Multiple Plane Phase Retrieval Based On Inverse Regularized Imaging and Discrete Diffraction Transform

Transcription

1 Mutipe Pane Phase Retrieva Based On Inverse Reguaried Imaging and Discrete Diffraction Transform Artem Migukin, Vadimir Katkovnik, and Jaakko Astoa Department of Signa Processing, Tampere University of Technoogy (TUT), Korkeakouunkatu 1, P.O.Box 553, Fi-337, Tampere, Finand Abstract. The phase retrieva is formuated as an inverse probem, where the forward propagation is defined by Discrete Diffraction Transform (DDT) [1], []. This propagation mode is precise and aiasing free for pixewise invariant (pixeated) wave fied distributions in the sensor and object panes. Because of finite sie of sensors DDT can be iconditioned and the reguariation is an important component of the inverse. The proposed agorithm is designed for mutipe pane observations and can be treated as a generaiation of the Gerchberg-Saxton iterative agorithm. The proposed agorithm is studied by numerica experiments produced for phase and ampitude moduated object distributions. Comparison versus the conventiona forward propagation modes such as the anguar spectrum decomposition and the convoutiona mode used in the agorithm of the same structure shows a cear advantage of DDT enabing better accuracy and better imaging. Keywords: phase retrieva, digita image processing agorithms, computer generated hoograms PACS: 4.3.Rx, 7.5.Pj, 4.4.Jv INTRODUCTION The phase of radiation scattered from an object carries important information about an object surface and its properties. Phase measurements are expoited in many fieds in materias and bioogica sciences. Because ony the magnitude (intensity) of radiation can be measured directy the probem of phase reconstruction from intensity measurements appears. Soutions fa into two arge categories: interferometric (hoographic) with a reference beam and non-interferometric (phase-retrieva), where a reference beam is not required. The atter soution is substantiay simper and, what is important, much more robust with respect to disturbances. The method proposed by Gerchberg and Saxton [3] is the first popuar beam-propagation-based phase reconstruction method. The idea is that the phases missing in observations are recovered iterativey appying the magnitude constraints in object and sensor panes. This technique has been studied, modified and deveoped in a fow of pubications (e.g. [4], [5], [6]). In this paper we consider a singe and mutipe pane observation scenarios where the magnitudes of the wave fied are measured in the panes parae to the object pane. It is assumed that the wave fied is generated by radiation from the object pane. The reconstruction of the phases missed in observations is produced through the object pane distribution considered as the ony unknown variabe. The proposed iterative agorithm is a generaiation of the Gerchberg and Saxton agorithm for mutipe pane observations and inverse imaging techniques. In this work we are restricted to numerica experiments and the forward wave fied propagation modeing based on the discrete diffraction transform (DDT) deveoped in two compementary forms: in the frequency (Fourier) domain (F-DDT) [1] and in the spatia domain as the matrix discrete diffraction transform (M-DDT) []. By its very nature this transform enabes the perfect forward propagation modeing for the pixeated wave fied. The "pixeated" assumes that the wave fied distributions are pixewise invariant in the object and sensor panes. This assumption is natura for a sorts of digita sensors and used as a pixewise approximation for the object pane. In numerica modeing we aways dea with the pixeated objects and in this way we can caim that at east for numerica experiment this wave fied modeing is accurate. The contribution of this paper concerns a demonstration that the deveoped agorithm enabes a better accuracy in comparison with the standard ones and better visuaiation. In this comparison we study as vauabe aternatives the agorithms using the anguar spectrum decomposition (ASD) and discrete convoution (DC) as modes for the

