Signal reconstruction algorithm based on a single intensity in the Fresnel domain

Transcription

1 Signal recnstructin algrithm based n a single intensity in the Fresnel dmain Hne-Ene Hwang Deartment Electrnic Engineering, Chung Chu Institute Technlgy, Yuan-lin 510, Changhua, Taiwan Pin Han Institute Precisin Engineering, atinal Chung Hsing University, Taichung 40, Taiwan in@dragn.nchu.edu.tw Abstract: A nvel algrithm that can recnstruct a symmetrical signal (bth the amlitude and the hase inrmatin with nly a single Fresnel transrm intensity is rsed. A new cmlex-cnvlutin methd is intrduced, which is needed in the algrithm. The essential rerties the discrete Fresnel transrm are resented as well. umerical results shw that this methd can successully rebuild the signal rm ne signal intensity, which is mre advantageus in seed and eiciency than the cnventinal methd that requires tw intensities t accmlish this task. 007 Otical Sciety America OCIS cdes: ( Otical data rcessing, ( Digital image rcessing, ( Otical rcessing. Reerences and links 1. H. A. Ferwerda, The hase recnstructin rblem r wave amlitudes and cherence unctins, in Inverse Surce Prblems in Otics, H. P. Baltes, ed. (Sringer-Verlag, Berlin, R. W. Gerchberg and W. O. Saxtn, A ractical algrithm r the determinatin hase rm image and diractin lane ictures, Otik 35, ( R. W. Gerchberg and W. O. Saxtn, Phase determinatin r image and diractin lane ictures in the electrn micrsce, Otik 34, ( P. Van Tn and H. A. Ferwerda, On the rblem hase retrieval in electrn micrscy rm image and diractin attern. IV. Checking algrithm by means simulated bjects, Otik 47, ( Z. Zalevsky and R. G. Drsch, Gerchberg Saxtn algrithm alied in the ractinal Furier r the Fresnel dmain, Ot. Lett. 1, ( W. J. Dallas, Digital cmutatin image cmlex amlitude rm image and diractin intensity: an alternative t hlgrahy, Otik 44, ( W. Kim and M. H. Hayes, Phase retrieval using tw Furier-transrm intensities, J. Ot. Sc. Am. A 7, ( akajima, Phase retrieval rm tw intensity measurements using the Furier series exansin, J. Ot. Sc. Am. A 4, ( W. X. Cng,. X. Chen, and B. Y. Gu, Phase retrieval in the Fresnel transrm system: a recursive algrithm, J. Ot. Sc. Am. A 16, ( C. Sng, R. J. Damien, Y. ishin, Y. Khmura, T. Ishikawa, C. C. Chen, T. K. Lee, and J. Mia, Phase retrieval rm exactly versamled diractin intensity thrugh decnvlutin, Phys. Rev. B. 75, 0110 ( M. A. Peier, G. J. Williams, I. A. Vartanyants, R. Harder, and I. K. Rbinsn, Three-dimensinal maing a dermatin ield inside a nancrystal, ature 44, ( J. Mia, P. Charalambus, J. Kir, and D. Sayre, Extending the methdlgy X-ray crystallgrahy t allw imaging micrmeter-sied nn-crystalline secimens. ature, 400, ( J. Mia, D. Sayre, and. H. Chaman, Phase retrieval rm the magnitude the Furier transrm nneridic bjects, J. Ot. Sc. Am. A. A ( Y. M. Bruck and L. G. Sdin, On the ambiguity the image recnstructin rblem, Ot. Cmmun. 30, ( M. H. Hayes, The recnstructin a multidimensinal sequence rm the hase r magnitude its Furier transrm, IEEE Trans. Acust., Seech, Signal Prcess. ASSP-30, (198 (C 007 OSA Aril 007 / Vl. 15,. 7 / OPTICS EXPRESS 3766

