2013 IEEE Conference on Computer Vision and Pattern Recognition Handling Noise in Single Image Deblurring using Directional Filters Lin Zhong1 Sunghyun Cho2 Dimitris Metaxas1 Sylvain Paris2 Jue Wang2 1 2 Rutgers University Adobe Research Abstract Many single image blind deconvolution methods have been recently proposed [4, 6, 8–10, 13, 14, 16, 20]. Although they generally work well when the input image is noise-free, their performance degrades rapidly when the noise level increases. Speciﬁcally, the blur kernel estimation step in previous deblurring approaches is often too fragile to reliably estimate the blur kernel when the image is contaminated with noise, as shown in Fig. 1. Even assuming that an accurate blur kernel can be estimated, the ampliﬁed image noise and ringing artifacts generated from the non-blind deconvolution also signiﬁcantly degrade the results [5, 11, 21, 22]. State-of-the-art single image deblurring techniques are sensitive to image noise. Even a small amount of noise, which is inevitable in low-light conditions, can degrade the quality of blur kernel estimation dramatically. The recent approach of Tai and Lin [17] tries to iteratively denoise and deblur a blurry and noisy image. However, as we show in this work, directly applying image denoising methods often partially damages the blur information that is extracted from the input image, leading to biased kernel estimation. We propose a new method for handling noise in blind image deconvolution based on new theoretical and practical insights. Our key observation is that applying a directional low-pass ﬁlter to the input image greatly reduces the noise level, while preserving the blur information in the orthogonal direction to the ﬁlter. Based on this observation, our method applies a series of directional ﬁlters at different orientations to the input image, and estimates an accurate Radon transform of the blur kernel from each ﬁltered image. Finally, we reconstruct the blur kernel using inverse Radon transform. Experimental results on synthetic and real data show that our algorithm achieves higher quality results than previous approaches on blurry and noisy images. 1 To handle noisy inputs in single image deblurring, Tai and Lin [17] ﬁrst apply an existing denoising package [1] as preprocessing, and then estimate the blur kernel and the latent image from the denoised result. This process iterates a few times to produce the ﬁnal result. However, applying existing denoising methods is likely to damage, at least partially, the detailed blur information that one can extract from the input image, thereby leading to a biased kernel estimation. In Sec. 2, we illustrate that standard denoising methods, from bilateral ﬁltering to more advanced approaches such as Non-Local Means [3] and BM3D [7], have negative impacts on the accuracy of kernel estimation. In this paper, we propose a new approach for estimating an accurate blur kernel from a noisy blurry image. Our approach still involves denoising and deblurring steps. However, we carefully design the denoising ﬁlters and deblurring procedures in such a way that the estimated kernel is not affected by the denoising ﬁlters. That is, we shall see that, unlike existing approaches, we can theoretically guarantee that our approach does not introduce any bias in the estimated kernel. 1. Introduction Taking handheld photos in low-light conditions is challenging. Since less light is available, longer exposure times are needed – and without a tripod, camera shake is likely to happen and produce blurry pictures. Increasing the camera light sensitivity, i.e., using a higher ISO setting, can reduce the exposure time, which helps. But it comes at the cost of higher noise levels. Further, this is often not enough, and exposure time remains too long for handheld photography, and many photos end up being blurry and noisy. Although many techniques have been proposed recently to deal with camera shake, most of them assume low noise levels. In this work, we do not make this assumption and aim to restore a sharp image from a blurry and noisy input. Our approach is derived from the key observation that if a directional low-pass linear ﬁlter is applied to the input image, it can reduce the noise level greatly, while the frequency content, including essential blur information, along the orthogonal direction is not affected. We use this property to estimate 1D projections of the desired blur kernel to the orthogonal directions of these ﬁlters. These projections, also known as the Radon transform, will not be affected by applying directional low-pass ﬁlters to the input image, except for the noise reduction. Based on this observation, 1 This work was performed when the ﬁrst author interned at Adobe Research. 