Revisiting Cross-channel Information Transfer for Chromatic Aberration Correction

Transcription

1 Reviiting Cro-channel Information Tranfer for Chromatic Aberration Correction Tiancheng Sun 2, Yifan Peng 3, Wolfgang Heidrich,3 King Abdullah Univerity of Science and Technology, Thuwal, Saudi Arabia 2 IIIS, Tinghua Univerity, Beijing, China 3 The Univerity of Britih Columbia, Vancouver, Canada Abtract Image aberration can caue evere degradation in image quality for conumer-level camera, epecially under the current tendency to reduce the complexity of len deign in order to hrink the overall ize of module. In implified optical deign, chromatic aberration can be one of the mot ignificant caue for degraded image quality, and it can be quite difficult to remove in pot-proceing, ince it reult in trong blur in at leat ome of the color channel. In thi work, we reviit the pixel-wie imilarity between different color channel of the image and accordingly propoe a novel algorithm for correcting chromatic aberration baed on thi cro-channel correlation. In contrat to recent weak prior-baed model, our ue trong pixelwie fitting and tranfer, which lead to ignificant quality improvement for large chromatic aberration. Experimental reult on both ynthetic and real world image captured by different optical ytem demontrate that the chromatic aberration can be ignificantly reduced uing our approach.. Introduction Modern optical ytem uually conit of dozen of heavy and complex lene, which are ued to compenate for aberration of all kind [5]. However, the increae of the computing power and the demand for more compact device, ha recently tarted a puh to optical deign with reduced complexity, that intead ue computational imaging method to recover high quality photograph [23, 7, 8, 9]. Thi i of ignificance particularly in mobile device, where form factor contraint mandate optical deign with a mall number of len element, and the mall ize of component complicate manufacture and alignment. Image captured by uncorrected lene may uffer from both monochromatic aberration (e.g. pherical, atigmatim, coma, field curvature), a well a chromatic aberration. Further, in the abence of optical image tabilizer (OIS), the captured image may be ubject to motion blur. With optical mean, chromatic aberration can only be corrected by involving multiple len element with different geometric hape and different refractive indice. Thu, there i a high demand on an effective deblurring approache pecially targeting the recontruction of color image captured by reduced complexity optic, which i alo the objective of our work. Specifically, we tackle thi problem by exploiting the pixel correlation between color channel. In thi way, we reduce the problem to ingle channel deblurring, and how that high-quality color image can be recovered with cro-channel information... Related work In the following we will ummarize two type of related work: general deconvolution/deblurring reearch, and work that pecifically target chromatic aberration. Image deblurring Early development of deconvolution algorithm include direct frequency invere, Wiener filter [29], which fail to tackle thoe frequencie that are zeroed out by the blur kernel, and thu may reult in ringing artifact. Later development of iterative algorithm [5] extract better reult but may amplify noie level. In recent year, reearcher have attempted to add extra prior knowledge into deconvolution to retore highfrequency information. Chan et al. [2] ued a total variation prior, where a global optimum can be obtained with convex optimization. Alternative non-convex regularization term have been invetigated [2, ], empirically giving improved reult at reaonable local optimum. Cho et al. [3] explored to match gradient ditribution in order to retore mid-frequency information. All thee method rely on weak tatitic prior for image retoration. General image deconvolution model require a known convolution kernel, which could be impractical to calibrate, for intance motion blur. Accordingly, blind deconvolution algorithm have been extenively tudied aiming to recontruct the latent image and etimate the kernel imultaneouly [24]. Similar to their non-blind counterpart, a wide

