Comparison of Optical Sparse Aperture Image Restoration with Experimental PSF and Designed PSF Zhiwei Zhou, Dayong Wang

Transcription

1 Comparison o Optical Sparse Aperture Image Restoration with Eperimental PSF and Designed PSF Zhiwei Zhou, Daong Wang Applied Science, Beijing Universit o Technolog, Beijing, 0024, P.R.China Juan Zhao, Yuhong Wan, Zhuqing Jiang, Shiquan Tao Applied Science, Beijing Universit o Technolog, Beijing, 0024, P.R.China Abstract The digital post-processing method is needed to restore the image qualit rom the blurring image output directl rom optical sparse aperture imaging sstems. The common method or post-processing is the Wiener ilter, where the important parameter is the point spread unction (PSF). The Wiener ilter will deconvolve the convolution eect i the PSF is the precise impulse response o the optical sparse aperture sstem. Usuall it is hard to measure the PSF eperimentall. On the contrar, it is eas to calculate the PSF based on telescope arra coniguration, which is given b design. In order to evaluate whether the calculated PSF is adaptable to the image restoration, a comparison o the image restoration has been done b using these two kinds o PSFs. The results show that the calculated PSF works well in the case that the co-phasing error is small, while the measured PSF is ineective. Ke words: sparse aperture; Wiener ilter; PSF. Introduction With the eploration o outer space getting deeper, a high angular resolution telescope is necessar to stud the speciic structure o universe, however, the quest or high angular resolution telescope inevitabl leads to large aperture, which is limited b the volume and mass constrain o current launch vehicles as well as the inancial support limitation[][2]. Since the diicult o manuacturing a large diameter monolithic prime mirror telescope increases rapidl, the optical sparse aperture telescope is proposed to resolve this problem. An optical sparse aperture telescope consists o several independent small telescopes, each o that will collect light separatel. This structure is called Fizeau intererometr telescope, which is also called Image-plane intererometr because it is the method o combining multiple beams, each ocused to make an image o the sk. Since the Fizeau intererometers produce direct images with ull spatial requenc coverage, and onl the light collection areas decreases, the picture obtained rom the optical sparse aperture telescope needs post-processing to retrieve the original image o the sk. The mostl common method used or the image restoration o the optical sparse aperture telescope is the Wiener ilter in the spatial requenc domain [3]. The most important parameter o the Wiener ilter is the optical transer unction (OTF) which is the Fourier transormation o point spread unction (PSF). There are two possible was to obtain the OTF o the sstem. Both methods are in need o PSF irst. One metho d calculates the Fourier transormation o designed aperture unction, named the calculated PSF. Another method detects eperimentall the PSF b using a CCD, when the sstem is illuminated with the quasi-monochromatic light. Such PSF is named the measured PSF, which is usuall considered to be the practical PSF o sstem. However, it is diicult to measure the PSF o optical sparse aperture telescope with large aperture. Although there are some alternate methods, the measurement o PSF is still diicult. On the contrar, the calculated PSF is ver eas to obtain through the Fourier transormation. In this paper, we present an analsis and an comparison o the image restoration with these two PSFs o the optical sparse aperture sstem, and demonstrate the easibilit o the calculated PSF and the measured PSF respectivel.