1063-6919/13 $26.00 © 2013 IEEE DOI 10.1109/CVPR.2013.85 610 612 (a) Input (c) Levin et al. [15] (b) Cho and Lee [4] (d) Our method Figure 1. Previous deblurring methods are sensitive to image noise. (a) Synthetic input image with 5% noise and the ground truth kernel (overlayed). It is cropped to better show blur and noise. (b) Estimated kernel and latent image by Cho and Lee [4]. (c) Results by Levin et al. [15]. (d) Results of our method. the relative error of k with respect to the noise in b using the condition number of the linear system, that is: we apply a series of directional low-pass ﬁlters at different orientations, and estimate a slice of kernel projection from each image. This yields an accurate estimate of the Radon transform. Finally, we reconstruct the blur kernel using the inverse Radon transform. Once a good kernel is obtained, we incorporate denoising ﬁltering into the ﬁnal deconvolution process to suppress noise and obtain a high-quality latent image. Results on synthetic and real noisy data show that our method is more robust and achieves better results than previous approaches. e(k) (LT L)−1 LT n/(LT L)−1 LT b = e(b) LT n/LT b ≤ (LT L) · (LT L)−1 = κ(LT L), where e(k) and e(b) are relative errors in k and b, respectively. Thus, the noise n in the input image will be ampliﬁed at most by the condition number κ(LT L) for kernel estimation, where LT L is often called the deconvolution matrix and has a block-circulant-with-circulant-block (BCCB) structure [12]. Eq. 4 shows that the upper bound on the error in the estimated kernel is proportional to the amplitude of the noise in input image. Building on this result, one can attempt to apply sophisticated denoising ﬁlter to the blurry image to reduce the noise amplitude, hoping that this will improve the kernel estimate. However, denoising ﬁlters also alter the proﬁle of edges, e.g., [2]. This information is critical to accurate kernel estimation, and as we shall see, the beneﬁts of the noise reduction are often outweighed by the artifacts caused by the proﬁle alteration. To illustrate it, we ﬁrst look at a simple noise reduction method, Gaussian smoothing. Convolving with a Gaussian Gg decreases the noise level. However, the kernel estimation then becomes: 2. Side effects of denoising as preprocessing Before introducing our approach, we ﬁrst analyze the negative impact of employing denoising as preprocessing on kernel estimation. In single image deblurring, a blurry and noisy input image b is usually modeled as: b = ∗ k + n, (1) where , k and n represent the latent sharp image, blur kernel, and additive noise, respectively, ∗ is the convolution operator. Solving and k from input b is a severely illposed problem, and the additional noise n makes this problem even more challenging. Assuming that is known, a common approach to solve for k is: k = argmink b − k ∗ 2 + ρ(k) , (2) kg where ρ(k) is the additional regularization term that imposes smoothness and/or sparsity prior on k. Without considering the regularization term, this becomes a leastsquares problem and the optimal k can be found by solving the following linear system: LT Lk = LT b = LT (b + n), (4) = argminkg b ∗ Gg − ∗ kg 2 = argminkg ( ∗ k + n) ∗ Gg − ∗ kg 2 ≈ argminkg ∗ (k ∗ Gg − kg )2 = k ∗ Gg , (5) where k is the blur kernel for the original input image and kg is the optimal solution after Gaussian denoising. Eq. 5 shows that the estimated kernel kg is a blurred version of the actual kernel k. Further, since Gg is a low-pass ﬁlter, the high frequencies of k are lost and recovering them from kg would be very difﬁcult, if possible at all. This result comes from the initial noise reduction and is independent of the kernel estimation method. (3) where k and b are the corresponding vector forms of k and b, respectively, and L is the matrix form of . We also introduce the noise-free blurry image b = b − n. We estimate 613 611 (a) Input (b) True kernel (c) No denoising (d) Gaussian ﬁlter (e) Bilateral ﬁlter (f) Non-local means (g) BM3D (h) Our method Figure 2. The side effects of employing different denoising methods as preprocessing step in single image deblurring. (a) the synthetic input image with 5% noise. (b) the ground truth kernel. (c) the blur kernel estimated without applying any denoising method to the input image (a). (e)-(g) the estimated blur kernels after applying different denoising ﬁlters. (h) the kernel estimated by our method. where I is an image, p is a pixel location, t is the spatial distance from one pixel to p, c is the normalization factor ∞ deﬁned as c = −∞ w(t)dt, and uθ = (cos θ, sin θ)T is a unit vector of direction θ. The proﬁle of the ﬁlter is determined by w(t), for which we use a Gaussian function: w(t) = exp(−t2 /2σf2 ), where σf controls the strength of the ﬁlter. Filtering the image affects the estimated kernel. With the same argument as for Eq. 5, the kernel that we estimate from the ﬁltered image bθ = b ∗ fθ is: Although more sophisticated denoising methods are better at preserving high frequencies, denoising remains an open problem for which no perfect solution exists. Since no information about the blur kernel can be observed in uniform regions of the blurry image, edges are the main source of information that drives deblurring algorithms either implicitly or explicitly, e.g., [4, 6, 10, 20]. Even small degradations introduced by state-of-the-art denoising techniques can have a strong impact on deblurring results as shown in Fig. 2. In this experiment, we apply bilateral ﬁltering [19], non-local