2 variety of prior are introduced to reolve reaonable reult in image enhancement [, 3, 26]. An invetigation on the ue of pare gradient prior for blind deconvolution via total variation can be found in [2]. Chromatic aberration correction The chromatic aberration oberved in image are caued by the wavelengthdependence of focal length. Different point pread function (PSF) in channel reult in color fringe on harp edge. Alternative low-level technique ha been applied to remove the color fringe in conventional complex optical ytem [9, 4]. However, thee method fail to tackle the large aberration induced by uncorrected lene. Schuler et al. [22] firt propoed an aberration removal algorithm for a ingle len in YUV color pace. Further, a convex cro-channel prior i developed and efficiently olved in [7]. Accordingly, everal implified optical application have been preented a low-end camera len i corrected via optical computing in [3]; DOE are optimized along with aberration removal algorithm to reolve high-quality image [8, 9]. Although challenging, it i practically of ignificance to remove aberration without the knowledge of kernel. Beide imply implementing blind deconvolution on each channel, the ymmetry of the convolution kernel are exploited in [23], and the geometric and viual prior are invetigated in [3]. While exiting model are ucceful for yielding reaonably good reult with chromatic aberration mitigated, they may over-implify the dratic patiallyvarying defocu effect (e.g. commonly at the borderline of harp foreground and blurred background, reulting in a failure of geometric ymmetry aumption). Thee method may reult in color fidelity lo of reolved image (a we will dicu in our experiment)..2. Motivation and contribution We bae our work on two primary obervation. Firt, a hown in Fig., the dependence of focal length on wavelength reult in evere chromatic aberration, which i often too challenging to robutly recovery with tate-of-theart gradient tranfer cheme. However, at leat one of the channel uually exhibit relatively harper focu (and could be recovered more eaily) when the camera ha been focued at a patial point. Second, depite the different pixel intenity between channel, the high-level tructure conveyed by the channel are motly identical. Thee fact have inpired u to exploit eential imilarity of a color image, and accordingly tranfer pixel information acro channel, to blindly remove the evere chromatic aberration introduced by uncorrected or complexity reduced optic. In particular, the technical contribution include: We invetigate the cro-channel content imilarity of natural image by deriving a novel image formation Figure : Relation between focal length and wavelength in two typical optical ytem. The focal length of a refractive len increae when the wavelength increae (left); while that of a diffractive len exhibit the oppoite way (right). The color diperon of a diffractive len i much larger than that of a refractive len. (Image credit at [27]) model to expre one channel uing another channel. Intead of relying on weak tatitical prior knowledge, our method ue a trong pixel-wie correlation, leading to a deconvolution that can tackle coniderable chromatic aberration efficiently. We compare our algorithm againt tate-of-the-art method, and our exhibit overall better performance in term of removing chromatic aberration while uppreing edging artifact. We alo work in a blind deconvolution mode without previouly known kernel. We verify our method on data captured by imple refractive len and diffractive len, uggeting it univerality and robutne acro divere optical ytem, making our method a promiing candidate for developing lightweight computational imaging olution. 2. Image formation model Cro-channel imilarity. To reviit the problem of tranferring information from a harper channel to other, we tart by exploring the cro-channel imilarity uing natural image with no (or only imperceptible) aberration. Different channel in natural image convey largely redundant information, ince trong edge and texture tend to be aligned between the channel. However, although the correlation between channel i trong due to joint tructure [4] in edge and texture feature, the exact nature of thi correlation can be complicated and difficult to expre in cloed form. Recent work have attempted to model pecific form of channel correlation, uch a pare hue change at object boundarie. However, there exit a myriad of other type of hue change in natural image, uch a color gradient, rainbow etc., thu reulting in relationhip that i in general highly non-linear. In our work, we follow the trategy of Taylor expanion to mode thi inter-channel dependency. Although it i hard

