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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 22, Iss. 7 — Apr. 7, 2014
  • pp: 8298–8308
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Influence of lenslet number on performance of image restoration algorithms for the TOMBO imaging system

Yuan Gao, Lizhi Dong, Ping Yang, Guomao Tang, and Bing Xu  »View Author Affiliations


Optics Express, Vol. 22, Issue 7, pp. 8298-8308 (2014)
http://dx.doi.org/10.1364/OE.22.008298


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Abstract

In this paper the influence of the number of lenslets on the performance of image restoration algorithms for the thin observation module by bound optics (TOMBO) imaging system was investigated, and the lenslet number was optimized to achieve thin system and high imaging performance. Subimages with different numbers of lenslets were generated following the TOMBO observation model, and image restoration algorithms were applied to evaluate the imaging performance of the TOMBO system. The optimal lenslet number was determined via theoretical performance optimization and verified via experimental comparisons of angular resolutions of two TOMBO systems and a conventional single-lens system.

© 2014 Optical Society of America

1. Introduction

Thin observation module by bound optics (TOMBO) based on a combination of imaging techniques and efficient image processing algorithms is a fast-developing area of computational photography [1

1. J. Tanida, T. Kumagai, K. Yamada, S. Miyatake, K. Ishida, T. Morimoto, N. Kondou, D. Miyazaki, and Y. Ichioka, “Thin observation module by bound optics (TOMBO): concept and experimental verification,” Appl. Opt. 40(11), 1806–1813 (2001). [CrossRef] [PubMed]

3

3. D. Mendlovic, “Toward a super imaging system,” Appl. Opt. 52(4), 561–566 (2013). [CrossRef] [PubMed]

]. The TOMBO imaging system is an optical system that achieves thinness and high-resolution imaging by replacing a conventional full-aperture lens with a lenslet array [4

4. K. Choi and T. J. Schulz, “Signal-processing approaches for image-resolution restoration for TOMBO imagery,” Appl. Opt. 47(10), B104–B116 (2008). [CrossRef] [PubMed]

]. The optical arrangement of the TOMBO imaging system is shown in Fig. 1
Fig. 1 The optical arrangement of the TOMBO imaging system.
. Regarding thinness, the focal length of each lenslet can be 1/n of the conventional single-lens system with equal f number, with n being the number of subimages of the scene in one direction across the image sensor. The reduction in the focal length of the lenslet (keeping the f number the same) would keep the focal resolution unaffected with the compromise of reduction in angular resolution [5

5. M. Shankar, R. Willett, N. Pitsianis, T. Schulz, R. Gibbons, R. Te Kolste, J. Carriere, C. Chen, D. Prather, and D. Brady, “Thin infrared imaging systems through multichannel sampling,” Appl. Opt. 47(10), B1–B10 (2008). [CrossRef] [PubMed]

]. In the view of high resolution, recovering the lost angular resolution by collecting subimages can be achieved. The lenslet array accumulates a series of subimages from the same scene which are undersampled. Each subimage resolution is determined by pixel size rather than the optical performance of the lenslet, and these subimages can be processed in a manner similar to multiframe superresolution processing to obtain a fully upsampled, high-resolution image [6

6. A. V. Kanaev, D. A. Scribner, J. R. Ackerman, and E. F. Fleet, “Analysis and application of multiframe superresolution processing for conventional imaging systems and lenslet arrays,” Appl. Opt. 46(20), 4320–4328 (2007). [CrossRef] [PubMed]

]. A number of TOMBO image restoration algorithms have been proposed and proven to be effective in the literature [7

7. Y. Kitamura, R. Shogenji, K. Yamada, S. Miyatake, M. Miyamoto, T. Morimoto, Y. Masaki, N. Kondou, D. Miyazaki, J. Tanida, and Y. Ichioka, “Reconstruction of a high-resolution image on a compound-eye image-capturing system,” Appl. Opt. 43(8), 1719–1727 (2004). [CrossRef] [PubMed]

12

12. S. Mendelowitz, I. Klapp, and D. Mendlovic, “Design of an image restoration algorithm for the TOMBO imaging system,” J. Opt. Soc. Am. A 30(6), 1193–1204 (2013). [CrossRef] [PubMed]

].

