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Optical eigenmode imaging with a sparse constraint Wei Wang,1,* Yan Pu Wang,1 Yao Wu,2 Xiaoxue Yang,1 and Ying Wu1 1
Wuhan National Laboratory for Optoelectronics and School of Physics, Huazhong University of Science and Technology, Wuhan 430074, China 2 College of Optoelectronic Science and Engineering, Huazhong University of Science and Technology, Wuhan 430074, China *Corresponding author: [email protected] Received January 28, 2014; revised March 20, 2014; accepted March 20, 2014; posted March 20, 2014 (Doc. ID 205079); published April 21, 2014 Optical eigenmode imaging (OEI) is an interesting nonlocal imaging method but has the drawback that the completeness of eigenmodes used in OEI is hard guarantee. This may lead to significant blurring of the reconstructed images. Here we show that in OEI with a sparse constraint, the correlation between the original target and the recovered image can be extremely close to 1 by retrieving lost information, and the compressibility is enhanced. In addition, highquality images can be received for a wide range of both object and system parameters. © 2014 Optical Society of America OCIS codes: (100.0100) Image processing; (270.0270) Quantum optics; (030.6140) Speckle. http://dx.doi.org/10.1364/OL.39.002614
Optical eigenmode imaging (OEI) is a new nonlocal imaging method [1] that is similar to ghost imaging (GI) [2–10]. Recently, compressive GI has been proved an important extension of GI for sparse objects whose compressibility depends on the sparsity of the image [6,7]. By applying a compressive-sensing-based reconstruction algorithm [11], compressive GI can reduce significantly the acquisition time and dramatically improve the signal-to-noise ratio [6,10]. Compared to GI that lies on spatial correlations of light beams, OEI is an inherent compressive-sensing technology that takes advantage of the orthogonality of light beams (called optical eigenmodes) on the target plane. However, since the completeness of eigenmodes is not guaranteed, OEI usually is associated with inherent inaccuracies and imperfect point-spread functions (PSFs) [1,8]. There always are blurred edges and unexpected smears in recovered images even with a correlation beyond 90%. Traditionally, it is more effective on focusing than imaging [8]. Optical eigenmode imaging with a sparse constraint (OEISC) is a modified version of OEI that exploits the spatial correlation of the target by assuming its sparsity. This assumption of sparsity is suitable for natural images in daily life [6,9], i.e., many coefficients are close or equal to zero when images are represented in an appropriate basis. In this Letter, we show OEISC is an effective imaging technology. Targets can be recovered with a sample number far below the Nyquist criteria, and the correlation between the original and recovered images is close to 100%. An additional compressive property and complementary ability of retrieving lost contents from the sparse priori decrease the acquisition duration and increase the image quality significantly, which can promise potential applications for remote tracking and sensing. Effects of different eigenmode numbers, resolutions decided by eigenmodes, and different sparse ratios of images are discussed in this Letter. A simplified schematic of the experimental setup is illustrated in Fig. 1, in which the CCD is used for detection and calibration of the reference light. In OEI, each reference light is identical to the corresponding signal light. A He–Ne laser beam (λ 633 nm) is expanded 0146-9592/14/092614-04$15.00/0
to fill the plane of a spatial light modulator (SLM) (Hamamatsu LCOSSLMX10468, the resolution is 600 × 800 pixels; the pixel size σ x0 is 20 μm × 20 μm) [1]. Here, we divide the region of interest (ROI) to be 151 × 151 pixels with pixel size 20 μm × 20 μm. The light field is focused by mirror F3 (focal length f ≈ 19 cm). When calculating it digitally using the theory of fast Fourier transform, the pixel size σ x0 on the SLM plane must be smaller than λf ∕Nσ x , where N is the pixel number along one direction of the ROI, for realizing aimed eigenmodes with this defined resolution. In addition, the minimum area of the light field on the SLM plane is N × σ x0 2 . Since the pixel size of our SLM is half of the upper limit of σ x0 , the pixel number in the minimum area is doubled to be 302 × 302 out of all the modulation units of the SLM. In OEI, orthogonal eigenmodes (E l ) are obtained by N 1 X vjl E j ; E l p λl j1
(1)
where E j represents the jth test mode. vjl and λl are the lth eigenvector and eigenvalue of M Rrespectively. The elements of M are defined as M jk ROI drE j r E k r.