2 1. Intrductin In mst tical systems, the measured inrmatin is usually the intensity the image (signal r its diractin attern. Hence, the substantial inrmatin encded in the hase vanishes. Recnstructing bth the magnitude and the hase inrmatin rm the measured intensity in an tical ield has been an imrtant widely investigated issue [1-8]. Mst the revius methds were based n the tw measured intensities r diractin atterns in the Furier transrm dmain. Fr examle, Gerchberg and Saxtn [] rsed an iterative hase-retrieval algrithm that bunces back and rth between tw lanes that have the Furier transrm relatin t each ther. Cng et al. [9] rsed a dierent recursive algrithm t errm the hase retrieval rm tw intensities in the Fresnel transrm system. This methd can er a multiarius chice tw intensities at arbitrary lcatins in the reesace diractin lanes and minimie the hardware requirement t imlement it withut the lenses needed in the Furier dmain. Als the versamling hasing methd was rsed and has been successully alied t the cherent X-ray micrscy r D r 3D images nancrystal r nncrystalline secimens [10-1]. Hwever, all arementined algrithms still need tw intensities even in dierent transrm systems. Fr this reasn, a simler and mre eicient algrithm that needs nly ne single Fresnel transrm intensity t recnstruct the signal (bth the hase and the magnitude inrmatin is resented. In the llwing analysis we cnsider, r simliicatin, the case ne dimensinal unctin and ne-dimensinal transrm. The generaliatin t tw dimensinal searable unctins is straightrward.. Thery The Fresnel transrm ( FrT a unctin ( x is given by iπ ex Z Z λ iπ FrT [ ( x ] ( ( ex ( = F x = x x x d x, iλ (1 where is the ragatin distance, λ is the wavelength, x is the variable the crdinate the inut lane, and x is the variable the crdinate the utut lane. As illustrated in Fig. 1, the tical ield in the utut lane xy is F ( x, but nly the intensity F ( x is recrded. The aim is t rebuild the signal ( x (bth the hase and the magnitude inrmatin rm nly ne F ( x. y y x Incident light x s ignal F resnel dmain Fig. 1. The cniguratin t illustrate the Fresnel transrm. # $15.00 USD Received 31 January 007; revised 3 March 007; acceted 3 March 007 (C 007 OSA Aril 007 / Vl. 15,. 7 / OPTICS EXPRESS 3767

3 In mst the realistic tical systems, the inut and utut signals can be regarded as being arximately bth sace limited and band limited. Assume the signal extensin is Δ x and there are samle ints with interval δ x = Δ x. This must satisy the yquist samle thery that δ x is less than the inverse twice the signal s bandwidth because the hase inrmatin is uniquely embedded inside the diractin attern when the attern is samled at a sacing iner than the quist requency (i.e. versamling [13-15]. The sequence { ( mδ x0 : m =,, 1} can be generated rm the cntinuus ( x. Cnsequently, we deine the discrete Fresnel transrm ( DFrT ( mδ x as / 1 m= / DFrT [ ( mδx ] = F ( nδx = δx κ( m, n, ( mδx, ( where δ and δ x x κ ( m, n, is the kernel the are the samling erid in x and x sace, m and n are integers, and DFrT as iπ ex λ iπ ( m, n, ex ( m x κ = δ nδx. i (3 The inverse DFrT ( IDFrT is given by / 1 ( δ δ κ (,, ( δ, n= / m x = x m n F n x (4 where δx = ( δx, and the suerscrit dentes the cmlex cnjugatin. The crrelatin rerty r DFrT can be reresented, rm Eqs. (-(4, as llws [9] Z / 1 m= / i πmk( δx ( ( + ex[ ] δ x / 1 iπ( kδx iπ nk = ex[ ] F ( n ex(, δx m m k n= / (5 where ( m and F ( n dente ( mδ x and F ( nδ x, resectively. We nw cnsider the rblem recnstructing a unctin ( m rm nly the single intensity its discrete Fresnel transrm F ( n. In a discrete transrm system, the er adding skill is emlyed t avid the time aliasing which results rm the eridic (circular crrelatin. In rder t btain the crrect linear crrelatin relatin, we set the er adding rm ( m as ( k = 0 r k =,, 1,,, 1, (6 # $15.00 USD Received 31 January 007; revised 3 March 007; acceted 3 March 007 (C 007 OSA Aril 007 / Vl. 15,. 7 / OPTICS EXPRESS 3768