3 original recontruct red recontruct blue original green original red original blue RGB catter plot.9 Blue Red Green Blue Red Green Figure 3: Reult of our imilarity model teted on a mall patch cropped from the image by olving Eq. 2. The red and blue channel are expreed parely uing the pixel intenitie of green channel. Notice that the recontructed window (upper middle and upper right) are almot identical to the original one (lower middle and lower right), indicating the effectivene of our cro-channel imilarity model. Channel Channel RGB catter plot from another channel I, imilar to a Taylor expanion:.9 Blue Red Green Blue Red Green It α + α I + α2 x I + α3 y I + Channel 2.7 = T(I ) α, ().5 where.3 T(I ) =. Channel RGB catter plot.9 Blue Red Green Blue Red Green Channel Channel Figure 2: The catter plot of pixel value from three patche elected from a ample image. The red, yellow, blue plot correpond to RG plot, GB plot, and BR plot of each patch. The linearity within mall patche can been clearly oberved (ditribution of color plot). to formulate the correlation from a global view, one can till linearize the problem within mall window cropped from an image. Specifically, we reaonably aume under a mall window patch which contain only few feature (e.g. edge, corner point, etc.), the pixel intenity of one channel It can be expreed a a weighted um of higher-order derivative I x I y I, and α are the coefficient to be fitted. Notice that thi i the model for an intrinic (i.e. aberration free) image. The propoed model i validated on elected image from dataet in []. A hown in Fig. 2, we crop three patche from the original image, and plot them out ubject to their RGB value. From the patche graph, we can oberve noticeable linearity which i ueful in correcting chromatic aberration. Then, the problem (Eq. ) can be treated a a fitting operation with it leat-quare olution being calculated efficiently uing peudo-invere, to find α by olving: 2 αopt = argmin kit T(I ) αk2. (2) α In practice, we only expand the erie to the econd order derivative of the image (i.e. xx I, xy I and yy I). The recontructed reult of a mall window patch by olving Eq. 2 i hown in Fig. 3. We ee that the red and blue channel are reaonably expreed uing the pixel intenitie of green channel, exhibiting very imilar content to the original one. Chromatic aberration formation. The chromatic aberration in the captured image are mainly caued by different focuing contribution in different channel. Generally,

4 we denote I, J, N a the underlying harp image, oberved image, and additive noie in channel, repectively. The oberved image in channel can be formatted a: J = B I + N, (3) where B i the point pread function (PSF) of channel. Although the content in I are trongly correlated acro different channel, the PSF (B ) may vary dratically between channel, which overall lead to the chromatic aberration uch a red or purple fringe. 3. Image recontruction We rely on the imilarity model from above to reolve the chromatically aberrated image captured by complexity reduced lene. We firt reolve a harp reference channel uing a mature deconvolution cheme, then develop a method for individual image patche where we aume a contant PSF a well a the validity of our cro-channel model derived from Eq. 2. After that, we incorporate thi patch-baed olution into an efficient algorithm at full image cale. Reference channel deblurring. Given a blurry color image, there i at leat one channel exhibiting relatively harper information, which i uually the green channel [7, 6]. For generality, we denote the relatively harper channel a I ref. We then apply a blind deconvolution algorithm on thi channel, in order to uppre the monochromatic aberration and poibly the motion blur. Section 4 detail thi ingle-channel deblurring algorithm. Cro-channel information tranfer. Given a clear image of reference channel I ref, we build our algorithm to deblur other channel by tranferring the harp information acro channel. By introducing the blur kernel into our imilarity model, we now formulate an optimization problem to deconvolve mall image patche, a follow: min J B (T(I ref ) α) 2 α,b 2 + µ B ν a a B 2 2, (4) where I ref i the image patch of the harp reference channel and, J i that of the blurred channel. The claical l 2 fuion data term and gradient prior are added to regularize PSF. Notice that the above problem i biconvex and can be olved by alternatively updating α and B. The B tep (PSF etimation tep) can be olved efficiently in Fourier domain, and the α tep (cro channel tranfer tep, aka. CCT tep) can be olved by uing peudo-invere. Compared to previou work [7, 3], we don t add any tatitical prior to the CCT term, but ue a et of appropriate bae (i.e. T(I ref )) which are derived from other channel to retore the blurred image. In other word, we mainly rely on Figure 4: Left: ground truth image; right: recovered image uing -order CCT initialization. Notice that in the zoom-in inet the black line have turn red. other harp channel rather than everely blurred channel itelf to extract effective information. Thu, more accurate pixel information can be tranferred with a impler model. Notice that there i a trade-off between the patch ize for PSF etimation and the cro-channel model. On the one hand, the window ize for PSF etimation hould be a large a poible, for