In this paper we focus on the optimization of the number of lenslets to achieve a thin system with high image restoration performance for the TOMBO system. For one TOMBO system with a fixed image sensor pixel number, a larger lenslet number leads to more subimages and fewer pixels for each subimage. More subimages only improve the resolution marginally if a special set of low-resolution pixels has been captured. On the other hand, subimages with fewer pixels demand a larger magnification factor, which leads to the performance deterioration of existing algorithms [13

13. Z. Lin and H. Y. Shum, “Fundamental limits of reconstruction-based superresolution algorithms under local translation,” IEEE Trans. Pattern Anal. Mach. Intell. 26(1), 83–97 (2004). [CrossRef] [PubMed]

]. Therefore, optimizing the lenslet number is an important practical problem regarding the design of TOMBO. However, due to the specific architecture of TOMBO, to the best of our knowledge, this problem has not been adequately discussed.

In this paper, the optimization of the number of lenslets for the TOMBO system is performed by analyzing the influence of lenslet number on the image restoration performance. The imaging process (comprising image global translation, blur, downsampling, and noise) of each subimaging system with different lenslet numbers is presented. Then iterative backprojection (IBP) [8

8. A. Stern and B. Javidi, “Three-dimensional image sensing and reconstruction with time-division multiplexed computational integral imaging,” Appl. Opt. 42(35), 7036–7042 (2003). [CrossRef] [PubMed]

], bilateral total variance (BTV) [16

16. S. Farsiu, M. D. Robinson, M. Elad, and P. Milanfar, “Fast and robust multiframe super resolution,” IEEE Trans. Image Process. 13(10), 1327–1344 (2004). [CrossRef] [PubMed]

], and l1 norm combined with simultaneous auto regressive (l1-SAR) [17

17. S. Villena, M. Vega, S. D. Babaccan, R. Molina, and A. K. Katsaggelos, “Bayesian combination of sparse and non-sparse priors in image super resolution,” Digit. Signal Process. 23(2), 530–541 (2013). [CrossRef]

] are applied to restore high-resolution images with actual translation parameters as well as translation parameters estimated by scale invariant feature transform [18

18. D. G. Lowe, “Distinctive image features from scale-invariant keypoints,” Int. J. Comput. Vis. 60(2), 91–110 (2004). [CrossRef]

] and random sample consensus [19

19. M. A. Fischler and R. C. Bolles, “Random sample consensus: a paradigm for model fitting with applications to image analysis and automated cartography,” Commun. ACM 24(6), 381–395 (1981). [CrossRef]

] (SIFT-RANSAC), respectively. Mean square error (MSE) and peak signal-to-noise ratio (PSNR) of restored images are calculated for image quality assessment. The optimal lenslet number is obtained by analyzing the influence of lenslet number on the restored images with 100 synthetic and real images as input. Experimental comparison of angular resolutions among two TOMBO systems and a single-lens system is performed to validate the optimal lenslet number.

2. Performance simulation

Consider a TOMBO system with n × n subimaging units. Each captured subimage can be modeled as [12

12. S. Mendelowitz, I. Klapp, and D. Mendlovic, “Design of an image restoration algorithm for the TOMBO imaging system,” J. Opt. Soc. Am. A 30(6), 1193–1204 (2013). [CrossRef] [PubMed]

]
Li,j=[bi,jti,j(ri,j)H]D+vi,j,
(1)
where Li,j represents the blurred, noisy, and low-resolution output image captured by subimaging system in the i row of the j column of TOMBO (i, j = 1, 2, …, n); H is the input high-resolution image; bi,j is a two-dimensional PSF representing the channel blur for each imaging unit; ti,j(ri,j) is a global translation shift operator which has ri,j = [Δxi, Δyj] translation with respect to the input image; ↓D is the downsampling operator; and vi,j is the noise, such as fixed-detector thermal noise, signal-dependent shot noise, and background noise [20

20. M. W. Haney, “Performance scaling in flat imagers,” Appl. Opt. 45(13), 2901–2910 (2006). [CrossRef] [PubMed]

].

