# Coded Exposure Deblurring: Optimized Codes for PSF Estimation and Invertibility

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3 1 0 Blur Profile α Coded k Code = Slope=1/k Slope = (n/s)(1/k) Sharp Blur Coded Blur Horizontal Motion Sharp Coded ( ) Blur profile α Figure 3. (Left) Motion from blur constraint holds for coded exposure for those parts of the blur which correspond to ones in the code. For traditional camera, slope of α equals 1/k. For coded exposure, slope increases by a factor of n/s. (Middle) Synthetic example showing a polygon shaped object moving horizontally. (Right) Corresponding blur profiles obtained from blurred synthetic images. cubic phase plate in front of the lens to make defocus PSF invariant to depth. This enables the use of a single deconvolution filter to recover the sharp image without knowing the depths in the scene. Nagahara et al. [14] move the sensor in the lateral direction during image capture to make the defocus PSF invariant to depth. However, the drawback is that the typical plane of focus due to lens is also blurred. By moving the camera in a parabolic fashion, Levin et al. [11] make the motion PSF approximately invariant to the speed of the object. Similarly, the drawback is that static parts of the scene are also blurred. Coded exposure [17] and coded aperture [23] techniques make the PSF invertible so that the resulting deconvolution process becomes well-posed. However, PSF estimation is still required. PSF estimation and deblurring: Recent interest in computational photography has spurred significant research in PSF estimation and deblurring algorithms. Fergus et al. [5] use natural image statistics to estimate the PSF from a single blurred image. Joshi et al. [8] estimate nonparametric, spatially-varying blur functions by predicting the sharp version of a blurry input image. Yuan et al. [24] use both a short exposure image and a long exposure image to estimate the motion PSF and use them simultaneously for deblurring to handle camera shake. Recent work on deblurring algorithms [21, 25] have shown excellent results on images corrupted due to camera shake. 2. Blur estimation using alpha matting Let s(x, y) denote the image of the object if it was static and h(x, y) be the motion PSF. Let M(x, y) be a binary indicator function for the object 1. When the object moves in front of the background b(x, y), the captured blurred image I is given by the sum of blurred foreground object and partial background [17] I = s h + (1 M h)b. (1) Comparing with the familiar matting equation I = αf + (1 α)b [22], we get B = b, α = M h, F = (s h)/(m h). (2) 1 We assume that the moving object is opaque and in sharp focus. Note that the foreground for the matting algorithm is not the actual object s, but the blurred object which depends on the PSF h. Although matting algorithms can handle complex α (such as hair, smoke etc.) and thus discontinuous I, they require both the foreground and background to be locally smooth or low frequency. For a traditional camera, PSF is a box function (h is low pass) and results in smooth foreground F. Previous motion blur estimation algorithms based on alpha matting have shown very good results on images captured using a traditional camera. However, deblurring is ill-posed due to h being low pass Coded exposure camera The key idea of coded exposure is to open and close the shutter according to a pseudo-random binary code to preserve high spatial frequencies in the captured blurred image. Thus, the motion PSF h becomes broadband and deblurring is well-posed. However, this results in high frequency variations in the blur profile. Thus, alpha matting is not robust due to non-smooth foreground and PSF estimation using transparency is also hard due to non-smooth alpha. Our goal is to design the code so that certain parts of the code result in smooth blur to help matting and PSF estimation, while overall the code is still invertible for good deblurring. For rest of the paper, let c(x) be the code, n be the code length, s be the total number of ones, t be the number of transitions, and r be the maximum number of continuous ones in the code. A traditional camera can also be characterized as a coded exposure camera with s = r = n and t = 0. Note that the coded exposure camera loses light by a factor of n s. The linear system corresponding to motion blur is given by Ax = b, where A is the motion smear matrix, x is the unknown sharp image, and b is the blurred photo. Similar to [17], we use f noise = mean(a T A) 1 for evaluating the increase in deconvolution noise Motion from blur We first show that motion from blur constraint also holds for coded exposure camera. The constraint is given by [2] α k = ±1, (3)