Fig. 1. Proposed experiment configuration of OEISC. Laser is expanded by focal mirrors F1 and F2 to cover the SLM. Light deflected from the SLM experiences different degrees of modulation controlled by the computer, then is focused by F3 on the target plane. Finally, the single pixel photodiode (PD) collects the coefficients between eigenmodes and the target (O). © 2014 Optical Society of America
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This procedure is realized in experiments by the phasesensitive lock-in method. sl carries the information that unknown object Or projects on the lth eigenmode. P Thus objects can be recovered as Tr N l1 sl × El r because of the orthogonality of these eigenmodes. Detailed procedures of OEI are illustrated in Ref. [1]. However, comparing to the concept of eigenfunction in quantum mechanics, these eigenmodes do not satisfy the spatial completeness. This means that the projection number is not enough, so contents of the object carried by missing eigenmodes are lost. This is easily observed from the theoretical PSF. The P PSF for a definite position r 0 is denoted as Fr; r 0 N l1 E l rE l r 0 , which is not equal to δr − r 0 . Fr; r 0 can also be regarded as the spatial transfer matrix that relates recovered images P Tr to original targets as Tr r0 Fr; r 0 Or 0 . Though it has a convoluted form consistent with traditional pixel-by-pixel imaging methods, OEI is a nonlocal imaging method whose PSF is changed with the spatial position. Because of the above error sources, the captured images always are blurred and distorted. It has been concluded that OEI is a weak imaging method, but it can otherwise be treated as a compressive sensing technology. If we rearrange all the measured coefficients sl into a vector J, only m elements of vector J dominate where m is smaller than the pixel number n (151 × 151). The recovered image Tr from dominate elements of J is an approximation of the object O. By adding a sparse constraint, OEISC provides a new computational framework of calculating T by solving the optimization problem given by
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Test modes will no longer be needed once the eigenmodes are decided. OEI has no obvious preference on test modes if they have well-organized and symmetric structures. We take Hermite–Gaussian modes (HG) as test modes here. What is different from traditional OEI is that the test modes are on the target plane, rather than on the SLM plane. In this situation, light fields backreflected from the SLM are the Fourier inversion of eigenmodes. This computational scheme of OEI can introduce sophisticated controls on the SLM and avoid the pollution from spotted eigenmodes in conventional schemes [6]. The single pixel detector collects the intensity signal of each eigenmode sequentially. The projection coefficients sl are related to the object and light beams by Z Or × E l rdr: (2) sl
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1 XX Oi; j − Ti; j2 ; N pix i j
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where T represents the recovered image, and O is the original image. OEI can measure both the amplitude and phase of a target, and there is no difference on OEI’s capacity for retrieving the real and imaginary parts of the object when eigenmodes have no imaginary part. We take three different kinds of samples as objects, none of which carries any phase information. The first object [Fig. 2(a)] is similar to the transmission function of simple slits, and the quantitative analysis in Fig. 5 is based on this target. Moreover, OEI is weak on recovering circlelike images, so we include the second object [Fig. 2(b)] in our discussion; its sparsity is verified in Figs. 2(c) and 2(d). If we perform a discrete cosine transform (DCT) on this target, only about 3,000 coefficients are not close to zero, while space coefficients are spread all over Fig. 2(c). We start with testing the capacities of both OEI and OEISC on recovering Fig. 2(a), of which partial data is extracted to plot Fig. 3. In OEI, the recovered image no longer maintain the rectangle shape of the original object, sharp edges are smoothed, and the peak value exceeds the initial one to a visible degree. Clearly, the blue line figure received by OEISC is a more precise representation of the original object than the black line figure received by OEI. To compare the abilities of OEI and OEISC directly, four “LOCK GATE” images are taken as the third kind of object because of these 1.4 Object 1.2
OEI
(3)
where DT is the discrete gradient of T, and u is a weighting factor between the first and the second terms (28 here) [7,10]. In the second term of Eq. (3), 2D matrix T of the target is reshaped into a 1D vector T 1 with m elements. Each row of A represents one eigenmode whose 1D form is calculated in the same way as T. We tackle the problem with “total variation minimization by augmented Lagrangian and alternating direction” (TVAL3) [9,11]. Ready-made codes can be found online [12]. The quality of a recovered image is measured by the mean-square error (MSE):
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Fig. 3. Red, 1D profile of one bar in Fig. 2(a). Blue, 1D profile of the bar recovered by OEISC. Black, 1D profile of the bar recovered by OEI.