the PSF to be recovered robutly. On the other hand, the approximation of one color channel by lower-order term of another channel work bet in relatively mall tile. We reolve thi trade-off by working at two different cale maller patche (CCT window) for the color model and larger window for PSF etimation. We dicu thee and other parameter in Section 4. Initial etimation. Since the alternating trategy can only converge to local optimum, a good initial etimation of harp image i of ignificance to peed up the convergence and avoid noticeable artifact. An intuitive olution for initial gue i the -order CCT: min α J (α + α I ref ) 2 2. (5) However, uing thi initial gue may till reult in artifact. A hown in Fig. 4, mot black line have turned red. Thi i becaue thoe black line or harp peak have been moothed away, epecially in red and blue channel where the blur kernel are of non-trivial ize. Thu, imply implementing CCT may over-fit the moothed ignal, a hown in Fig. 5. To fix thi iue, we eek to directly hift the ignal from reference channel to other channel rather than data fitting, indicated a follow: I = mean(j ) mean(i ref ) I ref. (6) where I i the harp image patch of channel to be etimated, and J i the correponding oberved image patch. By applying Eq. 6 in the initialization, the red line artifact can be dratically mitigated, improving the conditioning for

5 Pixel intenity Red groundtruth Green groundtruth Red oberved Red recontructed Pixel coordinate Figure 5: Pixel value plot of a -d lice of the image. The ditribution of the original red and green ignal are quite imilar, while the intenity peak in the ignal i moothed out in the oberved red channel (orange plot). Thu, imply relying on CCT can only recontruct a mall peak, which lead to the noticeable red line artifact. Algorithm Chromatic aberration correction algorithm : Blind deconvolution on the relatively harper channel I ref = BlindDeconv(J ref ) 2: for channel in other channel do 3: Initial gue of the harp image I baed on blurred image J and harp reference channel I ref on each CCT window, and then merge them: I CCT = mean(jcct ) mean(i CCT ref ) ICCT ref I = Merge ( I CCT 4: repeat 5: Etimate the blur kernel in each PSF window: B = argmin J PSF B I PSF 2 2 B ) further deblurring. Pleae refer to Fig. 8 in the next ection. Patch blending. Poible method to merge thee mall window patche include overlapping the mall window and extracting the mean or median of each pixel value, or jut extracting the value of the nearet window. Marwah et al. [7] have hown that the median-baed method outperform other alternative. However, the image recontructed by the tate-of-the-art uffer from vertical or horizontal edging artifact. Thu, we multiply a Hamming window before taking the average. Thi trategy lead to le edging artifact appearing and recover a higher image quality. Joint algorithm. The algorithm for correcting the chromatic aberration i ummarized in Alg.. Given a corrupted color image, we firtly apply an efficient blind deconvolution on the le blurred channel image (uually the green one). Next, relying on the deconvolved channel image, we implement PSF etimation and CCT alternatively on image from other channel. Here in the algorithm, we ue I, J to denote the full-ize image, and I PSF /I CCT, J PSF /J CCT to denote the patch image from PSF/CCT etimation window. 4. Implementation and dicuion In thi ection, we preent elected experimental reult to verify the effectivene of our algorithm. Refer to upplementary document for full reolution reult. 4.. Parameter and analyi Blind deconvolution pre-proceing. We ue the algorithm in [] a the firt tep proceing to deconvolve the reference channel. The applied algorithm reduce achromatic aberration of mot kind, and more importantly al- µ B ν a B 2 2 a ( = F F ( ) I PSF ( ) ) F J PSF F (I PSF ) 2 + µ a F ( a) 2 + ν 6: Calculate coefficient in each CCT window α = argmin J CCT B (T(I CCT G ) α) 2 2 α 7: Derive the harp image patche baed on the coefficient and merge them: 8: until Converge 9: end for I CCT = T(I CCT ref ) α I = Merge ( I CCT ) low to control the aggreivene of blind deconvolution o a to uppre artifact that may affect the retored image quality of other channel. PSF etimation window ize. Intuitively, the window ize of PSF etimation hould be et large to contain ufficient feature in order to yield a good etimation. Practically, PSF ditribution may vary dratically due to the patial variance of the cene. Empirically, thi window ize i et between 33% to 5% of image ize. In order to avoid edging artifact, both CCT window and PSF etimation window hould overlap, which we empirically et to /5 of the window ize. We et the two regularizer parameter in etimating PSF (µ and ν) to.3 in the following.

