Image restoration using HOS and the Radon transform

IMAGE RESTORATION USING HOS AND THE RADON TRANSFORM Elisa Say ol*, Ch ysos omos L. Nikias*, and Toni Gad** Signal and Image P ocessing Ins i u e Uni e si y o Sou he n Cali o nia Los Angeles CA 90089-2564 el: (213) 7404654, ax: (213) 7404651 email: [email p o ec ed] *Depa men o Signal ~beo y and ~ommunica im. Uni e si a Polibica de Ca alunya 08080 Ba celona, Spain * i El3400 Apdo. 3O002 ABSTRACT^ We p opose he use o highe -o de s a is ics (HOS) o s udy he p oblem o image es o a ion. We conside images deg aded by linea o ze o phase blu ing poin sp ead i ne ions (PSF) and addi i e Gaussian noise. The complexi y associa ed wi h he combina ion o wo- dimensional signal p ocessing and highe -o de s a is ics is educed by means o he Radon T ans o m The p ojec ion a each angle is an one-dimensional signal ha can be p ocessed by any exis ing 1-D highe - o de s a is ics-based me hod. We apply wo me hods ha ha e p o en o a ain good one-dimensional signal econs uc ion, especially in he p esence o noise. A e he ideal p ojec ions ha e been es ima ed, he In e se Radon T ans o m gi es he es o ed image. Simula ion esul s a e p o ided. 1 INTRODUCTION Highe -O de s a is ics ha e been success ully applied be o e U, he p oblem o 2-D signal econs uc ion. In [l] he phase o he bispec um is used o econs uc images deg aded by a ji e y channel wi h addi i e one in e e ence. In [2] he bispec um is used o es ima e a andomly ansla ing and o a ing objec om a sequence o noisy images. Ano he applica ion is desc ibed in [3] whe e in a ian Highe -O de Spec a ea u es o p ojec ions a e used o objec classi ica ion. The Bispec um B lwl,w2) o a one-dimensional, de e minis ic. disc e e- ime signal in), is de ined as he Fou ie T ans o m o i s iple co ela ion unc ion and is gi en by [41 151 whe e F(w) is he Fou ie T ans o m o in). The phase o he Bispec um is he e o e his wo k was suppo ed by he O ice o Na al Resea ch unde con ac N0oO14-92-J-1034 and he Spanish Minis y o Educa ion and Science. dBpwpw2) = NWl) + W2) - Nq+w2)Q) whe e #(w)=.CF(w). The Bispec a, and HOS in gene al, a e insensi i e o addi i e Gaussian noise and p ese e he Fou ie phase o he signal up o a linea phase ac o . Ou goal is o exploi such p ope ies o he "a ion o 2-D blu ed and noisy images. We ha e de eloped echniques ha conside wo HOS-based me hods ha ha e p o en o a ain good 1-D signal econs uc ion. The m one is he Bispec um I e a i e Recons uc ion Algo i hm @IRA) desc ibed in [6] ha was u ilized o signal econs uc ion om he phase o he bispec um. Al hough i was applied o es o a ion o images, he p ocessing was made on a line by line basis. When he image is a ec ed by a ocus blu he dis o ing il e is 2-D and he econs uc ion o lines does no lead o good esul s. This me hod canno be ex ended o he wo dimensional case since i is based on he ceps al coe icien s ha can no be de ined in 2-D. Addi ionally, BIRA does no pe o m well when he Z- ans o m o he signal has ze os on he uni ci cle, unless an exponen ial window o he o m an mo es he ze os ou wa ds o inwa ds. An adequa e pa ame e U is usually di icul o ind. Fo una ely, as we will see, we can o e come bo h p oblems by econs uc ing om p ojec ions. The second app oach we s udy is he Weigh Slice me hod (WS) [7] ha eco e s he signal o in em e en in he case o ze os on he uni ci cle. Howe e , al hough possible, i s ex ension o 2-D will inc ease eno mously i s analy ical complexi y. We p esen an algo i hm using he WS me hod o e he p ojec ions o he image. In Sec ion 2 we desc ibe he imaging sys em ha we assume. We e iew he p ojec ion heo em ha allows he es o a ion o images om hei p ojec ions. In Sec ion 3 we de elop wo di e en me hods o he es o a ion ei he om he phase o he Bispec um o om he phase o he Fou ie T ans o m. In Sec ion 4 some examples a e gi en. Finally, Sec ion 5 is de o ed o conclusions and ema ks. 0-7803-1238493 $3.00 0 1993 IEEE 76 .. . . I_-. 