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Fig. 6. Upper row, original object and recovered images when w1 is between 9 and 11. Bottom row, recovered images when w1 is between 17 and 20. Eigenmode number N equals 400. Fig. 4. Upper row, original images with different K1 numbers. Middle row, images from OEI. Bottom row, images from OEISC, w1 10, N 400.
Letters’ divergent shapes and edges. As seen from Fig. 4, distortions in all four images received by OEI clearly implicate OEI’s weakness for imaging. In contrast with this, these smears are mostly erased and the details of these letters emerge in OEISC. Hermite–Gaussian modes are defined as E mn p p H m wxH n wy exp−wx2 y2 . In OEI, the bigger w is, the smaller the light spot will be, which means a bigger resolution and a smaller field of view.pThe nondimensional parameter (w1 ) is defined as 1∕ w× pixelsize, which has an inverse relationship with resolution, so we discuss MSE’s relationship with w1 instead of its relationship with the resolution. According to Figs. 5(a) and 5(b), there are two tendencies between MSE and w1 . If w1 is smaller than first critical point (w1 21), ideal MSEs can be reached. Next, the MSE increases exponentially [verified by the relationship between LogMSE and w1 ] until w1 reaches the second critical point (w1 49). In this situation, images are totally distorted. In contrast with OEI, OEISC enlarges the first critical point, and ideal MSEs are realizable for w1 up to 40. But the two tendencies still hold. As for w1 beyond 49, vast content of images are still lost. In this point, OEISC decreases the influence of w1 on the ability of recovery. 0.06
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It is also implied that our OEISC is effective for increasing the image resolution, which provides possible application in superresolution, as in Refs. [13–15]. Abnormal MSEs in the left of Fig. 5(a) are caused by the intersection of recovered images since the field of view is smaller than the real image size. Images recovered by OEISC with different values of w1 are shown in Fig. 6. In OEI, the PSFs of different locations are disparate, so its capacity for recovering images with different content fluctuates irregularly. K 1 in Fig. 5(c) is defined as the non-zero pixel number of the object, which has a close relationship with object’s sparsity. Different values of K 1 are generated through the erasion of some non-zero pixels in the first object [Fig. 2(a)]. In OEISC, the MSE has a positive relationship with K 1 in general, but the fluctuation still exists [red line in Fig. 5(c)]. In OEI, this fluctuation limits its effectiveness [blue line in Fig. 5(c)], however all MSEs in OEISC are so small that they would not affect OEISC’s capacity. A direct observation of OEI’s and OEISC’s capacities on images with different values of K 1 can be found in Fig. 4. However, it is difficult to discern the fluctuation. OEI itself is also a compressive imaging method, which is shown by Fig. 5(d), in which the MSE no longer increases once eigenmode number N reaches 210 (far below whole pixel number). But this compressive property cannot guarantee the image quality. In contrast with that, the MSE in OEISC is lowered to be under 1∕10 of the original one, and the resulting relationship between the MSE and N seems to follow a simple rule [red line in Fig. 5(d)] rather than the irregular shape of the blue line. Ten circle-like images in Fig. 7 are those recovered
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Fig. 5. (a) Relationship between MSE and w1 , N 400. (b) Log(MSE) versus w1 . (c) Relationship between MSE and effective pixel number K 1 . (d) MSE versus N. All results here correspond to the object in Fig. 2(a).
Fig. 7. Upper and bottom rows, images from OEI and OEISC, respectively, with eigenmode number N 50, 100, 200, 300, and 400.