5 k = [-28.78,5.36] k = [-54.8,5.08] Coded C 1 k = [-42.09,10.88] Coded C 2 Figure 5. Motorcycle moving at an angle. (Top) Blurred photos. (Middle) Alpha maps with inliers. (Bottom) Deblurred results. PSF estimation for traditional camera is good but deblurring is poor due to non-invertible PSF. Bad PSF estimation for code C 1 leads to poor deblurring. For C 2, estimated PSF is good, as proved by the deblurring result. The ratio between the lengths of motion vectors k for coded and traditional exposure should be n/s=31/21=1.47. It is 1.48 for C 2, 1.88 for C 1. Input images are rotated using the estimated motion angle before deblurring to bring the motion horizontal. For C 1, incorrectly estimated angle cannot be used to rectify the input image. ground truth, as shown by the good deblurring result. PSF estimation for traditional camera is also good but deblurring is bad due to PSF being non-invertible. Figure 5 (middle row) also shows inliers (different color for each cluster) obtained from MFB algorithm. For traditional camera, inliers span all parts of the blur as expected; while, for coded blur, the α-motion blur constraint only holds for those parts of the blur that correspond to 1 s in the code, as described in Section 2.2. Note that for C 2, most of the inliers are present on one end of the blur corresponding to the long string of 1 s in C 2. However, for C 1, inliers are scattered all over the blur which shows that alpha estimation and MFB algorithm was not successful. Figure 6 also shows the ground truth photo and the deblurring result for C 1 if the PSF estimated Ground Truth Coded C 1 Figure 6. Motorcycle. (Left) Ground truth sharp image. (Right) Deblurring result for C 1 using the motion PSF estimated from C 2 shows that the deblurring performances are similar for C 1 and C 2, but PSF estimation fails using C 1 (see Figure 5 middle image in the bottom row.) using C 2 is used. This clearly demonstrates that the deblurring performances for C 1 and C 2 are similar. However, C 2 assists in PSF estimation, while C 1 does not. Non-uniform background: Figure 7 shows an example of a moving sticker in front of a non-uniform background. Again note that the estimated inliers for C 2 are restricted to those parts of the blur, which correspond to the long string of 1 s. The deblurring results demonstrate that the motion estimation is good for C 2, but poor for C 1. Complex object shape: Figure 8 shows another example on an complex shaped action figure. Even though the shape is complex, our algorithm successfully estimates the PSF using C 2 since it produces partial smooth blur. Fine features are recovered on the action figure using C 2 code. Outdoor scene: Our approach also works on challenging outdoor scene as shown in Figure 1. Since the car is far away, it is assumed to be moving parallel to the image plane. A n = 15 code with r = 7 was used to capture the photo. Note that the deblurring result recovers sharp features on the car. 4. Implementation and analysis In [17], coded exposure was implemented using an external ferro-electric shutter placed in front of the lens of a SLR camera. The ferro-electric shutter from DisplayTech

7 C best C 2 Ground Truth Coded C best Coded C Figure 10. (Left) Visual deblurring comparison of C 2 versus C best on real datasets. Note that the proposed code C 2 gives similar deblurring performance compared to optimal code C best. (Right) PSF estimation capability for n = 31 codes with increasing r (decreasing number of 0 1 transitions). For small r, PSF estimation fails leading to poor deblurring results. Note that r = 5 for C best, and thus optimal invertible code may not give good PSF estimation. As r increases, PSF estimation improves, but PSF invertibility degrades. r maximum r in the sorted list. A second faster approach is to first set r, the continuous number of ones in the code. For simplicity, let the first r bits be ones. The search