever Ma: Higher-order Differenial Srucure Of mages. mages And Vision Compuing 1(): 17-5 JU-AU (199).. Romen BMT: Fron-End Vision and Muliscale mage Analsis. Kluwer Academic Publishers (00).. John Sporring Mads Nielsen uc Florack and Peer Johansen (Eds.):aussian Scale-Space Theor. Kluwer Academic Publishers (1997).. [J]: hp:// (las access: 0 June 00) 5. ena images: hp:// (las access: 0 Januar 00). Sco E Umbaugh: Compuer Vison and mage Processing..Prenice-Hall (1998). 7. uc Florack: mage Srucure. Kluwer Academic Publishers (1997). 8. Milan Sonka Vaclav Hlavac Roger Bole: mage Processing Analsis and Machine Vision. Brooks/Cole Publishing Compan (1999). 9. Delores M. Eer: Engineering Problem Solving wih Malab. Prenice Hall (1997). 10. Alfons H.Salden Bar M. Ter Haar Romen and Ma A. Viergever: inear Scale- Space Theor from Phsical Principles. Journal of Mahemaical maging and Vision Kluwer Academic Publisher. 17