2 PRELIMINARY DEFINITIONS To add ess he p oblem o eco e ing images om deg aded e sions o i , we assume a linea imaging sys em wi h space in a ian poin sp ead unc ion and addi i e Gaussian noise. Thus. o he con inuos model (3) whe e g(x,y) is he obse ed image, h(x,y) he poin sp ead unc ion, (sy) he o iginal image and n(x,y) he addi i e Gaussian noise. The 2-D signal eco e y p oblem can be uniquely decomposed in o many l-D signal econs uc ion p oblems. The Radon ans o m o a 2-D unc ion, lx,y), deno ed pe(s), is de ined as i s line in eg al along a line inclined a an angle 8 om he y-axis and a a dis ance s om he o igin. A undamen al ela ionship, he P ojec ion Theo em, ela es he l-D Fou ie ans o m o he p ojec ion pds), wi h he cen al slice, a angle e, o he 2-D Fou ie ans o m o he objec l y) gp&) = hp&) * ps(S)+Fl,S+G@ ) = H(e, F(6@ (4) whe e gpds). hpe(s), pds) a e he p ojec ions o g(x,y), h(x,y), (x,y) a angle 8, and G(e, ), H(e. ), F(8, ) a e he Fou ie ans o ms in pola coo dina es o g(x,y), h(x,y), and lx,y), espec i ely. We conside ha he Op ical T ans e Func ion (OF), he Fou ie ans o m o he PSF, has linea o ze o phase. As we see om (4). when he phase o he OTF is ze o, he phase o he l-D Fou ie ans o m o each p ojec ion G(O, ), co esponds o he phase o he Fou ie ans o m o he p ojec ion o he o iginal image, In mos applica ions, blu deg ada ions due o a numbe o sou ces a e conside ed o ha e linea o ze o phase. Fo example in [8] models o eal ou -o - ocus images a e in es iga ed. I is shown ha he phase is ei he ze o o linea . Fu he mo e, a unique disc e e ep esen a ion esul s in a PSF suppo ha co e s he minimum numbe o pixels and whe e he OTF has ze o phase. I is well known ha he disc e iza ion o he in e se o mula o he Radon T ans o m is no easily de ined. Finding accu a e and as algo i hms is a cu en opic o esea ch. On he o he hand, i he PSF suppo is no an in ege numbe , in e ms o numbe o pixels, some dis o ion occu s. Despi e hese p oblems, es o a ion om p ojec ions is s ill easible. F(e.5). 3 RESTORATION FROM THE PHASE OF THE PROJECTIONS I was shown in [9] ha a sequence o leng h N can be uniquely speci ied i he phase unc ion is known a (&I) dis inc equencies in he in e al O<w<z, excep o a cons an ac o . Likewise in [6], i is shown ha a sequence is uniquely speci ied i€ he phase o he bispecuum is known a N{N-l)n dis inc equencies in he egion O<WI+W~< z, W~<WI, wl 0, excep o a cons an ac o and a linea shi . This is ue unde he assump ions ha he Z- ans o m o he signal has no ze os in ecip ocal pai s. The e o e, i he p ojec ion o he blu ing il e has leng h da and he blu ed p ojec ion has leng h Lb he e is a unique signal o leng h Lbe = Le - (de -1) wi h he same phase unc ion (o phase bispec um) han he blu ed p ojec ion. This signal is no hing bu he p ojec ion o he o iginal image. We nex desc ibe wo HOS-based me hods ha can be used o he econs uc ion o he p ojec ions. Fo mo e de ails on he algo i hms see 161 and 171 3.1 Recons uc ion wi h BIRA As we men ioned be o e, BIRA econs uc s a signal om he phase o i s bispecuum. I employs he ceps al coe icien s ha o a signal