2 forward propagation modeing. We consider the reconstruction of the object wave fieds with ampitude or phase moduations. It is shown that the spatiay adaptive reguariation used in the inverse imaging resuts in the further improvement of the agorithm performance. y u [ ] u [ ] u [ ] [k] k 1 k k u x FIGURE 1. Mutipe pane phase (wave fied) reconstruction: u is an object pane, u are measurement panes, =1,,. 1 et u (x) and u (x), x OBSERVATION MODES R, =1,...,, denote the compex-vaued wave fied distributions in the object and sensor panes respectivey. indicates a distance between the parae object and -th panes, and is a number of the observation panes (sensors) (see Fig. 1). The pixeated modeing means that the continuous arguments x are repaced by integer ones with the foowing repacements of the continuous distributions by their discrete modes: u (x) u [k], u (x) u [k], k Z. Thus, the argument k is a two dimensiona vector with integer components, i.e. beongs to the set Z. For a finite sie square image it means that k=(k₁,k₂), k₁,k₂ N-1. In DDT the vaues u [k], u [k] are defined as the mean vaues of the corresponding continuous variabes with the means cacuated over the square or rectanguar pixes. The ink between these u [k] and u [k] is accurate for the pixewise object distribution and the discrete sensor. In the frequency domain this ink can be given in the form [1] U [ f ] A [ f ] U [ ], f where Ũ [f] and Ũ [f] are cacuated as the fast Fourier transform (FFT) of ũ [k] and ũ [k], f (1) Z and the transfer function Ã, [f] is FFT of the pixewise averaged vaues of the kerne of the Rayeigh-Sommerfed integra. The cacuation of Ã, [f] in [1] is produced for the Fresne approximation of this kerne. In what foows ũ [k] and ũ [k] are extended doube sie versions of u [k] and u [k] ero padded to the doube sie of the images, and a frequency domain operations are produced for the doube sie images. There are two reasons for these doube sie operations. First, we need this doube sie in order to the DDT forward propagation be accurate. u [k] is cacuated from the doube sie inverse FFT ( u FFT 1 { U [ f ]}) by taking the midde part of this doube sie image. Remind that these sort of doube sie cacuations are typica for accurate convoutiona techniques (e.g. [7]). Second, the extra area of images appeared due to ero padding is essentia for the recursive inverse performed in the frequency domain [1]. Concerning f in (1) note, that it is a vector f=(f₁,f₂), -N f₁,f₂ N-1. The integer D reguar grid for f is circuar-shifted in such a way that the origin point f₁=, f₂= is in the eft upper corner of the grid. The frequency modes simiar to (1) can be used for a forward propagation modes. Here we mention ony two of them used in our numerica experiments. The transfer function for ASD is cacuated anayticay as foows [8]: ASD, [ f ] exp[ j / 1 f /( N) ], /( N) 1 The transfer function for the discrete convoutiona (DC) mode is cacuated as [7]: DC, [ f ] FFT{ g [ k ]}, g ( x) / r [1/( j ) 1/( r )] exp( j r / ), r x In these formuas λ is the waveength, and Δ is a parameter of the pixe sie (we assume that the pixes are square Δ Δ). The transfer functions ()-(3) are expoited for cacuations of the spectrum in the sensor panes based on the formua (1). These cacuations can be produced for doube and singe sie images. () (3)

3 PHASE RETRIEVA AGORITHM et us assume for a moment that the compex-vaued u are known. Then the reconstruction of u can be produced in the frequency domain according to the foowing optimiation formuation: U [ f ] arg min J, J U [ f ] A [ f ] U [ f ] U [ f ], U [ f ] 1 where the Eucidean norm means Ũ [f] =Σ f Ũ [f] and α² is a reguariation parameter. The routine cacuations give the estimate of Ũ [f] in the form * U [ f ] ( A [ f ] U [ f ]) /( A [ f ] ) 1 1 where '*' stands for a compex-conjugate variabe. The criterion in (4) corresponds to the standard Tikhonov's quadratic reguariation of the i-conditioned inverse probems [9]. We wi use the formua (5) in order to derive the agorithm for the case, when ony magnitude data are avaiabe instead of the compex vaued one. et the observations in the sensor panes be given as o u, where ε [k] are measurement noises. Substitute these noisy magnitude observations in (5) instead of u, then we arrive to the foowing expressions: U [ f ] ( 1 A * v [ o k] ange( u ), u [ f ] FFT{ v }) /( u (1 1 FFT { A 1 k])] exp( j ), [ f ] U [ f ]}. Here k] is the indicator of the image area, i.e. k]=1 for the image area and equa to otherwise. Then the formua õ [k] k]+ ũ [k] (1-k]) repaces the magnitude of u [k] by the observations o [k] in the midde part of the doube sie image and preserves the magnitudes of ũ [k] cacuated for the outside of this midde part. The formuas in (7) are equations with respect to unknown u [k]. These equations are noninear because the phases in the observation panes depend on u [k]. The foowing iterative procedure is natura for these equations and defines the proposed iterative agorithm: { ( t 1) * U [ f ] ( A [ f ] FFT v }) /( A [ f ] ), ( v t) 1 [ o k] ange( u ), u u (1 1 FFT { A A 1 [ f ] ), k])] exp( j ), [ f ] U [ f ]}, t,1,... We name this agorithm Mutipe Pane Frequency DDT (MF-DDT). This iterative procedure can be used aso with the ASD and DC frequency domain modes (transfer functions () and (3)). The cacuations become simper because in this case Ã, [f] = 1, Ã, * [f] = Ã -, [f], and we do not need reguariation. We wi refer to the agorithms generated from (8) as to the ASD or DC agorithms depending on which forward propagation mode is used. It is not difficut to reaie that the agorithm (8) can be treated as a generaiation of the Gerchberg-Saxton agorithm for mutipe observation panes and the DDT forward propagation requiring the reguaried inverse for the backward propagation. The singe-beam mutipe-intensity phase reconstruction (SBMIR) technique using ony the intensity measurements of a voume specke fied and wave propagation equation has been proposed and studied in [1], [11]. The phase distributions for the observation panes are reconstructed successivey in circuar iterations from one observation pane to the foowing one starting from the first observation pane. The agorithm (8) is different from SBMIR in principe because the observations from a panes are processed simutaneousy with estimation of u as the ony unknown variabe. The simuation confirms the advantage of this parae processing of a observations. A further deveopment of the proposed approach is produced by using a varying spatiay adaptive reguariation instead of the simpe Tikhonov`s one. It is shown in [1] that this sort of reguariation can be impemented as (4) (5) (6) (7) (8)