6 R e d B l u e Figure 6: PSNR heatmap of red and blue channel of a retored image (Figure 8) ubject to different parameter etting. The horizontal axi denote CCT window ize while the vertical one denote PSF etimation window ize. Cro-channel tranfer window ize. Conceptually, the CCT window hould contain few feature in order to exploit the content linearity. In practice, we alo oberve that an overly large CCT window may reult in a pale or low color fidelity image. It can t be too mall a well, otherwie, the content of the mall window patch may fail to contain ufficient color information ubject to the full pectrum, making the problem trongly ill-poed. For an image with, 4, pixel, the ize of CCT window i et 7 7 pixel. Robutne. The ize of PSF etimation window and CCT window can be etimated from the ize of image. To ae the robutne ubject to different parameter, we run our algorithm on the upper image of Fig. 8 with different etting of PSF and CCT window (Fig. 6). We found that the image quality doen t change much when window ize change. The optimal PSNR lie on a plateau (ee the red color block) rather than on individual harp peak, uggeting our algorithm doe not need to fine-tune many parameter. Convergence. Although the topping criteria in Alg. i not determinitic, the alternating optimization converge very quickly in practice. Empirically, there i little perceptible change after the firt iteration, and the imilarity metric continue to improve for a few iteration. Thu we fix thi parameter to 3 iteration. Efficiency. Denote N, N p, N c a the ize of full image, PSF etimation window, and CCT window, repectively. The time complexity for a ingle iteration can be indicated a O ( (N N p ) 2 log Np 2 + ( )) N 2 + N 3 /Nc 3. Notice that the firt term i almot the ame a that of [6]. Without a convergence problem when retoring the harp image, the econd term i relatively mall compared to the tateof-the-art. In term of practical computational efficiency, the propoed algorithm retore one blurry channel image (, 4, pixel) within 7 econd, uing a Matlab Pixel intenity Pixel coordinate Figure 7: PSF illutration for the ynthetic image (left), with it cro-ection intenity in three channel (right). Notice that the green channel ha the harpet PSF ditribution. Table : Quantitative comparion on ynthetic image, indicated a averaged PSNR and SSIM value. PSNR SSIM Blurred image Krihnan et al. [] Heide et al. [6] Our Non-blind [7] implementation on a PC with Intel Core i7 CPU at 4.GHz. Pleae refer to upplementary document for detailed analyi Synthetic reult We tet our algorithm on the image from dataet [] for quantitative evaluation. All 29 ynthetic image without cro-channel difference are ued for evaluation. The image convolved with a PSF that i derived from a real Frenel len (Fig. 7), whoe chromatic aberration (corrupted with trong noie) i much more evere than that of conventional refractive lene. We compare our algorithm againt 2 blind method and non-blind method with averaged PSNR and SSIM [28] aement (Tab. ). Specifically, Krihnan method imply applie blind deconvolution on three channel individually. Intead, Heide method have conidered the cro-channel correlation to further improve the color fidelity. Our algorithm outperform thee blind aberration correction. Moreover, compared with the non-blind deconvolution [7], our reult are till competitive. Pleae refer to upplemental document for full reolution reult. Selected cropped region are preented to highlight thoe detail tructure in Fig. 8. Specifically, the dek cene (top row) how that our algorithm i capable of eliminating thoe red fringe even though the color are cattered over a large region becaue of defocu. Epecially, the red line artifact (bottom row) ha been reolved with the aid of our initialization gue. We encourage reader to refer to the upplemental document for more reult.