In our simulation, a rotationally symmetric Gaussian low-pass filter of size [5

5. M. Shankar, R. Willett, N. Pitsianis, T. Schulz, R. Gibbons, R. Te Kolste, J. Carriere, C. Chen, D. Prather, and D. Brady, “Thin infrared imaging systems through multichannel sampling,” Appl. Opt. 47(10), B1–B10 (2008). [CrossRef] [PubMed]

] with standard deviation σ2 = 1 is used as channel blur bi,j, variance σ2 of two-dimensional zero mean white Gaussian noise is set to 2, and downsampling factor d equals the lenslet number n.

Because the sufficient number of low-resolution subimages is M2 when the translation between adjacent subimages is 1/d (M is an integer magnification factor of superresolution algorithms) [13

13. Z. Lin and H. Y. Shum, “Fundamental limits of reconstruction-based superresolution algorithms under local translation,” IEEE Trans. Pattern Anal. Mach. Intell. 26(1), 83–97 (2004). [CrossRef] [PubMed]

], global translation for input image is set to ri,j = [(i – 1), (j – 1)] (i.e., Δxi = i – 1 and Δyj = j – 1), and magnification factor M equals the lenslet number n in our simulation. For instance, translation between two low-resolution pixels is 1/3 in horizontal and vertical as shown in Fig. 2
Fig. 2 The 1/3 translation between two low-resolution pixels of subimages when downsampling factor d is 3. The dotted squares represent high-resolution pixels from one input image, whereas the red square represents a low-resolution pixel from the top left corner of subimages L2,2, and the green square represents a low-resolution pixel from the top left corner of subimages L3,3.
when downsampling factor d is 3. Take the system in [5

5. M. Shankar, R. Willett, N. Pitsianis, T. Schulz, R. Gibbons, R. Te Kolste, J. Carriere, C. Chen, D. Prather, and D. Brady, “Thin infrared imaging systems through multichannel sampling,” Appl. Opt. 47(10), B1–B10 (2008). [CrossRef] [PubMed]

]. as an example: the translation between adjacent subimages is manufactured as 70 μm, and this corresponds to 2 1/3 pixels of the sensor. Figure 3
Fig. 3 (a) Input image. (b) Simulated image captured by the single-lens system. Simulated subimages with lenslet numbers (c) 2 × 2, (d) 3 × 3, (e) 4 × 4, (f) 5 × 5, (g) 6 × 6, (h) 7 × 7, (i) 8 × 8, (j) 9 × 9, and (k) 10 × 10.
shows simulated subimages with different lenslet numbers. All these subimages are resized to resolution of the input image with nearest interpolation.

Figure 4
Fig. 4 (a) Input image. (b) Simulated image captured by the single-lens system. Restored images with actual translation parameters and lenslet numbers (c) 2 × 2, (d) 3 × 3, (e) 4 × 4, (f) 5 × 5, (g) 6 × 6, (h) 7 × 7, (i) 8 × 8, (j) 9 × 9, and (k) 10 × 10.
presents the restored images using the l1-SAR algorithm with actual translation parameters. In the l1-SAR algorithm, the magnification factor M equals the lenslet number n, and the number of iterations is 20. Furthermore, for consideration of practical applications, restored images using the same algorithm with estimated translation parameters by SIFT-RANSAC are shown in Fig. 5
Fig. 5 (a) Input image. (b) Simulated image captured by the single-lens system. Restored images with estimated translation parameters and lenslet numbers (c) 2 × 2, (d) 3 × 3, (e) 4 × 4, (f) 5 × 5, (g) 6 × 6, (h) 7 × 7, (i) 8 × 8, (j) 9 × 9, and (k) 10 × 10.
. Matched pixel pairs between any two subimages are generated by SIFT, and then RANSAC is used to remove incorrect matches.

Restored images with 100 synthetic and real images as input are tested for assessment of the influence of the lenslet number. IBP, BTV, and l1-SAR are utilized as restored algorithms. Moreover, MSE and PSNR of restored images are averaged for quantitative evaluation of the influence. PSNR is defined as
PSNR=20log10MAXMSE,
(2)
where MAX is the maximum pixel intensity of an input image and MSE is the mean square error between the input image and the restored image. For the sake of simplicity, temporal noise is not taken into account in Eq. (2).