4 where the riginal data are still stred in k =,, 1. With the hel Eqs. (5 and (6, the crrelatin rerty can be rewritten as k 1 m= i πmk( δx ( ( ex[ ] m m+ k δ x 1 iπ( kδx iπ nk = ex[ ] F ( n ex(, δx n= (7 where δx = ( δx, and let Rk ( be the entire right-hand side Eq. (7 as δ x 1 iπ( kδx iπ nk Rk ( = ex[ ] F ( n ex(. (8 δx n= I the autcrrelatin lag is set as k = 1, rm Eqs. (7 and (8, we have iπ( 1( δx ( ( 1 = R( 1ex[ ]. (9 ext, r the lag k =, iπ( ( δx ( ( ex[ ] + iπ( ( ( δx ( + 1 ( 1ex[ ] = R (. (10 Similarly, we derive the relatin, r the lags k = m r m 3, as iπ( m( δx ( ( ex[ + iπ( m( j( δx ( + ( + ex[ ] + m m j = 1 j m j iπ( m( m+ ( δx ( + 1 ( 1 ex[ ] m (11 = R ( m. We can srt and arrange all the rduct actrs withut the hase term r dierent lags k = m as in art A belw. Each rw dentes the same lag. Fr examle, the irst rw is r k = 1 ; the secnd rw is r k =, and s n. The symbl A(i,j (the actr in ith rw, jth term is used t hel identiying the actr; r examle, A(,1 is the rduct actr ( ( because it is n the secnd rw and the irst term. It is bvius that r all rduct actrs, nly ( ( 1 [=A(1,1] in the irst rw can be btained directly rm Eq. (9; thus the ther actrs (nt t mentin the # $15.00 USD Received 31 January 007; revised 3 March 007; acceted 3 March 007 (C 007 OSA Aril 007 / Vl. 15,. 7 / OPTICS EXPRESS 3769

5 sequence{ ( m: m =,, 1 } can nt be und using Eqs. (10 and (11. This is why the revius study claimed that the hase retrieval algrithm needs tw intensities r diractin atterns t recnstruct the signal [9]. Part A k = 1 ( ( 1 A(1,1 k = ( (, ( + 1 ( 1 A(,1, A(, ( ( 3, ( + 1 (, ( + ( 1 ( (1,.. ( k = /+ 1.., ( ( 1 ( (0, ( + 1 (1,...( k = /., ( (, ( 1 ( 1 ( ( 1, ( + 1 (0, ( k = / 1, ( 1 (, (0 ( 1 T create mre relatins r the rduct terms t recnstruct the signal sequence, a rerty called the cmlex-cnvlutin is develed as + k m= i πmk( δx ( ( ex[ ] m k m δ x 1 iπ( kδx iπ nk = ex[ ] F ( n ex(. δx n= (1 The abve equatin can be btained rm Eq. (7 and using the symmetrical rerty the signal ( ( m = ( m. Setting k = in Eq. (1, and let R (k be the entire right-hand side Eq. (1 as δ x 1 iπ( kδx iπ nk R ( k = ex[ ] F ( n ex(. (13 δx We have the llwing equatin as n= iπ( δx ( ( = ( ex[ ] R = (. (14 ext, r k = + 1, # $15.00 USD Received 31 January 007; revised 3 March 007; acceted 3 March 007 (C 007 OSA Aril 007 / Vl. 15,. 7 / OPTICS EXPRESS 3770

6 iπ( + 1( δx ( ( + 1ex[ ] + iπ( ( + 1( δx ( + 1 ( ex[ ] (15 = R ( + 1. Fr the lags k = + m with m, the generalied exressin is m m 1 j = 1 iπ( m( δx ( ( + ex[ + iπ( m( j( δx ( + j ( + m jex[ ] + iπ( m( m( δx ( + m ( ex[ ] = R ( + m. (16 As we d in art A, rm Eqs.(14-(16, the cmlex-cnvlutin actrs withut hase term can be srted and arranged r dierent lags k = + m as in art B belw. Each rw dentes the same lag, such as the irst rw r k =, secnd rw r k = + 1, and s n; and the symbl B(i,j is used t identiy the indicated term. Again, nly B(1,1 (= ( ( in the irst rw can be btained directly by Eq.(14, but these rduct actrs are very imrtant r ur aim as exlained later. Part B k = ( ( B(1,1 k = + 1 ( ( + 1, ( + 1 ( B(,1, B(, ( ( +, ( + 1 ( + 1, ( + ( ( ( + 3, ( + 1 ( +, ( + ( + 1, ( + 3 ( 3. The recursive algrithm w, we intrduce the signal recnstructin algrithm and the rcedures are exlained as in the llwing. Ste 1: The exact values A(1,1 (= ( ( 1 and B(1,1 (= ( ( are calculated rm Eqs. (9 and (14. It is und that B(,1 (= ( ( + 1 is equal t A(1,1 (= ( ( 1 because ( m is symmetrical and the cnditin ( + 1 = ( 1 is satisied. Ste : It is als nted that B(, = B(,1; therere the cmlex cnjugate eratin can be used t gain the actr B(, as r ( + 1 ( = [ ( ( + 1]. Ater that, the actr # $15.00 USD Received 31 January 007; revised 3 March 007; acceted 3 March 007 (C 007 OSA Aril 007 / Vl. 15,. 7 / OPTICS EXPRESS 3771