4 spatiay adaptive fitering. In the deveoped modification of the agorithm (8) the phase and the magnitude of each iterative estimate ũ (t) [k] are subjects of the specia fitering. For this fitering we use the powerfu adaptive BM3D agorithm [13]. Simuation demonstrates an essentia improvement of these phase and magnitude estimates visuay and numericay. NUMERICA EXPERIMENTS We tested our agorithm in mutipe experiments with ampitude (AM) or phase (PM) moduated object distributions. The images are square N N, N =56 with square pixes Δ Δ, Δ =6.7μm, the waveength λ =63.8nm. The "in-focus" distance is cacuated as f = NΔ²/λ =18.16mm. It is assumed that the additive noise in (6) is eromean Gaussian with σ=.1. The number of measurement panes is varying from =1 to =. The number of iterations is fixed to 1. As an image used for moduation of the object wave fied distribution we use the standard test-image ena. The distances between the measurement panes are constant and equa to Δ =.5mm. 1 is the distance from the object to the first measurement pane (see Fig. 1). It is appeared that the DC agorithm fais for 1 < f due to strong aiasing effects and neary equivaent to the ASD agorithm for 1 f. In what foows we show the resuts ony for MF-DDT and ASD agorithm and drop the resuts for the DC agorithm. FIGURE. The accuracy (RMSE) of the object phase reconstruction by the MF-DDT and ASD agorithms versus the distance 1. This distance is given as a ratio to the "in-focus" distance f.. The accuracy of the phase reconstructions in the object pane is shown in Fig. for different distances 1. The accuracy of MF-DDT is aways better than that for the ASD agorithm with the reative improvement in RMSE vaues about 3 5%. The adaptive reguariation embedded in the MF-DDT agorithm (BM3D fitering) improves the accuracy and visua perception of images. FIGURE 3. The phase reconstruction by the MF-DDT agorithm ( 1 =1.5 f ): (a) =1, PSNR=15.9 db; (b) =, PSNR=16.3 db; (c) =1, PSNR=16.8 db.

5 FIGURE 4. The phase reconstruction by the ASD agorithm ( 1 =1.5 f ): (a) =1, PSNR=14.6dB; (b) =, PSNR=15.5dB; (c) =1, PSNR=16.3dB. FIGURE 5. The fitering effects in the phase reconstruction by the MF-DDT agorithm ( 1 =1.5 f ): (a) =1, PSNR=15.9dB (no fitering); (b) =1, PSNR=16.4dB (with fitering); (c) =1, PSNR=16.8dB (no fitering); (d) =1,PSNR=17dB (with fitering). Exampes of images obtained from the phase reconstruction by MF-DDT and ASD agorithms are presented in Fig. 3 and Fig. 4 respectivey. The image quaity is characteried by PSNR vaues cacuated in db as og 1 (max u₀ /RMSE). For the MF-DDT agorithm the enhancement in the image quaity is quite cear visuay and numericay with PSNR improvement equa to about.4 db for = as compared with =1. =1 gives a further improvement in PSNR of about.5 db as compared with =. For the ASD agorithm the improvements in PSNR vaues even more essentia for arger vaues of as it can be seen in Fig. 4. The fitering (adaptive reguariation) in the MF-DDT agorithm is vauabe for imaging as it is demonstrated in Fig. 5 for =1 and =1. A arger number of the observation panes resuts in a better accuracy for the MF-DDT agorithm, and this improvement is very vauabe for =. Higher vaues of monotonicay improve the accuracy but not so efficienty. The dependences of RMSE on the number of the observation panes for MF-DDT agorithm with fitering and without fitering are shown in Fig. 6a. A resuts in this figure are given for the observations obtained by the DDT forward propagation modeing. For comparison we show aso the accuracy achieved by the ASD agorithm. We can see that the accuracy of the MF-DDT agorithm is much better for a numbers of the observation panes from =1 up to =. This difference in the performance of the agorithms is defined by the fact that the MF- DDT agorithm is a more appropriate fit (than the ASD agorithm) to the forward DDT propagation mode used to generate the observations. In this way MF-DDT is abe to give better resuts. The RMSE curves in Fig. 6a show that this potentia is we reaied by the MF-DDT agorithm. FIGURE 6. The object magnitude reconstruction, RMSE accuracy of the MF-DDT, ASD and SBMIR agorithms versus the number of panes, 1 =1.5 f. Observed data are generated according to the forward propagation modes: (a) DDT, (b) ASD.