2 2. Theor o imaging and image restoration The imaging process o the optical sparse aperture sstems can be simpliied to be incoherent optical sstem described with the Eq.(). (, ) i (, )* PSF(, ) + n( ) i =, g where the * is convolution operand, i(, ) stands or the light intensit distribution o image plane, i g (, ) is the light intensit distribution o the geometrical ideal image, PSF(, ) is the point spread unction o sstem, and n(, ) is the noise distribution. According to the Eq. (), i the PSF(, ) is known, the geometrical ideal image can be retrieved through the deconvolution process. The PSF o an optical sparse aperture sstem is the square o modulus o Fourier transormation o sstem eit pupil, as shown in Eq. (2): PSF (, ) I[ P(, ) ] 2 = 2 which is called the impulse response[4]. Because the eit beam o each sub-telescope is still parallel, the sstem eit pupil is determined b sub-telescope arra coniguration [5]. There are three major tpes o telescope arra conigurations: Tri-Arm, Gola and Annulus. The digital image restoration is based on Wiener ilter in the requenc domain. OTF which is the which is the Fourier transormation o PSF is the most important parameter in Wiener ilter as shown in Eq.(3a) and Eq.(3b). W(, ) is the Wiener ilter epression in requenc do main, N(, ) is the noise spectrum and Ĩ g (, ) is the spectrum o the geometrical ideal image. Here the * is comple conjugate operand. In act, since the N(, ) is usuall unknown, a constant K is used or substitute o N(, )/ Ĩ g (, ) 2. is the distance rom the eit pupil to the image plane. The Wiener ilter essence is an inverse ilter, i the constant K equals zero. W OTF (, ) = OTF OTF (, ) ( ) 2 ~, + N(, ) I (, ) 2 ( ) = I[ PSF(, )] 3a g =, = λ λ, 3b 3. Measured and Calculated PSF The PSF o the optical sparse aperture imaging sstem is obtained rom two main was: eperiment and calculation [6]. ) Measured PSF Because the input and output beam are both parallel, the telescope arra coniguration is simpliied to a pupil mask as shown in Fig.. These pupil masks are the eit pupil unction o the sstem. The Fig.2 shows our eperimental setup or the measurement o PSF [7]. A pinhole with a diameter o 0µm, is used to simulate a point source. The pinhole was put in the orward ocal plane o the achromatic doublet lens L so as to produce parallel lights. The mask is placed in ront o the achromatic doublet lens L 2, and the CCD was put at the ocal plane o the L 2. The lenses o L and L 2 are set to be as close as possible. The image captured b the CCD is the pattern o the Fourier transormation o the mask s transmission unction, which is the PSF o the sstem.

3 (a) Tri-Arm (b) Gola-6 (c) Anuulus Fig. Masks used in the eperiments 2) Calculated PSF Fig.2 Setup or the measurement o the PSF The PSF can also be calculated rom the eit pupil unction. The arra pupil unction P(,) is demonstrated in Eq.(4) [8]: P N j n ( ) ( ) = ( ),, P an, bn e n= φ 4 where a n, b n are the central coordinates o a sub-telescope, and φ n (,) is the phase error o the nth aperture. Supposed the phase error changes little, the φ n (,) is simpliied to a constant φ n. Eq.(5) is the PSF o an optical sparse aperture sstem with phase error: PSF N ( N ) / 2 (, ) PSF (, ) N + 2 cos[ 2 ( + + φ )] = + sub k= π 5 k k k k, k are the vector separation components between pairs o sub-aperture centers, φ k is the phase dierence between pairs o sub-aperture, λ is the wavelength, and is the distance rom the pupil to the image plane. Eq.(6) reveals the OTF o the sparse aperture sstem with little phase error: ( ) = ( ) ( ) + + OTF, OTFsub, δ, N k= where * is the convolution operand. ) Results othe calculated PSF N ( N ) 2 δ k ±, λ 4. Eperimental results k ± λ ep j2π φ k According to the eperimental setup shown in Fig.2, =840mm, the central wavelength is 532nm, and the CCD valid area is 37*035. The sampling interval o CCD is 6.8µm*6.8µm with a nominall linear 2 bit response. The eplicit parameters o pupil masks are included in Table, and their calculated PSFs are shown in Fig.3. Table the eplicit parameters o three tpe s masks k + Sub-aperture diameter (mm) Equivalent illed aperture diameter (mm) Filled Factor Tri-Arm % Gola % Annulus % k 6

4 (a) Tri-Arm (b) Gola-6 (c) Annulus Fig.3 The calculated PSFs o three tpical arra conigurations 2) Results othe measured PSF The PSF images captured b the CCD are shown in Fig.4, where the sstem is illuminated b a 532nm diode laser. In order to obtain the incoherent light, a random phase plate is put in the ront o the pinhole. (a) Tri-Arm (b) Gola-6 (c) Annulus Fig.4 The measured PSFs o three tpical arra conigurations Compared with Fig.3, the dierence between the measured PSFs and the calculated PSFs is not apparent, which means that ater the ine adjustment the simpliied optical sparse aperture sstem now have ver little phase errors when combining the beams rom the sub-apertures. 3) Imaging eperiment with resolution target In order to investigate the imaging capabilit o the optical sparse aperture sstem, a USAF 95 resolution target is deploed into the eperiment. The imaging eperiment setup is illustrated in Fig.5. The source is a white light source output through a iber. The wave ilter s central wavelength is 532nm with a bandwidth o 0nm. The USAF 95 resolution target is used to replace the pinhole in the orward ocal plane o the achromatic doublet lens L so as to simulate the remote etended target. Fig.5 Setup o an optical sparse-aperture imaging sstem