The influence of the lenslet number on the averaged image quality (MSE and PSNR) of restored images is present in Figs. 6
Fig. 6 Averaged MSE of restored images with the actual translation parameters versus lenslet number.
, 7
Fig. 7 Averaged PSNR of restored images with the actual translation parameters versus lenslet number.
, 8
Fig. 8 Averaged MSE of restored images with the estimated translation parameters by SIFT-RANSAC versus lenslet number.
, and 9
Fig. 9 Averaged PSNR of restored images with the estimated translation parameters by SIFT-RANSAC versus lenslet number.
. As shown in these figures, l1-SAR has higher restoration performance than BTV, whereas BTV is better than IBP. Moreover, under the actual translation conditions, the restoration performance remains almost unchanged with the increasing lenslet number. On the other hand, under the estimated translation conditions the performance deteriorates with the increasing lenslet number. The registration error is a key factor that leads to the deterioration of restoration performances.

Figures 10
Fig. 10 The averaged horizontal registration errors versus lenslet number.
and 11
Fig. 11 The averaged vertical registration errors versus lenslet number.
present the averaged horizontal and vertical registration errors estimated using maximum likelihood [7

7. Y. Kitamura, R. Shogenji, K. Yamada, S. Miyatake, M. Miyamoto, T. Morimoto, Y. Masaki, N. Kondou, D. Miyazaki, J. Tanida, and Y. Ichioka, “Reconstruction of a high-resolution image on a compound-eye image-capturing system,” Appl. Opt. 43(8), 1719–1727 (2004). [CrossRef] [PubMed]

] and SIFT-RANSAC. As shown in these two figures, in general SIFT-RANSAC has smaller registration errors than maximum likelihood. However, the registration errors of both registration algorithms increase as the lenslet number increases. On the contrary, in the image restoration procedure, as actual translation parameter is equal to 1/n, higher registration accuracy is required as the lenslet number increases. Under this condition, larger registration errors lead to restoration performance deterioration.

As a larger lenslet number leads to thinner system dimensions, and the simulation results show that MSE or PNSR with 4 × 4 lenslets is almost the same as that in the single-lens system, the optimal lenslet number for TOMBO system is determined to be 4 × 4.

3. Experiment

In this section, the optimal lenslet number for TOMBO systems is verified by experimental comparison of angular resolutions among a 4 × 4 lenslet TOMBO system, a 5 × 5 lenslet TOMBO system, and a conventional single-lens system using the same monochrome image sensor. The sensor size is 10.9 mm × 10.9 mm with 1024 × 1024 pixels. For the 4 × 4 lenslet TOMBO system, lenses are aligned with a pitch of 2.6 mm. The focal length fl and the diameter of each lens are 20 mm and 2.6 mm, respectively. For the 5 × 5 lenslet TOMBO system, lenses are aligned with a pitch of 2 mm. The focal length fl and the diameter of each lens are 16 mm and 2 mm, respectively. For the single-lens system, we utilize Nikon Nikkor lens whose focal lens fs is 85 mm and f number ranges from 1.8 to 16.

The angular resolutions of the three systems with approximately the same f number ( = 8) are measured by a collimator. In the collimator, an illuminated USAF 1951 resolution target is positioned at the front focal plane of the objective lens as shown in Fig. 12
Fig. 12 The optical arrangements of experimental setups for (a) the TOMBO system and (b) the single-lens system.
. With this configuration, all light beams passing a point in the resolution target plane form a collimated light bundle behind the objective lens. The focal length fc and clear aperture of the collimator are 1000 mm and 100 mm, respectively. Figure 13
Fig. 13 Photo of the experimental setup for a TOMBO system.
presents a photo of the experimental setup for a TOMBO system.