6 The ony difference between the SBMIR and ASD agorithms is in their structure: pane-to-pane phase reconstruction in SBMIR and parae pane processing in the ASD as it is discussed in Section "Phase retrieva agorithm". In Fig.6b we produce a comparison of these agorithms. This comparison is done for observations generated by the ASD forward propagation mode. Remind that the wave fied propagation in the SBMIR agorithm is impemented according to the considered ASD mode [11]. In this case the wave fied propagation used in the both agorithms accuratey corresponds to the observations generated by the ASD mode. Thus, we obtain the scenario for an accurate comparison of two different ideas: the parae versus successive pane-by-pane data processing. The RMSE curves in Fig.6b show uniformy smaer RMSE vaues for the ASD agorithm. We consider it as an evidence in favor of the parae processing with respect to the successive one. For the data generated according to the forward DDT propagation mode the SBMIR agorithm demonstrates the performance which is simiar but a bit worse than that shown in Fig.6a for the ASD agorithm. Once more note that for the pixeated object and sensor panes the forward DDT propagation modeing is accurate. It is the main reason behind evauation of the agorithms on the data generated according to this mode. CONCUSIONS In this work we present a mutipe pane phase retrieva agorithm based on simutaneous (parae) data processing and DDT as the forward propagation mode. Both things are essentia to expain the advance performance of the proposed agorithm. Mutipe experiments have been produced in order to study the agorithm behavior for different test images and for reconstruction of phase/magnitude/compex-vaued object distributions. Ony a sma part of these resuts are shown in the paper. Our further work concerns the adaptive seection of the reguariation parameter as we as a tuning of the adaptive fitering, in particuar for noisy data. We wish to note that the deveoped approach is quite fexibe. In particuar, it can be appied for wave fied reconstruction from mutipe waveength data. ACKNOWEDGMENTS This research is supported by the Academy of Finand, project No (Finnish Centre of Exceence program 6-11), and the post graduate work of Artem Migukin is funded by the Tampere Graduate Schoo in Information Science and Engineering (TISE). REFERENCES 1. V. Katkovnik, J. Astoa, and K. Egiaarian, App. Opt. 47, (8).. V. Katkovnik, A. Migukin, and J. Astoa, App. Opt. 48, (9). 3. R. W. Gerchberg and W. O. Saxton, Optik 35, 7 46 (197). 4. J. R. Fienup, Optics ett. 3, 7-9 (1978). 5. J. R. Fienup, App. Opt. 1, (198). 6. M. Guiar-Sicairos and J. R. Fienup, Opt. Express 16, (8). 7. F. Shen and A. Wang, App. Opt. 45, (6). 8. J. W. Goodman, Introduction to Fourier Optics, New York: McGraw-Hi Inc, Second Edition, A.N. Tikhonov and V.Y. Arsenin. Soution of i-posed probems, New York: Wiey, G. Pedrini, W. Osten, and Y. Zhang, Opt. ett. 3, (5). 11. P. Amoro, G. Pedrini and W. Osten, App. Opt (6). 1. K. Dabov, A. Foi, and K. Egiaarian, Proc. SPIE Eectronic Imaging '8, no , San Jose, Caifornia, USA, K. Dabov, A. Foi, V. Katkovnik, and K. Egiaarian, Proc. SPIE Eectronic Imaging '6, no. 664A-3, San Jose, Caifornia, USA, 6.