7 (a) Groundtruth (b) Blurred (c) Non-blind (d) Krihnan et al. (e) Yue et al. (f) Heide et al. (g) Our Figure 8: Image recovered via applying different algorithm. Refer to the upplemental document for more reult Real world reult We alo apply our algorithm to the image captured uing a refractive len and a diffractive len, repectively. Refractive len data. The image captured by conventional plano-convex lene are hown in Fig. 9. In the firt two column, we compare with the work in [7], which belong to non-blind deconvolution cheme. The comparion ugget that our algorithm remove the aberration without acrificing fine-grain detail (econd column). Particularly, in the firt column the color fringe have been eraed even though partial image i out-of-focu. Surpriingly, our reult are lightly harper than that of the non-blind verion in ome cae. However, we till oberve artifact in our retored image, uch a the color fidelity lot in number 6. The reult in the third column come from [3], notice that the color fringe along the window and the color pot are completely removed. Although our intention wan t dealing with motion blur, we tet our algorithm on data from []. In the fourth column, we how that our algorithm i alo capable of removing a decent amount of motion blur. Diffractive len data. Recently, relying on diffractive optic to hrink len volume ha been an attractive direction [25, 8]. However, the induced large color fringe till need to be corrected. The image captured by a diffractive Frenel len [6] are evaluated (Fig. ). From the econd and fourth column, we ee that large chromatic fringe are eliminated correctly by our algorithm. A well, even for the color fringe on high reflection component (firt column) and large hue hift (rubic cube in third column), our algorithm i ufficiently robut to correct them properly Dicuion In our algorithm we don t add additional gradient prior on latent image a regularizer when implementing the nonblind deconvolution part. We empirically found that the image quality doen t improve harply with adding a cro- channel gradient prior like in [7]. Thi ugget that our trong pixel-wie tranfer in data fitting term ha fully contained the information provided by the weak tatitical prior. Refer to the upplementary document for detail. Although high image quality ha been achieved in our implementation, mall artifact remain. In thoe cenario where none of the channel image i ufficiently harp, ringing artifact could be introduced during the deconvolution proce of the reference channel and pread from there into other channel. Thi may reult in achromatic ringing, which i viually le noticeable than chromatic ringing, but till worth invetigating more. Alo, we approximate the cro-channel correlation via a imple math cheme (Eq. ) and move it into the data fitting term, which hall be computationally efficient but may not be very precie for all natural cene. One poible improvement i to apply learningbaed trategie to exploit a more comprehenive model. 5. Concluion We propoe a pot-proceing algorithm to correct chromatic aberration for color image and demontrate bet-incla performance on removing uch chromatic aberration without calibrating optical ytem. We have validated the robutne of our algorithm with a variety of yntheized image, where we obtain competitive reult a tate-of-theart method. The implementation on real world image captured by two different complexity reduced lene have further verified the effectivene and the univeral. We enviion thi method be an effective tool to enhance image quality epecially in cenario where high-quality optic are not feaible due to weight or form-factor conideration, uch a in mobile device. Importantly, our algorithm i capable of reolving image captured by a wide range of optical ytem, including diffractive optic which exhibit evere chromatic aberration where mot of tate-ofthe-art chromatic aberration correction method fail.

8 original tate-of-the-art our (a) Heide et al. [7] (b) Heide et al. [7] (c) Yue et al. [3] (d) Krihnan et al. [] our tate-of-the-art original Figure 9: Retored image captured by refractive lene from tate-of-the-art work. Zoom in for detail. Figure : Retored image captured by a diffractive len, uffering from evere chromatic aberration, with tate-of-the-art recontruction [6]. Zoom in for detail. Acknowledgement Thi work wa upported by KAUST baeline funding, a well a a UBC 4YF Doctoral Fellowhip. The author thank Tao Yue, Qiang Fu, and Felix Heide for the help on ynthetic reult. Reference [] A. Chakrabarti and T. Zickler. Statitic of Real-World Hyperpectral Image. In Proc. IEEE Conf. on Computer Viion and Pattern Recognition (CVPR), page 93 2, 2. [2] S. H. Chan, R. Khohabeh, K. B. Gibon, P. E. Gill, and T. Q. Nguyen. An augmented lagrangian method for total variation video retoration. Image Proceing, IEEE Tranaction on,