The data captured by the 4 × 4 and 5 × 5 lenslet TOMBO systems are shown in Fig. 14
Fig. 14 The data captured by (a) the 4 × 4 lenslet TOMBO system and (b) the 5 × 5 lenslet TOMBO system.
. Figure 15
Fig. 15 The restored images in the (a) 4 × 4 and (b) 5 × 5 lenslet TOMBO systems and (c) the image captured by the single-lens system.
shows the restored images and the image captured by the single-lens system. The restored images are acquired by SIFT-RANSAC and l1-SAR with magnification factors of 4 for Fig. 15(a) and 5 for Fig. 15(b). In contrast to that in Fig. 15(a), the images shown in Fig. 15(b) suffer deterioration of restoration performance as demonstrated in the simulation. Compared with Fig. 15(c), the contrast between white bar and black section is lower in Fig. 15(a), but according to Eq. (3) the angular resolutions of these two images are the same.
α=2b×103fc×206,265,
(3)
where b is the minimal interval of white bars that can be distinguished, fc is the focal length of the collimator, and the value 206,265 is the arcseconds for one radian. As shown in Figs. 15(a) and 15(c), b in yellow squares corresponds to the resolution target’s group 1 element 3, which indicates 2.52 line pairs/mm. So the angular resolution for the 4 × 4 lenslet TOMBO system and the single-lens system is 81.85″, whereas the 5 × 5 lenslet TOMBO system has a poorer angular resolution performance. These results confirm that 4 × 4 is the optimal lenslet number for TOMBO systems, as the simulations showed.

4. Conclusion and future work

The subimage capturing processes of the TOMBO imaging system with different lenslet numbers have been simulated, and restoration algorithms have been applied to restore high-resolution images. MAE and PSNR of the restored images have been calculated for quantitative image quality assessment to optimize the lenslet number. The optimal lenslet number has been determined to be 4 × 4 by the simulations and confirmed by imaging experiments with 4 × 4 and 5 × 5 TOMBO imaging systems and a single-lens imaging system.

Future studies for an observation model of the TOMBO imaging system and more accurate registration between subimages should be performed. An ideal imaging process was assumed in the observation models introduced in Section 2. The consideration of the optical aberration in a real imaging system could improve the model. Furthermore, larger optimal lenslet numbers could be expected for thinner system dimensions with more accurate registration.

Acknowledgments

This work was supported by the Preeminent Youth Fund of Sichuan Province under grant 2012JQ0012 and the National Natural Science Foundation of China under grant 11173008.

References and links

1.

J. Tanida, T. Kumagai, K. Yamada, S. Miyatake, K. Ishida, T. Morimoto, N. Kondou, D. Miyazaki, and Y. Ichioka, “Thin observation module by bound optics (TOMBO): concept and experimental verification,” Appl. Opt. 40(11), 1806–1813 (2001). [CrossRef] [PubMed]

2.

J. W. Duparré and F. C. Wippermann, “Micro-optical artificial compound eyes,” Bioinspir. Biomim. 1(1), R1–R16 (2006). [CrossRef] [PubMed]

3.

D. Mendlovic, “Toward a super imaging system,” Appl. Opt. 52(4), 561–566 (2013). [CrossRef] [PubMed]

4.

K. Choi and T. J. Schulz, “Signal-processing approaches for image-resolution restoration for TOMBO imagery,” Appl. Opt. 47(10), B104–B116 (2008). [CrossRef] [PubMed]

5.

M. Shankar, R. Willett, N. Pitsianis, T. Schulz, R. Gibbons, R. Te Kolste, J. Carriere, C. Chen, D. Prather, and D. Brady, “Thin infrared imaging systems through multichannel sampling,” Appl. Opt. 47(10), B1–B10 (2008). [CrossRef] [PubMed]

6.

A. V. Kanaev, D. A. Scribner, J. R. Ackerman, and E. F. Fleet, “Analysis and application of multiframe superresolution processing for conventional imaging systems and lenslet arrays,” Appl. Opt. 46(20), 4320–4328 (2007). [CrossRef] [PubMed]

7.

Y. Kitamura, R. Shogenji, K. Yamada, S. Miyatake, M. Miyamoto, T. Morimoto, Y. Masaki, N. Kondou, D. Miyazaki, J. Tanida, and Y. Ichioka, “Reconstruction of a high-resolution image on a compound-eye image-capturing system,” Appl. Opt. 43(8), 1719–1727 (2004). [CrossRef] [PubMed]

8.