7 B(3, (= ( + 1 ( + 1 in third rw art B can be calculated using the irst tw rws as ( B(3, = B(,1 B(,/B(1,1, r exlicitly as ( + 1 ( + 1 = ( ( + 1 ( + 1 (. (17 ( ( With the hel the symmetrical cnditin ( + 1 = ( 1, A(, is equal t the B(3,. Ste 3: A(,1 (= ( ( can be btained when A(, (= ( + 1 ( 1 is substituted in Eq. (10. Once this is und, the actrs B(3,1 (= ( ( + and B(3,3 (= ( + ( in art B can be btained immediately r the symmetrical and cnjugatin rerties (A(,1=B(3,1= B(3,3 as mentined abve. The actrs in the third rw art B are knwn and then cntinue the same rcedure t ind the actrs B(4, (= ( + 1 ( + and B(4,3 (= ( + ( + 1 in the rth rw art B as B(4, = B(3,1 B(3,/B(,1 and B(4,3 = B(3, B(3,3/B(,. Return these tw values int the third rw art A (A(3, = B(4,, A(3,3 = B(4,3 and utilie Eq. (11 r k = 3 t slve the actr A(3,1 (= ( ( 3 in the third rw art A because A(3, and A(3,3 are already knwn. This technique can be used reeatedly and the algrithm lw chart is deicted in Fig.. This recursive rcess cntinues until the value k = ( + 1 is reached, which is the last rw (the k = 1 rw as shwn in art A triangle and every rduct term in art A is und. Finally we can take the rduct the tw knwn terms ( (0 and (0 ( 1 (they are the irst tem in k = rw and the last term in k = 1 rw as shwn in art A triangle resectively and use Eq. (9 t get. π δ ( (0 (0 ( 1 (0 ( 1ex[ ]. i ( 1( x = R (18 Therere (0 can be determined using the abve equatin, and nly its hase is ree. Chsing an arbitrary value r its hase des nt have any bservable eect n the recvered signal since it can be used as a reerence. w (0 is knwn and reer t the irst term and last term rm bttm t head in art A. We can divide ( (0 (the irst term in k = by (0 t btain ( (thus (. And we can als divide (0 ( 1 (the last term in k = 1 rw by (0 t btain ( 1. Fr the same reasn, the value (1 can be und when ( (1 (the irst term in k = + 1 rw is divided by (. Als the value ( 1 (thus ( 1 is und when (1 ( 1 (the last term in k = rw is divided by ( 1. This rcess is reeated and all the sequence [ ( m : m =, +,, 1 ] can be calculated. The signal is thus recnstructed. # $15.00 USD Received 31 January 007; revised 3 March 007; acceted 3 March 007 (C 007 OSA Aril 007 / Vl. 15,. 7 / OPTICS EXPRESS 377

8 Find ( ( by Eq.(14 S et initial I = 0 F ind ( ( I 1 by Eq.(9 ( ( Find ( + I + 1 = ( ( I 1 + I + 1 ( = [ ( ( + I + 1] I = I + 1 Set II = 1:1:I + 1, ind ( + II ( II + I + = ( + II 1 ( II + I + ( + II ( II + I + 1 ( + II 1 ( II + I + 1 and ( + II ( + II I = ( + II ( II + I + Find ( ( I by Eq.(10 r I = 0 r by Eq. (11 r I 1 I I = + 1? Yes End Fig.. Signal recnstructin algrithm based n a single Fresnel transrm intensity 4. umerical results A symmetrical signal ( x = ex[ 16x + j0.5sin(64 π x] is emlyed as an examle r demnstrating the racticability the rsed algrithm. The cnditins r wavelength λ = 63.8 nm and the ragatin distance = 0.5 m are taken and the number samling ints is = 64. Only ne intensity F ( x is used in ur algrithm t recnstruct ( x. The recvered magnitude and hase inrmatin are indicated in Figs. 3(a and 3(b, resectively. In Fig. 3, the slid curves crresnd t the riginal signal ( x. The en circles indicate signal recvered by ur recursive methd. Frm the results, it is evident that this algrithm errms very well and the recnstructed signal is almst the same as the riginal. This algrithm is als alied t anther mre cmlicated tw-dimensinal signal with the rm ( x, y = ex[ 10( x + 3 y + j0.sin(48 πx cs(3 π y], which has a D Gaussian unctin but with dierent riles and chir requencies in each directin. Figs. 4(a, 4(b and 4(c and 4(d are the riginal and rebuilt signals resectively. The rebuilt signal is still cnsistent and well agrees with the riginal ne. The nise eect n ur algrithm is als studied. Fig. 5 shws the inluence the nise t ur retrieved signal when dierent extent randm nise is added n ur signal ( x = ex[ 16x + j0.5sin(64 π x] which is used as an examle in Fig. 3. It can be und rm the Figs. 5(a and 5(b that the recvered signal is gd r signal t nise ratin (S/ is equal t 10, but it wuld have large errrs when the S/ is equal t 3. Thus this methd has gd nise resistance rerty. # $15.00 USD Received 31 January 007; revised 3 March 007; acceted 3 March 007 (C 007 OSA Aril 007 / Vl. 15,. 7 / OPTICS EXPRESS 3773