pe(n) a e de ined as L, whe e aipa bipe a e he ze os inside and ou side he uni ci cle espec i ely, and Cipe a e he poles inside he uni ci cle o he Z- ans o m o pe(n). The las e m is no included o ini e sequences (L2 =O). The ceps al coe icien s a e ela ed o he phase o he bispec um as and o he powe ceps um as whe e Aipe is a cons an . I s ela ion o he signal is pe(n) = F1/eF/C P~m)l), n=O,..,N-1 (9) 77 Wh C m) = P e( -l/mA e(") D0 op m=O l/mB 8-m) md) P is he ceps um o he signal. The algo i hm s a s om he ue sequence o he phase o he bispec um o a leas N(N-1)/2 samples and an ini ial guess o he powe ceps um. I con e ges o a unique solu ion o he ceps al coe icien s and consequen ly o he signal, excep o a cons an ac o and a space shi . An addi ional p ocedu e is added o conml he space shi . The scale ac o is calcula ed aking in o accoun ha he p ojec ion is a posi i e unc ion and ha all he p ojec ions ha e he same a ea, he olume o he image. As we see in (5) and (6) in he case ha he e a e ze os on he uni ci cle, he ceps al coe icien s a e in ini e sequences. I s unca ion leads o inco ec solu ions. Two di e en al e na i es a e p oposed when p ojec ions wi h ze os on he uni ci cle a e de ec ed. A) When he numbe o such p ojec ions is low : In his case we es o e he image om he es o he p ojec ions since he omission o ew p ojec ions does no al e he es o a ion signi ican ly. B) When he numbe o such p ojec ions is high : We y o iden i y he coe icien s o he blu ing il e using he ollowing p ocedu e: B1) Recons uc he p ojec ions ha ha e nei he ze os on he uni ci cle no ecip ocal pai s. B2) F om (4) we see ha i we know gpB(s) and pe(s) we can ob ain hpHs) by decon olu ion. The sequence is he p ojec ion o he blu ing il e a an angle Band i s coe icien s a e weigh ed sums o he coe icien s o he blu ing il e . We build a sys em o equa ions o he ollowing o m xa=y, (11) whe e X is he ma ix o he weigh s o he coe icien s a he speci ied angles, a is he coe icien ec o and y is he ec o o he es ima ed blu ing il e p ojec ed a he speci ied angles. As an example suppose we ha e only he p ojec ion a angle B = 0"and a squa e 3x3 PSF wi h coe icien s named om a o i. Then, he sys em o equa ions will be 111000000 000111000 000000111 [ .I a b C d c 8 h i .I a+b+c d+c+ In gene al, we can sol e he sys em o equa ions b ind he Coe icien ec o i he ank o he ma ix is equal o g ea e han Nc. Using a LS app oxima ion he solu ion will be 21 = (XHX)-' XH y (12) B3) Once we know he blu ing il e coe icien s we can ei he decon ol e he 2-D noise- ee image o he Id noise ee p ojec ions. 3.2 Recons uc ion wi h he WS me hod We can uniquely econs uc he p ojec ions om he phase only o he Fou ie ans o m e en in he case ze os lie on he uni ci cle [9]. Howe e , in he p esence o noise he phase is deg aded. The e o e, we decompose he econs uc ion o a blu ed noisy p ojec ion in wo s eps. In he i s s ep, a me hod based on HOS educes Gaussian noise p ese ing he phase unc ion. We choose he WS me hod because i wo ks when hen? a e oo s on he uni ci cle and i has be e pe o mance han o he me hods. In he second s ep we deblu he signal om i s Ph. S ep 1) I is known ha o a causal and exponen ially s able sys em wi h inpu assumed o be independen , iden ically dis ibu ed and non Gaussian, wi h skewness p. he ou pu bispec um BX(wl,w2) exis s and is gi en by [31, [dl BX(wpw$ = B H(w1) WW~) H*(wl +w2) (13) As we see, inding he coe icien s o he il e H(w) is equi alen o es ima e he sequence F(w) in (1). The Weigh Slice WS) algo i hm has been p e iously used o ob ain he pa ame e s o a non-minimum FIR sys em. The pa ame e s can be exp essed as a linea combina ion o cumulan slices. I was shown ha i he e exis s a se o weigh s ha gi es a causal slice, hen he FIR sys em can be iden i ied. The sys em o equa ions is exp essed as SW = bo (14) whe e S is he ma ix o cumulan s s 1 C,is he es ima ed n h o de cumulan o he ou pu signal, w is he weigh ec o w = ( w2, w3cj), w4(j,k), ... ) (16) and bo is he coe icien ec o bo = (0 ... 