A. Stern and B. Javidi, “Three-dimensional image sensing and reconstruction with time-division multiplexed computational integral imaging,” Appl. Opt. 42(35), 7036–7042 (2003). [CrossRef] [PubMed]

9.

R. Horisaki, S. Irie, Y. Ogura, and J. Tanida, “Three-dimensional information acqusition using a compound imaging system,” Opt. Rev. 14(5), 347–350 (2007). [CrossRef]

10.

A. V. Kanaev, J. R. Ackerman, E. F. Fleet, and D. A. Scribner, “TOMBO sensor with scene-independent superresolution processing,” Opt. Lett. 32(19), 2855–2857 (2007). [CrossRef] [PubMed]

11.

A. A. El-Sallam and F. Boussaid, “Spectral-based blind image restoration method for thin TOMBO imagers,” Sensors (Basel Switzerland) 8(9), 6108–6124 (2008). [CrossRef]

12.

S. Mendelowitz, I. Klapp, and D. Mendlovic, “Design of an image restoration algorithm for the TOMBO imaging system,” J. Opt. Soc. Am. A 30(6), 1193–1204 (2013). [CrossRef] [PubMed]

13.

Z. Lin and H. Y. Shum, “Fundamental limits of reconstruction-based superresolution algorithms under local translation,” IEEE Trans. Pattern Anal. Mach. Intell. 26(1), 83–97 (2004). [CrossRef] [PubMed]

14.

S. Baker and T. Kanade, “Limits on super-resolution and how to break them,” IEEE Trans. Pattern Anal. Mach. Intell. 24(9), 1167–1183 (2002). [CrossRef]

15.

D. Robinson and P. Milanfar, “Statistical performance analysis of super-resolution,” IEEE Trans. Image Process. 15(6), 1413–1428 (2006). [CrossRef] [PubMed]

16.

S. Farsiu, M. D. Robinson, M. Elad, and P. Milanfar, “Fast and robust multiframe super resolution,” IEEE Trans. Image Process. 13(10), 1327–1344 (2004). [CrossRef] [PubMed]

17.

S. Villena, M. Vega, S. D. Babaccan, R. Molina, and A. K. Katsaggelos, “Bayesian combination of sparse and non-sparse priors in image super resolution,” Digit. Signal Process. 23(2), 530–541 (2013). [CrossRef]

18.

D. G. Lowe, “Distinctive image features from scale-invariant keypoints,” Int. J. Comput. Vis. 60(2), 91–110 (2004). [CrossRef]

19.

M. A. Fischler and R. C. Bolles, “Random sample consensus: a paradigm for model fitting with applications to image analysis and automated cartography,” Commun. ACM 24(6), 381–395 (1981). [CrossRef]

20.

M. W. Haney, “Performance scaling in flat imagers,” Appl. Opt. 45(13), 2901–2910 (2006). [CrossRef] [PubMed]

OCIS Codes
(110.1758) Imaging systems : Computational imaging
(110.3010) Imaging systems : Image reconstruction techniques

ToC Category:
Imaging Systems

History
Original Manuscript: January 21, 2014
Revised Manuscript: March 7, 2014
Manuscript Accepted: March 17, 2014
Published: April 1, 2014

Citation
Yuan Gao, Lizhi Dong, Ping Yang, Guomao Tang, and Bing Xu, "Influence of lenslet number on performance of image restoration algorithms for the TOMBO imaging system," Opt. Express 22, 8298-8308 (2014)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-7-8298