9 2():397 3, 2. [3] T. S. Cho, C. L. Zitnick, N. Johi, S. B. Kang, R. Szeliki, and W. T. Freeman. Image retoration by matching gradient ditribution. IEEE Tranaction on Pattern analyi and machine intelligence, 34(4): , 22. [4] S.-W. Chung, B.-K. Kim, and W.-J. Song. Detecting and eliminating chromatic aberration in digital image. In Image Proceing (ICIP), 29 6th IEEE International Conference on, page IEEE, 29. [5] R. E. Ficher, B. Tadic-Galeb, P. R. Yoder, and R. Galeb. Optical ytem deign. Citeeer, 2. [6] F. Heide, Q. Fu, Y. Peng, and W. Heidrich. Encoded diffractive optic for full-pectrum computational imaging. Scientific Report, 6, 26. [7] F. Heide, M. Rouf, M. B. Hullin, B. Labitzke, W. Heidrich, and A. Kolb. High-quality computational imaging through imple lene. ACM Tranaction on Graphic (TOG), 32(5):49, 23. [8] B. K. Horn and B. G. Schunck. Determining optical flow. Artificial intelligence, 7(-3):85 23, 98. [9] S. B. Kang. Automatic removal of chromatic aberration from a ingle image. In Computer Viion and Pattern Recognition, 27. CVPR 7. IEEE Conference on, page 8. IEEE, 27. [] D. Krihnan and R. Fergu. Fat image deconvolution uing hyper-laplacian prior. In Advance in Neural Information Proceing Sytem, page 33 4, 29. [] D. Krihnan, T. Tay, and R. Fergu. Blind deconvolution uing a normalized parity meaure. In Computer Viion and Pattern Recognition (CVPR), 2 IEEE Conference on, page IEEE, 2. [2] A. Levin, R. Fergu, F. Durand, and W. T. Freeman. Image and depth from a conventional camera with a coded aperture. ACM tranaction on graphic (TOG), 26(3):7, 27. [3] A. Levin, Y. Wei, F. Durand, and W. T. Freeman. Undertanding and evaluating blind deconvolution algorithm. In Computer Viion and Pattern Recognition, 29. CVPR 29. IEEE Conference on, page IEEE, 29. [4] Y. Liu, D. Zhang, G. Lu, and W.-Y. Ma. A urvey of contentbaed image retrieval with high-level emantic. Pattern recognition, 4(): , 27. [5] L. B. Lucy. An iterative technique for the rectification of oberved ditribution. The atronomical journal, 79:745, 974. [6] V. N. Mahajan and V. N. Mahajan. Aberration theory made imple. SPIE optical engineering pre Bellingham, WA, 99. [7] K. Marwah, G. Wetztein, Y. Bando, and R. Rakar. Compreive Light Field Photography uing Overcomplete Dictionarie and Optimized Projection. ACM Tran. Graph. (Proc. SIGGRAPH), 32(4):, 23. [8] Y. Peng, Q. Fu, H. Amata, S. Su, F. Heide, and W. Heidrich. Computational imaging uing lightweight diffractiverefractive optic. Optic expre, 23(24): , 25. [9] Y. Peng, Q. Fu, F. Heide, and W. Heidrich. The diffractive achromat full pectrum computational imaging with diffractive optic. [2] D. Perrone and P. Favaro. Total variation blind deconvolution: The devil i in the detail. In Proceeding of the IEEE Conference on Computer Viion and Pattern Recognition, page , 24. [2] C. J. Schuler, H. Chritopher Burger, S. Harmeling, and B. Scholkopf. A machine learning approach for non-blind image deconvolution. In The IEEE Conference on Computer Viion and Pattern Recognition (CVPR), June 23. [22] C. J. Schuler, M. Hirch, S. Harmeling, and B. Schölkopf. Non-tationary correction of optical aberration. In Computer Viion (ICCV), 2 IEEE International Conference on, page IEEE, 2. [23] C. J. Schuler, M. Hirch, S. Harmeling, and B. Schölkopf. Blind correction of optical aberration. In European Conference on Computer Viion, page Springer, 22. [24] Q. Shan, J. Jia, and A. Agarwala. High-quality motion deblurring from a ingle image. In ACM Tranaction on Graphic (TOG), volume 27, page 73. ACM, 28. [25] D. G. Stork and P. R. Gill. Optical, mathematical, and computational foundation of lenle ultra-miniature diffractive imager and enor. International Journal on Advance in Sytem and Meaurement, 7(3):4, 24. [26] L. Sun, S. Cho, J. Wang, and J. Hay. Edge-baed blur kernel etimation uing patch prior. In Computational Photography (ICCP), 23 IEEE International Conference on, page 8. IEEE, 23. [27] P. Wang, N. Mohammad, and R. Menon. Chromaticaberration-corrected diffractive lene for ultra-broadband focuing. Scientific report, 6, 26. [28] Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli. Image quality aement: from error viibility to tructural imilarity. IEEE tranaction on image proceing, 3(4):6 62, 24. [29] N. Wiener. Extrapolation, Interpolation, and Smoothing of Stationary Time Serie. The MIT Pre, 964. [3] T. Yue, J. Suo, J. Wang, X. Cao, and Q. Dai. Blind optical aberration correction by exploring geometric and viual prior. In The IEEE Conference on Computer Viion and Pattern Recognition (CVPR), June 25. [3] T. Yue, J. Suo, Y. Xiao, L. Zhang, and Q. Dai. Image quality enhancement uing original len via optical computing. Optic expre, 22(24): , 24.