5 (a) Tri-Arm (b) Gola-6 (c) Annulus Fig.6 The resolution target imaging result The original image is a 37*035 picture. In order to demonstrate the limit resolution o sparse aperture sstem, the high resolution portion in the picture is shown in igure 6. It is easil to recognize the 6th element o the 5th group, whose line width is 8.77µm. The Gola-6 tpe mask even demonstrates the st element o 6th group which line width is 7.8µm. As a comparison, the image o the equivalent illed aperture is illustrated in Figure 7. Without an post-process, the picture o equivalent aperture is much brighter than sparse apertures because o the dierent light-collecting area. It is eas to distinguish the st element o 6th group whose line width is 7.8µm, not like with Gola-6. Fig.7 The resolution target imaging o equivalent aperture 4) Restoration o the sparse aperture sstem image The Wiener ilter needs the PSF as the parameter to deconvolve the image. The results are demonstrated in Fig.8. (a) Restored with the measured PSF (b) Restored with the calculated PSF

6 (c) Restored with the measured PSF (d) Restored with the calculated PSF (e) Restored with the measured PSF () Restored with the calculated PSF Fig.9 (a),(b) are the results o Tri-Arm mask. (c),(d) are the results o Gola-6. (e),() are the results o Annulus The comparison illustrates clearl that with post-processing the resolution o optical sparse aperture sstem is improved largel. The previous work had alread shown that, the higher illed actor was, the more requenc would pass the sstem [3]. Compared with the results rom the measured PSF, the image restoration with the calculated PSF generates a more clearl picture. The eplanation or the phenomena is that the PSF captured b the CCD contains the noise meanwhile there is still some little phase error between the combined beams. The detected image output directl rom the optical sparse aperture sstem includes inherentl the eect o the noise and the phase error. When restored with the measured PSF, the eect o the noise and the phase error on the measured PSF will interere with the eect o the noise and the phase error on the direct image, and this will make the deconvolution process less perect. In the worst case it even will introduce more blur into the picture. However, since the calculated PSF is clean, such intererence does not appear, and the deconvolution process will produce much better results as long as the phase error is ver small. 5. Conclusion Based on the eperimental results and analsis, the calculated PSF rom telescope arra coniguration ehibits a more eective abilit in retrieving the geometrical ideal image rom the optical sparse aperture sstem. The measured PSF due to the noise and the residual phase error, is not suitable or Wiener ilter used in post-processing o the optical sparse aperture sstem. I the phase error is small enough, the calculated PSF will work well or the

7 image restoration. In act, to control the phase error is the most important ke technique in the design o the optical sparse aperture telescope, because the light interered in image plane is incoherent light. The calculated PSF is easier to obtain and more eective in the restored process. Possibl the measured PSF is no longer necessar or the sparse aperture sstem. 6. Acknowledgement The authors acknowledge inancial support rom the Funding Project or Academic Human Resources Development in Institutions o Higher Learning Under the Jurisdiction o Beijing Municipalit (PHRIHLB). Financial support b the National Natural Science Foundation o China (NSFC) (Contract No ), and partial inancial support b The Science Foundation o Education Commission o Beijing, China (Contract No.KZ ). 7. Reerence. Soon-Jo Chung, David W. Miller, et al, Design and Implementation o Sparse Aperture Imaging Sstems, Proceedings o SPIE, Vol. 4849, 2002, p Soon-Jo Chung, David W. Miller, et al, ARGOS testbed: stud o multidisciplinar challenges o uture spaceborne intererometric arras, Optical Engineering, Vol.43 (9), 2004, p Robert D. Fiete, Theodore A. Tantalo, Jason R. Calus, et al, Image qualit o sparse-aperture designs or remote sensing, Optical Engineering, 2002, Vol.4(8), p Harve J E, Kotha A, Phillips R L, Image characteristics in applications utilizing dilute subaperture arras, Applied Optics, 995, Vol.34(6), p 2983~ Meinel A B and Meinel M P, Large sparse-aperture space optical sstems, Optical Engineering, 2002, Vol.4(8), p 983~ R.L. Kendrick, Jean-Noel Aubrun, Ra Bell, et al, Wide-ield Fizeau imaging telescope: eperimental results, Applied Optics, 2006, Vol.45(8), p Daong Wang, Ji Han, et al, Eperimental stud on imaging and image restoration o optical sparse aperture sstems, Optical Engineering, Vol. 46 (0), 2007, p Nicholas J. Miller, Matthew P. Dierking, Bradle D. Duncan, Optical sparse aperture imaging, Applied Optics, Vol. 46 (23), 2007, p