space is reduced to 2 n r 2 (the (r + 1) th bit has to be 0 and the last bit has to be 1). Among these codes, we choose those whose f noise fnoise th and s = sth, and pick the one with minimum t. If no code satisfies the criteria, r is decreased by one and the search is repeated. The code C 2 described in Section 2.3 is found using the second approach for n = 31 and r = 13 by testing only = 65, 536 codes in 6.7 seconds on a standard PC. For this code, s = 21 and searching the code using the first approach requires testing ( ) =20.03 million codes, which is times more than that of the second approach Analysis We compare the proposed codes with the optimal code for PSF invertibility for the same light level. The optimal invertible code [17] simply minimizes f noise without considering PSF estimation (r and t). Obviously, the proposed codes will lead to more deconvolution noise, but the increase in deconvolution noise is small and visually the deblurring results are comparable. For example, for n = 31, the optimal invertible code C best = was found using [17]. The deconvolution noise for optimal code C best is 18.52dB compared to 20.05dB for C 2. Figure 9 compares the f noise for the proposed codes and optimal invertible codes for r varying from 1 to n. For a given r, we obtain our code using the approach described in Section 4.1. We record the s value and then find the optimal invertible code which has the same s value (for the same light level). The f noise for a traditional exposure with the same light level is also plotted in black. When r = n, there is only one code (all ones) and all three curves meet. In general, the optimal invertible codes have r around 3 5, so the green and red curves meet at low r values. The plot shows that the increase in f noise using proposed codes is small and the proposed codes are significantly better than the traditional camera in terms of deconvolution noise. Figure 10 (left) shows visual deblurring comparisons on real datasets for C best and C 2 using the same motion PSF (estimated from photos captured using C 2 ). Note that the deblurring results are visually similar. PSF estimation: We analyze the PSF estimation capability of the proposed codes for different values of r for a given n. As r increases, the code becomes similar to a traditional camera (r = n) and becomes favorable for PSF estimation, but f noise increases significantly. Smaller values of r (r 5) result in significant noise in the estimation of alpha values. In Figure 10 (right), we show results using codes having different r values. In general, we found that codes with r n/3 work well for PSF estimation. 5. Discussions We have focused on binary valued codes; however continuous valued codes can improve both PSF estimation and invertibility. As shown in [23], continuous valued codes perform better than binary codes in terms of deconvolution noise, since they could avoid the sharp transitions of a binary code and result in smoother blur. In fact, optimizing such codes will be easier using continuous optimization compared to the discrete search used for binary codes. To enforce smooth blur, a penalty on the spatial gradients of the code can be applied, similar to the regularization techniques. However, their implementation is not straightforward using external shutters or trigger-based cameras. It could be achieved by controlling the A/D gain during the

8 exposure time according to the code, but would require changes at the chip level. We have focused on spatiallyinvariant PSF, but the proposed codes could also be used for affine motion using variations of the MFB algorithm. Our approach shares the same limitations of the alpha matting algorithm (e.g., low-frequency background) and requires a few brushes for matting initialization. Combining information from multiple images captured with same or different codes will further help in matting and PSF estimation. Conclusions: PSF estimation is as important as PSF invertibility for motion deblurring. A traditional camera results in smooth blur which is easier to estimate, but makes the PSF non-invertible. A coded exposure camera makes the PSF invertible but results in sharp discontinuities in the blur and degrades PSF estimation. We showed that both criteria of PSF estimation and invertibility can be achieved by carefully