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of the second object through OEI and OEISC. When N is very small [50 or 100 in Fig. 7], sparse constraints no longer function well. As for N larger than 200, images recovered by OEISC are similar to the original ones, other than a glow effect on edges. As for OEI, there are many small holes in the white zone for different N. According to the discussion above, the image quality through OEISC keeps the same critical behaviors and tendencies as in OEI, but OEISC leads to an MSE within the acceptable limit, which is much more important for nonlocal imaging technologies than conventional imaging schemes. As the MSE in OEISC is within the acceptable limit, some potential applications of OEI are better realizable with OEISC. These figures also present two small flaws of the TVAL3 algorithm. First, this algorithm malfunctions for a small interval of w1 , as in Fig. 5(a). Second, it tends to cause glow effects on edges [Fig. 7], especially when the number of measurements is small. In conclusion, we have demonstrated that OEI is a weak imaging method and have proposed an improved version, i.e., OEISC, to deal with such a problem. It is clearly shown that OEISC can be effective for a wide range of both object and system parameters. The work is supported in part by the National Fundamental Research Program of China (Grant No. 2012CB922103), and the National Natural Science Foundation of China (NSFC, Grant Nos. 11375067 and 11275074). We also thank Professor Michael Mazilu at the University of St Andrews for explaining some details in the OEI experiment.
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References 1. A. C. D. Luca, S. Kosmeier, K. Dholakia, and M. Mazilu, Phys. Rev. A 84, 021803(R) (2011). 2. F. Ferri, D. Magatti, L. A. Lugiato, and A. Gatti, Phys. Rev. Lett. 104, 253603 (2010). 3. N. Tian, Q. C. Guo, A. Wang, D. L. Xu, and L. Fu, Opt. Lett. 36, 003302 (2011). 4. A. Valencia, G. Scarcelli, M. D’Angelo, and Y. Shih, Phys. Rev. Lett. 94, 063601 (2005). 5. B. Sun, M. P. Edger, R. Bowman, L. E. Vittert, S. Welsh, A. Bowman, and M. J. Padgett, Science 340, 844 (2013). 6. P. Zerom, K. W. C. Chan, J. C. Howell, and R. W. Boyd, Phys. Rev. A 84, 061804(R) (2011). 7. J. Du, W. L. Gong, and S. S. Han, Opt. Lett. 37, 001067 (2012). 8. X. Tsampoula, M. Mazilu, T. Vettenburg, F. Gunn-Moore, and K. Dholakia, Photon. Res. 1, 000042 (2013). 9. O. S. Magana-Loaiza, G. A. Howland, M. Malik, J. C. Howell, and R. W. Boyd, Appl. Phys. Lett. 102, 231104 (2013). 10. O. Katz, Y. Bromberg, and Y. Silberberg, Appl. Phys. Lett. 95, 131110 (2009). 11. M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laske, T. Sun, K. F. Kelly, and R. G. Baraniuk, IEEE Signal Process. Mag. 10(83), 1109 (2008). 12. C. B. Li, W. T. Yin, and Y. Zhang, http://www.caam.rice.edu/ ~optimization/L1/TVAL3/. 13. T. Sun, J. G. Liu, H. S. Yan, G. Morgan, and W. L. Chen, Opt. Lett. 38, 003471 (2013). 14. K. Piche, J. Leach, A. S. Johnson, J. Z. Salvail, M. I. Kolobov, and R. W. Boyd, Opt. Express 20, 026424 (2012). 15. E. Mudry, K. Belkebir, J. Girard, J. Savatier, E. L. Moal, C. Nicoletti, M. Allain, and A. Sentenac, Nat. Photonics 6, 312 (2012).