9 x x Fig. 3. umerical results the rsed recursive algrithm. (a The magnitude the recvered signal (the en circles and the magnitude riginal signal ( x (the slid curve; (b The hase the recvered signal (the en circles and the hase riginal signal ( x (the slid curve. x (a y x (b y x (c y x (d y Fig. 4. umerical results the rsed recursive algrithm r a tw dimensinal signal ( x, y = ex[ 10( x + 3 y + j0.sin(48 πx cs(3 πy]. (a The magnitude the riginal signal (b The hase the riginal signal (c The magnitude the recvered signal (d The hase the recvered signal. # $15.00 USD Received 31 January 007; revised 3 March 007; acceted 3 March 007 (C 007 OSA Aril 007 / Vl. 15,. 7 / OPTICS EXPRESS 3774

10 (a x (b x (c x (d Fig. 5. umerical results the rsed recursive algrithm with dierent extent randm nise. (a and (b: The magnitude and hase the riginal signal (the slid curve and the recvered signal (the en circles with S/ = 10 (c and (d: The magnitude and hase the riginal signal (the slid curve and the recvered signal (the en circles with S/ = 3. x # $15.00 USD Received 31 January 007; revised 3 March 007; acceted 3 March 007 (C 007 OSA Aril 007 / Vl. 15,. 7 / OPTICS EXPRESS 3775

11 5. Summary The discrete Fresnel transrm ( DFrT and its essential rerties were resented in this aer. A new cncet called cmlex cnvlutin relatin was intrduced. Using bth the crrelatin and the cmlex cnvlutin relatins, we develed an imrved recursive algrithm that can recnstruct a symmetrical signal using nly ne Fresnel transrm intensity. The numerical results shwed that this scheme is successul and bth the signal s magnitude and hase inrmatin can be rebuilt very well. The advantages this methd are that it is very eicient and nly ne Fresnel-intensity is required, but it can nly be alied t symmetric signals r igures. Cmaring it with the versamling hasing methd [10-1], the latter can be alied t any rms images r bjects and nly ne Furier-transrmintensity needed which is easier t btain the Fraunher image r the X-ray diractin micrscy, althugh it is still an iterative methd and sme cnstraints are needed. This algrithm can be nly alied t a symmetrical signal. Hwever, any signal and its wn mirrr signal can be synthesied int a symmetrical signal, making this methd racticable. As r the seed ur recursive algrithm, the authrs have estimated careully the amunt the arithmetic eratins invlved in ur rgram. We cmare it t that anther revius recursive wrk [9] and und that ur seed shuld be at least twice aster. The cmutatin time n a ersnal cmuter als shwed the same trend r these tw algrithms. Cmared with revius wrks, the rsed methd is advantageus in sme ways; such as, n lenses are needed in the rcess and this methd is eicient and ecnmical t imlement because nly ne intensity is necessary. Acknwledgments This study was surted by Chung Chu Institute Technlgy and the natinal Chung Hsing University. Als it was surted by the atinal Science Cuncil the R.O.C under the cntract. SC 95-1-E and SC 95-1-E # $15.00 USD Received 31 January 007; revised 3 March 007; acceted 3 March 007 (C 007 OSA Aril 007 / Vl. 15,. 7 / OPTICS EXPRESS 3776