means [3] and BM3D [7] to a test image with 5% noise, i.e., noise of standard deviation 0.05 when the intensity range is [0, 1], and then use Cho and Lee’s method [4] to estimate the blur kernel. The estimated kernels are not accurate due to the side effects of denoising. The recent approach of Tai and Lin [17] ﬁrst applies an existing commercial denoising package (NeatImage [1]) to the input image, then iteratively applies a motion-aware non-local mean ﬁltering and deblurring to reﬁne the results. Although special treatment has been added into the process, both the commercial denoising package and the non-local means ﬁlter have the same negative impacts on kernel estimation as we will show in Sec. 4. kθ = k ∗ fθ . (7) Similarly to ﬁltering with a 2D Gaussian Gg , applying fθ averages pixels and reduces the noise level. Since fθ ﬁlters only along the direction θ, it has nearly no inﬂuence on the blur information in the orthogonal direction. We exploit this property to estimate the projection of the original kernel k along the direction θ. The projection can be formulated as Radon transform [6, 18], which is the collection of integrals of a signal (i.e., k) along projection lines. The particular value on Radon transform corresponding to one projection line ρ = x sin(θ) + y cos(θ) is: Rθ (ρ) = k(x, y)δ(ρ − x sin(θ) − y cos(θ))dxdy, 3. Our approach (8) where k(x, y) indicates the value at the coordinate (x, y) on kernel k. θ and ρ are the angle and offset of the projection line, respectively. Thus, the projection of kernel kθ along the projection direction θ is: In the previous section, we have shown that there is a tension between noise reduction and edge preservation. The former helps to estimate a more accurate kernel, but the latter hinders it. Our experiments showed that even state-ofthe-art denoising ﬁlters do have negative impacts on kernel estimation. In this section, we resolve this problem by using directional blur and the Radon transform to estimate the kernel. Our approach reduces the noise without degrading blur information, thereby producing better kernels. Rθ (kθ ) = Rθ (k ∗ fθ ) = Rθ (k) ∗ Rθ (fθ ) = Rθ (k), (9) where Rθ (·) is the Radon transform operator to the direction θ , and θ = θ + π/2. It is a linear operator, and one can verify that Rθ (fθ ) is a 1D delta function, given the deﬁnition of fθ (Eq. 6). Eq. 9 shows fθ has no impact on the Radon transform of the blur kernel to the orthogonal direction of the ﬁlter. This is the foundation of the proposed approach. An example is shown in Fig. 3. 3.1. Applying directional ﬁlters We now show that directional low-pass ﬁlters can be applied to an image without affecting its Radon transform, while decreasing its noise level. We consider the directional low-pass ﬁlter fθ : 1 ∞ w(t)I(p + tuθ )dt, (6) I(p) ∗ fθ = c −∞ 3.2. The algorithm We now explain how we recover the sharp image, with the kernel estimation ﬁrst, and then the deconvolution step. 614 612 No noise Noisy + filter Noisy Algorithm 1 Multiscale noise-aware blind deconvolution Input: The pyramid {b0 , b1 , ..., bn } by down-sampling the input blurry and noisy image b, where b0 = b. Output: blur kernel k0 and latent image 0 . Directional filter 1: (a) (b) 2: (c) 3: PSF PSF 4: PSF 5: y y x (d) x (e) x 6: (f) Figure 3. Directional denoising mechanism in single image deblurring. (a)(b)(c) are the synthetic image before adding noise, after adding noise, and after applying a directional ﬁlter(θ = 3π/4), respectively. (d)(e)(f) are the corresponding estimated blur kernels and their Radon transforms in the same direction. Note that the estimated kernel in (f) is largely damaged by the directional ﬁlter, but its Radon transform is the same as the one in (d). 7: 8: 9: 10: 3.2.1 Noise-aware kernel estimation Based on the above analysis, we apply a directional blur fθ , estimate the combined blur kernel kθ , and then project it along the same direction of the ﬁlter to get the corresponding Radon transform. We repeat this process to get a set of projections. Finally, we compute the 2D kernel using the inverse Radon transform [18]. The advantage of this strategy is that it greatly reduces noise when applying fθ , while keeping the computed Radon transform intact. However, so far, we have assumed that the latent image is known when estimating the blur kernels. This is not the case in practice, and even with state-of-the-art kernel estimation techniques, recovering kθ from bθ , which is a blurry image convolved with an additional directional blur, has proven to be challenging. The additional ﬁlter tends to make nearby edges “collide” with each other, which in turn introduces errors in the estimated kernel. For a more reliable kernel estimation, we adopt the multiscale blind deconvolution framework commonly used in previous approaches [4, 20]. We create an image pyramid of the input image b as {b0 , b1 , ..., bn }, where b0 is the original resolution, and estimate the blur kernel in a bottom-up fashion from bn to b0 . Since noise is largely removed by image downsizing, we apply an existing approach by Cho and Lee [4] to estimate the blur kernels ki and latent images i from