OPTICS LETTERS / Vol. 39, No. 9 / May 1, 2014
Optical eigenmode imaging with a sparse constraint Wei Wang,1,* Yan Pu Wang,1 Yao Wu,2 Xiaoxue Yang,1 and Ying Wu1 1
Wuhan National Laboratory for Optoelectronics and School of Physics, Huazhong University of Science and Technology, Wuhan 430074, China 2 College of Optoelectronic Science and Engineering, Huazhong University of Science and Technology, Wuhan 430074, China *Corresponding author: [email protected] Received January 28, 2014; revised March 20, 2014; accepted March 20, 2014; posted March 20, 2014 (Doc. ID 205079); published April 21, 2014 Optical eigenmode imaging (OEI) is an interesting nonlocal imaging method but has the drawback that the completeness of eigenmodes used in OEI is hard guarantee. This may lead to significant blurring of the reconstructed images. Here we show that in OEI with a sparse constraint, the correlation between the original target and the recovered image can be extremely close to 1 by retrieving lost information, and the compressibility is enhanced. In addition, highquality images can be received for a wide range of both object and system parameters. © 2014 Optical Society of America OCIS codes: (100.0100) Image processing; (270.0270) Quantum optics; (030.6140) Speckle. http://dx.doi.org/10.1364/OL.39.002614
Optical eigenmode imaging (OEI) is a new nonlocal imaging method [1] that is similar to ghost imaging (GI) [2–10]. Recently, compressive GI has been proved an important extension of GI for sparse objects whose compressibility depends on the sparsity of the image [6,7]. By applying a compressive-sensing-based reconstruction algorithm [11], compressive GI can reduce significantly the acquisition time and dramatically improve the signal-to-noise ratio [6,10]. Compared to GI that lies on spatial correlations of light beams, OEI is an inherent compressive-sensing technology that takes advantage of the orthogonality of light beams (called optical eigenmodes) on the target plane. However, since the completeness of eigenmodes is not guaranteed, OEI usually is associated with inherent inaccuracies and imperfect point-spread functions (PSFs) [1,8]. There always are blurred edges and unexpected smears in recovered images even with a correlation beyond 90%. Traditionally, it is more effective on focusing than imaging [8]. Optical eigenmode imaging with a sparse constraint (OEISC) is a modified version of OEI that exploits the spatial correlation of the target by assuming its sparsity. This assumption of sparsity is suitable for natural images in daily life [6,9], i.e., many coefficients are close or equal to zero when images are represented in an appropriate basis. In this Letter, we show OEISC is an effective imaging technology. Targets can be recovered with a sample number far below the Nyquist criteria, and the correlation between the original and recovered images is close to 100%. An additional compressive property and complementary ability of retrieving lost contents from the sparse priori decrease the acquisition duration and increase the image quality significantly, which can promise potential applications for remote tracking and sensing. Effects of different eigenmode numbers, resolutions decided by eigenmodes, and different sparse ratios of images are discussed in this Letter. A simplified schematic of the experimental setup is illustrated in Fig. 1, in which the CCD is used for detection and calibration of the reference light. In OEI, each reference light is identical to the corresponding signal light. A He–Ne laser beam (λ 633 nm) is expanded 0146-9592/14/092614-04$15.00/0
to fill the plane of a spatial light modulator (SLM) (Hamamatsu LCOSSLMX10468, the resolution is 600 × 800 pixels; the pixel size σ x0 is 20 μm × 20 μm) [1]. Here, we divide the region of interest (ROI) to be 151 × 151 pixels with pixel size 20 μm × 20 μm. The light field is focused by mirror F3 (focal length f ≈ 19 cm). When calculating it digitally using the theory of fast Fourier transform, the pixel size σ x0 on the SLM plane must be smaller than λf ∕Nσ x , where N is the pixel number along one direction of the ROI, for realizing aimed eigenmodes with this defined resolution. In addition, the minimum area of the light field on the SLM plane is N × σ x0 2 . Since the pixel size of our SLM is half of the upper limit of σ x0 , the pixel number in the minimum area is doubled to be 302 × 302 out of all the modulation units of the SLM. In OEI, orthogonal eigenmodes (E l ) are obtained by N 1 X vjl E j ; E l p λl j1
(1)
where E j represents the jth test mode. vjl and λl are the lth eigenvector and eigenvalue of M Rrespectively. The elements of M are defined as M jk ROI drE j r E k r.