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References

  1. J. Tanida, T. Kumagai, K. Yamada, S. Miyatake, K. Ishida, T. Morimoto, N. Kondou, D. Miyazaki, Y. Ichioka, “Thin observation module by bound optics (TOMBO): concept and experimental verification,” Appl. Opt. 40(11), 1806–1813 (2001). [CrossRef] [PubMed]
  2. J. W. Duparré, F. C. Wippermann, “Micro-optical artificial compound eyes,” Bioinspir. Biomim. 1(1), R1–R16 (2006). [CrossRef] [PubMed]
  3. D. Mendlovic, “Toward a super imaging system,” Appl. Opt. 52(4), 561–566 (2013). [CrossRef] [PubMed]
  4. K. Choi, T. J. Schulz, “Signal-processing approaches for image-resolution restoration for TOMBO imagery,” Appl. Opt. 47(10), B104–B116 (2008). [CrossRef] [PubMed]
  5. M. Shankar, R. Willett, N. Pitsianis, T. Schulz, R. Gibbons, R. Te Kolste, J. Carriere, C. Chen, D. Prather, D. Brady, “Thin infrared imaging systems through multichannel sampling,” Appl. Opt. 47(10), B1–B10 (2008). [CrossRef] [PubMed]
  6. A. V. Kanaev, D. A. Scribner, J. R. Ackerman, E. F. Fleet, “Analysis and application of multiframe superresolution processing for conventional imaging systems and lenslet arrays,” Appl. Opt. 46(20), 4320–4328 (2007). [CrossRef] [PubMed]
  7. Y. Kitamura, R. Shogenji, K. Yamada, S. Miyatake, M. Miyamoto, T. Morimoto, Y. Masaki, N. Kondou, D. Miyazaki, J. Tanida, Y. Ichioka, “Reconstruction of a high-resolution image on a compound-eye image-capturing system,” Appl. Opt. 43(8), 1719–1727 (2004). [CrossRef] [PubMed]
  8. A. Stern, B. Javidi, “Three-dimensional image sensing and reconstruction with time-division multiplexed computational integral imaging,” Appl. Opt. 42(35), 7036–7042 (2003). [CrossRef] [PubMed]
  9. R. Horisaki, S. Irie, Y. Ogura, J. Tanida, “Three-dimensional information acqusition using a compound imaging system,” Opt. Rev. 14(5), 347–350 (2007). [CrossRef]
  10. A. V. Kanaev, J. R. Ackerman, E. F. Fleet, D. A. Scribner, “TOMBO sensor with scene-independent superresolution processing,” Opt. Lett. 32(19), 2855–2857 (2007). [CrossRef] [PubMed]
  11. A. A. El-Sallam, F. Boussaid, “Spectral-based blind image restoration method for thin TOMBO imagers,” Sensors (Basel Switzerland) 8(9), 6108–6124 (2008). [CrossRef]
  12. S. Mendelowitz, I. Klapp, D. Mendlovic, “Design of an image restoration algorithm for the TOMBO imaging system,” J. Opt. Soc. Am. A 30(6), 1193–1204 (2013). [CrossRef] [PubMed]
  13. Z. Lin, H. Y. Shum, “Fundamental limits of reconstruction-based superresolution algorithms under local translation,” IEEE Trans. Pattern Anal. Mach. Intell. 26(1), 83–97 (2004). [CrossRef] [PubMed]
  14. S. Baker, T. Kanade, “Limits on super-resolution and how to break them,” IEEE Trans. Pattern Anal. Mach. Intell. 24(9), 1167–1183 (2002). [CrossRef]
  15. D. Robinson, P. Milanfar, “Statistical performance analysis of super-resolution,” IEEE Trans. Image Process. 15(6), 1413–1428 (2006). [CrossRef] [PubMed]
  16. S. Farsiu, M. D. Robinson, M. Elad, P. Milanfar, “Fast and robust multiframe super resolution,” IEEE Trans. Image Process. 13(10), 1327–1344 (2004). [CrossRef] [PubMed]
  17. S. Villena, M. Vega, S. D. Babaccan, R. Molina, A. K. Katsaggelos, “Bayesian combination of sparse and non-sparse priors in image super resolution,” Digit. Signal Process. 23(2), 530–541 (2013). [CrossRef]
  18. D. G. Lowe, “Distinctive image features from scale-invariant keypoints,” Int. J. Comput. Vis. 60(2), 91–110 (2004). [CrossRef]
  19. M. A. Fischler, R. C. Bolles, “Random sample consensus: a paradigm for model fitting with applications to image analysis and automated cartography,” Commun. ACM 24(6), 381–395 (1981). [CrossRef]
  20. M. W. Haney, “Performance scaling in flat imagers,” Appl. Opt. 45(13), 2901–2910 (2006). [CrossRef] [PubMed]

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