designing the code. We proposed design rules based on minimizing the transitions and maximizing the number of continuous ones in the code for good PSF estimation and described two schemes for searching such codes. We analyzed the performance of the proposed codes in comparison with the optimal invertible codes. We also described how coded exposure can be implemented on machine vision sensors without any additional cost and presented real experimental results that showed the effectiveness of the proposed codes for PSF estimation and invertibility. Acknowledgements We thank Ramesh Raskar for stimulating discussions. We also thank Jay Thornton, Keisuke Kojima, and Haruhisa Okuda, Mitsubishi Electric, Japan, for help and support. References [1] R. Accorsia, F. Gasparinib, and R. C. Lanza. Optimal coded aperture patterns for improved snr in nuclear medicine imaging. Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 474: , [2] S. Dai and Y. Wu. Motion from blur. In Proc. Conf. Computer Vision and Pattern Recognition, pages 1 8, June [3] E. R. Dowski and W. Cathey. Extended depth of field through wavefront coding. Appl. Optics, 34(11): , Apr [4] E. Fenimore and T. Cannon. Coded aperture imaging with uniformly redundant arrays. Appl. Optics, 17: , [5] R. Fergus, B. Singh, A. Hertzmann, S. T. Roweis, and W. T. Freeman. Removing camera shake from a single photograph. ACM Trans. Graph., 25(3): , [6] P. Jansson. Deconvolution of Image and Spectra. Academic Press, 2nd edition, [7] J. Jia. Single image motion deblurring using transparency. In Proc. Conf. Computer Vision and Pattern Recognition, pages 1 8, June [8] N. Joshi, R. Szeliski, and D. Kriegman. PSF estimation using sharp edge prediction. In Proc. Conf. Computer Vision and Pattern Recognition, June [9] A. Levin, R. Fergus, F. Durand, and W. T. Freeman. Image and depth from a conventional camera with a coded aperture. ACM Trans. Graph., 26(3):70, [10] A. Levin, D. Lischinski, and Y. Weiss. A closed-form solution to natural image matting. IEEE Trans. Pattern Anal. Mach. Intell., 30(2): , [11] A. Levin, P. Sand, T. S. Cho, F. Durand, and W. T. Freeman. Motion-invariant photography. ACM Trans. Graph., 27(3):1 9, [12] C.-K. Liang, T.-H. Lin, B.-Y. Wong, C. Liu, and H. H. Chen. Programmable aperture photography: multiplexed light field acquisition. ACM Trans. Graph., 27(3):1 10, [13] L. Lucy. An iterative technique for the rectification of observed distributions. J. Astronomy, 79: , [14] H. Nagahara, S. Kuthirummal, C. Zhou, and S. Nayar. Flexible Depth of Field Photography. In Proc. European Conf. Computer Vision, Oct [15] S. K. Nayar, V. Branzoi, and T. Boult. Programmable imaging using a digital micromirror array. In Proc. Conf. Computer Vision and Pattern Recognition, volume 1, pages , [16] H. Poor. An Introduction to Signal Detection and Estimation. Springer-Verlag, [17] R. Raskar, A. Agrawal, and J. Tumblin. Coded exposure photography: motion deblurring using fluttered shutter. ACM Trans. Graph., 25(3): , [18] N. Ratner and Y. Y. Schechner. Illumination multiplexing within fundamental limits. In Proc. Conf. Computer Vision and Pattern Recognition, June [19] W. Richardson. Bayesian-based iterative method of image restoration. J. Opt. Soc. of America, 62(1):55 59, [20] Y. Y. Schechner, S. K. Nayar, and P. N. Belhumeur. A theory of multiplexed illumination. In Proc. Int l Conf. Computer Vision, volume 2, pages , [21] Q. Shan, J. Jia, and A. Agarwala. High-quality motion deblurring from a single image. ACM Trans. Graph., 27(3):1 10, [22] A. R. Smith and J. F. Blinn. Blue screen matting. In SIG- GRAPH, pages , [23] A. Veeraraghavan, R. Raskar, A. Agrawal, A. Mohan, and J. Tumblin. Dappled photography: Mask enhanced cameras for heterodyned light fields and coded aperture refocusing. ACM Trans. Graph., 26(3):69, [24] L. Yuan, J. Sun, L. Quan, and H.-Y. Shum. Image deblurring with blurred/noisy image pairs. ACM Trans. Graph., 26(3):1, [25] L. Yuan, J. Sun, L. Quan, and H.-Y. Shum. Progressive interscale and intra-scale non-blind image deconvolution. In SIG- GRAPH 08: ACM SIGGRAPH 2008 papers, pages 1 10, New York, NY, USA, ACM. [26] A. Zomet and S. Nayar. Lensless imaging with a controllable aperture. In Proc. Conf. Computer Vision and Pattern Recognition, pages , 2006.

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