Fig. 1. Proposed experiment configuration of OEISC. Laser is expanded by focal mirrors F1 and F2 to cover the SLM. Light deflected from the SLM experiences different degrees of modulation controlled by the computer, then is focused by F3 on the target plane. Finally, the single pixel photodiode (PD) collects the coefficients between eigenmodes and the target (O). © 2014 Optical Society of America
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This procedure is realized in experiments by the phasesensitive lock-in method. sl carries the information that unknown object Or projects on the lth eigenmode. P Thus objects can be recovered as Tr N l1 sl × El r because of the orthogonality of these eigenmodes. Detailed procedures of OEI are illustrated in Ref. [1]. However, comparing to the concept of eigenfunction in quantum mechanics, these eigenmodes do not satisfy the spatial completeness. This means that the projection number is not enough, so contents of the object carried by missing eigenmodes are lost. This is easily observed from the theoretical PSF. The P PSF for a definite position r 0 is denoted as Fr; r 0 N l1 E l rE l r 0 , which is not equal to δr − r 0 . Fr; r 0 can also be regarded as the spatial transfer matrix that relates recovered images P Tr to original targets as Tr r0 Fr; r 0 Or 0 . Though it has a convoluted form consistent with traditional pixel-by-pixel imaging methods, OEI is a nonlocal imaging method whose PSF is changed with the spatial position. Because of the above error sources, the captured images always are blurred and distorted. It has been concluded that OEI is a weak imaging method, but it can otherwise be treated as a compressive sensing technology. If we rearrange all the measured coefficients sl into a vector J, only m elements of vector J dominate where m is smaller than the pixel number n (151 × 151). The recovered image Tr from dominate elements of J is an approximation of the object O. By adding a sparse constraint, OEISC provides a new computational framework of calculating T by solving the optimization problem given by
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Test modes will no longer be needed once the eigenmodes are decided. OEI has no obvious preference on test modes if they have well-organized and symmetric structures. We take Hermite–Gaussian modes (HG) as test modes here. What is different from traditional OEI is that the test modes are on the target plane, rather than on the SLM plane. In this situation, light fields backreflected from the SLM are the Fourier inversion of eigenmodes. This computational scheme of OEI can introduce sophisticated controls on the SLM and avoid the pollution from spotted eigenmodes in conventional schemes [6]. The single pixel detector collects the intensity signal of each eigenmode sequentially. The projection coefficients sl are related to the object and light beams by Z Or × E l rdr: (2) sl
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1 XX Oi; j − Ti; j2 ; N pix i j
(4)
where T represents the recovered image, and O is the original image. OEI can measure both the amplitude and phase of a target, and there is no difference on OEI’s capacity for retrieving the real and imaginary parts of the object when eigenmodes have no imaginary part. We take three different kinds of samples as objects, none of which carries any phase information. The first object [Fig. 2(a)] is similar to the transmission function of simple slits, and the quantitative analysis in Fig. 5 is based on this target. Moreover, OEI is weak on recovering circlelike images, so we include the second object [Fig. 2(b)] in our discussion; its sparsity is verified in Figs. 2(c) and 2(d). If we perform a discrete cosine transform (DCT) on this target, only about 3,000 coefficients are not close to zero, while space coefficients are spread all over Fig. 2(c). We start with testing the capacities of both OEI and OEISC on recovering Fig. 2(a), of which partial data is extracted to plot Fig. 3. In OEI, the recovered image no longer maintain the rectangle shape of the original object, sharp edges are smoothed, and the peak value exceeds the initial one to a visible degree. Clearly, the blue line figure received by OEISC is a more precise representation of the original object than the black line figure received by OEI. To compare the abilities of OEI and OEISC directly, four “LOCK GATE” images are taken as the third kind of object because of these 1.4 Object 1.2
OEI
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where DT is the discrete gradient of T, and u is a weighting factor between the first and the second terms (28 here) [7,10]. In the second term of Eq. (3), 2D matrix T of the target is reshaped into a 1D vector T 1 with m elements. Each row of A represents one eigenmode whose 1D form is calculated in the same way as T. We tackle the problem with “total variation minimization by augmented Lagrangian and alternating direction” (TVAL3) [9,11]. Ready-made codes can be found online [12]. The quality of a recovered image is measured by the mean-square error (MSE):
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Fig. 6. Upper row, original object and recovered images when w1 is between 9 and 11. Bottom row, recovered images when w1 is between 17 and 20. Eigenmode number N equals 400. Fig. 4. Upper row, original images with different K1 numbers. Middle row, images from OEI. Bottom row, images from OEISC, w1 10, N 400.