0 I ... b(q-I) b(9)). (17) The unknowns a e he ec o w, and he las q elemen s The ma ix equa ion is sol ed in wo s eps : 1) Compu a ion o he minimum no m weigh s ha gi e a causal W-slice o bo. su wm = (0, .. , 0,I) wm = SU# 1 whe e Su is he ma ix o med om he uppe q+l ows o S, and Su# deno es he pseudo in e se o Sua 2) Compu a ion o he coe icien s as bo=Swm=SSu#l S ep 2) The ec o bo is he sequence o he noise- educed blu ed p ojec ion. Once bo has been calcula ed, we deblu he signal using a simila algo i hm han he one desc ibed in [9]. I is equi alen o BIRA, i s a s wi h he Vue sequence o he phase o he Fou ie T ans o m and an ini ial guess o i s magni ude. 4 EXAMPLES In he nex examples, he Radon T ans o m is calcula ed om an in e pola ion o he Ca esian sampling g id o he 2-D FFT o he image o a pola g id. The In e se Radon T ans o m is implemen ed in a e e se way. Fig. la shows a 24x24 image. The econs uc ed image a e applying Radon T ans o m ollowed by he In e se Radon ans o m is shown on Fig. Id. The dimensions o he 2-D FIT and 2-D IFFT a e 64x64 o a oid dis o ion in he high equencies. Fig. lb p esen s a blu ed e sion o his image when a 3x3 Gaussian il e wi h pa ame e a = 0.4 is used. Fig. le shows he es o ed image om he Fou ie phase o he p ojec ions. In Fig. IC he image is blumd wi h he same il e and Gaussian noise o SNR=25 dbs is added. The es o ed image is shown in Fig. I whe e he Ws me hod is applied using one slice only, he hi d o de cumulan is es ima ed hm 50 ealiza ions. Ano he example is shown in Fig. 2a and Fig. 24 whe e he o iginal image is 64x64 and he 2-D FFTs a e 256x128. Fig. 2b is a blu ed e sion o his image wi h a 5x5 Gaussian il e and a = 0.4. Fig. 2c is he es o ed image om he Fou ie phase. F om his image we see ha high equencies ha e been eco e ed. Howe e , we obse e some dis o ion. This is due o ela i e shi ing among p ojec ions. Fig. 3 shows a ypical p ojec ion, i s blu ed e sion, he econs uc ed signal and he di e ence be ween he o iginal and he es o ed signal. 5 CONCLUSIONS AND REMARKS In his pape , we ha e shown ha i is possible o es o e a blu ed and noisy image om HOS o i s p ojec ions. Two di e en app oaches we e gi en o econs uc he 1- D signals om ei he he phase o he Fou ie T ans o m o he phase o he bispec um. Al hough he complexi y using HOS o e 2-D signals is educed, he compu a ional load is s ill high. Some o he applica ions o HOS o e he p ojec ions a e unde in es iga ion. REFERENCES [l] S.A. Diana , M. R. Raghu ee . “Fas Algo i hms o phase and magni ude econs uc ion om bispec a”, Op ical Enginee ing, May 1990. [2] B.M.Sadle , “Shi and Ro a ion In a ian Objec Recons uc ion using Bispec um”, Wo kshop on HOS analysis, Vail, Colo ado, I989,pag 106 [3] V. Chand an, S. Elga , “Posi ion. o a ion, and scale in a ian ecogni ion o images using Highe -o de spec a”, P oc. ICASSP ‘92, S. F.,pp -213-216. [4] C. L. 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Id Fig. le Fig. I Fig. 2a Fig. 2b Fig. 2c e) RcmmWcIed om b wFl 0 ...... ... -50 . 02l4060~ Fig. 2d 80 Fig.3