Letters’ divergent shapes and edges. As seen from Fig. 4, distortions in all four images received by OEI clearly implicate OEI’s weakness for imaging. In contrast with this, these smears are mostly erased and the details of these letters emerge in OEISC. Hermite–Gaussian modes are defined as E mn p p H m wxH n wy exp−wx2 y2 . In OEI, the bigger w is, the smaller the light spot will be, which means a bigger resolution and a smaller field of view.pThe nondimensional parameter (w1 ) is defined as 1∕ w× pixelsize, which has an inverse relationship with resolution, so we discuss MSE’s relationship with w1 instead of its relationship with the resolution. According to Figs. 5(a) and 5(b), there are two tendencies between MSE and w1 . If w1 is smaller than first critical point (w1 21), ideal MSEs can be reached. Next, the MSE increases exponentially [verified by the relationship between LogMSE and w1 ] until w1 reaches the second critical point (w1 49). In this situation, images are totally distorted. In contrast with OEI, OEISC enlarges the first critical point, and ideal MSEs are realizable for w1 up to 40. But the two tendencies still hold. As for w1 beyond 49, vast content of images are still lost. In this point, OEISC decreases the influence of w1 on the ability of recovery. 0.06
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It is also implied that our OEISC is effective for increasing the image resolution, which provides possible application in superresolution, as in Refs. [13–15]. Abnormal MSEs in the left of Fig. 5(a) are caused by the intersection of recovered images since the field of view is smaller than the real image size. Images recovered by OEISC with different values of w1 are shown in Fig. 6. In OEI, the PSFs of different locations are disparate, so its capacity for recovering images with different content fluctuates irregularly. K 1 in Fig. 5(c) is defined as the non-zero pixel number of the object, which has a close relationship with object’s sparsity. Different values of K 1 are generated through the erasion of some non-zero pixels in the first object [Fig. 2(a)]. In OEISC, the MSE has a positive relationship with K 1 in general, but the fluctuation still exists [red line in Fig. 5(c)]. In OEI, this fluctuation limits its effectiveness [blue line in Fig. 5(c)], however all MSEs in OEISC are so small that they would not affect OEISC’s capacity. A direct observation of OEI’s and OEISC’s capacities on images with different values of K 1 can be found in Fig. 4. However, it is difficult to discern the fluctuation. OEI itself is also a compressive imaging method, which is shown by Fig. 5(d), in which the MSE no longer increases once eigenmode number N reaches 210 (far below whole pixel number). But this compressive property cannot guarantee the image quality. In contrast with that, the MSE in OEISC is lowered to be under 1∕10 of the original one, and the resulting relationship between the MSE and N seems to follow a simple rule [red line in Fig. 5(d)] rather than the irregular shape of the blue line. Ten circle-like images in Fig. 7 are those recovered
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Fig. 5. (a) Relationship between MSE and w1 , N 400. (b) Log(MSE) versus w1 . (c) Relationship between MSE and effective pixel number K 1 . (d) MSE versus N. All results here correspond to the object in Fig. 2(a).
Fig. 7. Upper and bottom rows, images from OEI and OEISC, respectively, with eigenmode number N 50, 100, 200, 300, and 400.
May 1, 2014 / Vol. 39, No. 9 / OPTICS LETTERS
of the second object through OEI and OEISC. When N is very small [50 or 100 in Fig. 7], sparse constraints no longer function well. As for N larger than 200, images recovered by OEISC are similar to the original ones, other than a glow effect on edges. As for OEI, there are many small holes in the white zone for different N. According to the discussion above, the image quality through OEISC keeps the same critical behaviors and tendencies as in OEI, but OEISC leads to an MSE within the acceptable limit, which is much more important for nonlocal imaging technologies than conventional imaging schemes. As the MSE in OEISC is within the acceptable limit, some potential applications of OEI are better realizable with OEISC. These figures also present two small flaws of the TVAL3 algorithm. First, this algorithm malfunctions for a small interval of w1 , as in Fig. 5(a). Second, it tends to cause glow effects on edges [Fig. 7], especially when the number of measurements is small. In conclusion, we have demonstrated that OEI is a weak imaging method and have proposed an improved version, i.e., OEISC, to deal with such a problem. It is clearly shown that OEISC can be effective for a wide range of both object and system parameters. The work is supported in part by the National Fundamental Research Program of China (Grant No. 2012CB922103), and the National Natural Science Foundation of China (NSFC, Grant Nos. 11375067 and 11275074). We also thank Professor Michael Mazilu at the University of St Andrews for explaining some details in the OEI experiment.
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