layer bn to b1 . Only for the full resolution layer b0 , we apply the directional ﬁlter fθ and then estimate the Apply an existing nonblind approach ( [4] in our implementation) to estimate ki and i for bi ,i = n, ..., 1. Upsample 1 to generate initial 0 . repeat Apply Nf directional ﬁlters to the input image b0 , each ﬁlter has a direction of i · π/Nf , i = 1, ..., Nf , where Nf is the number of directional ﬁlters. For each ﬁltered image bθ , use 0 as the latent image to estimate kθ . For each optimal kernel kθ , compute its Radon transform Rθ (kθ ) as in Eq. 9, along the direction θ = θ + π/2. Reconstruct k0 from the series of Rθ (kθ ) using inverse Radon transform. Update 0 based on the new k0 using a noise-aware nonblind deconvolution approach. until k0 converges. With the ﬁnal estimated kernel k0 , use the ﬁnal deconvolution method described in Sec. 3.2.2 to generate the ﬁnal output 0 . kernel using the robust deconvolution technique described later in this section. The process is described in Algorithm 1. Steps 4 to 7 are also illustrated in Fig. 4. Speciﬁcally, in Step 5, although each ﬁltered image bθ is severely blurred with the additional ﬁltering, the latent image 0 , initialized from the multiscale process, is relatively sharp and clean, which allows us to estimate kθ as: kθ = argminkθ ∇bθ − kθ ∗ ∇0 2 + ρ(kθ ) , (10) where ∇ is the gradient operator. This process is robust to noise because ∇bθ is a low-pass ﬁltered image. In Step 8, nonblind deconvolution is employed to update 0 based on the new k0 . However, existing methods do not work well in this case since we need to estimate a clean 0 from a noisy image b0 , and the results of previous methods are prone to inaccuracy. To generate a noise-free 0 , we minimize the following energy function that aims for limiting the impact of noise on the result: ∇0 ∗ k0 − ∇b0 2 + w1 ∇0 − u(∇1 )2 + w2 ∇0 2 , (11) where u(·) is the upsampling function, and w1 and w2 are pre-deﬁned weights. The second term encourages the gradient of 0 to be similar to the upsampled gradient ﬁeld of 615 613 PSF Reconstruction (a) Input image Figure 4. Illustration of applying directional ﬁlters for blur kernel estimation from a noisy input image. We apply directional ﬁlters in different orientations to the input image. From each ﬁltered image a corresponding kernel is computed ﬁrst, then projected along the same direction to generate the correct radon transform of the true kernel. The ﬁnal blur kernel k0 is reconstructed using inverse Radon transform [6]. (c) Zoran&Weiss [23] (b) Estimated kernel (d) Cho et al. [5] (e) Our method Figure 5. Comparison results of our ﬁnal noise-aware nonblind deconvolution with other recent nonblind deconvolution methods. The results are obtained using the same input image and the estimated kernel. (c),(d),(e) show the zoom-in results. 1 , which is from the previous level in the pyramid. Since 1 contains much less noise due to image downsizing, incorporating this term can effectively reduce the noise level in 0 . This non-blind deconvolution step is an intermediate step in blur kernel estimation that produces sufﬁciently accurate images at a limited computational cost. In the next section, we describe a more sophisticated non-blind deconvolution algorithm for generating high-quality ﬁnal latent image given the estimated kernel. It is worth mentioning that for simplicity, in the above discussion we assume b1 is almost noise-free after downsizing the image by half. However, this will not be true, if severe noise presents in b0 . To deal with severe noise, we will only use previous methods to estimate blur kernels from bn to b2 in Step 1 of the algorithm, and then apply noise-aware kernel estimation from Step 2 to 9 to the last two layers b1 and b0 . We applied this modiﬁed version of the algorithm to examples with 10% noise (Gaussian noise with standard deviation of 0.1) in Sec. 4. denoising methods in the process. This is in sharp contrast to Tai and Lin’s method [17] where denoising and kernel estimation interfere with each other. In our approach, we minimize the following energy function to estimate the ﬁnal 0 : 0 ∗ k0 − b0 2 + w3 0 − NLM(0 )2 , (12) where NLM(·) is the non-local means denoising operation [3], and w3 is a balancing weight. Minimizing this energy function will ensure that the deblurred result is noisefree, and can best ﬁt with k0 and b0 as well. Directly minimizing this energy is hard because NLM(0 ) is highly nonlinear. We found that iterating the following two steps yields a good result in practice: 0 = NLM(0 ), (13a) 2 2 0 = argmin0 0 ∗ k0 − b0 + w3 0 − 0 . (13b) Discussion Cho et al. [6] also use the Radon transform to recover the blur kernel. However, their approach to compute the kernel projection is different from ours. They rely on heuristics to identify straight edges in the images, and extract the projections from these edges. Because this process relies on a few arbitrary thresholds to locate and analyze the edges, it is sensitive to noise. We also show that it performs poorly on noisy inputs in the experimental section. In comparison, our approach does not rely on such arbitrary thresholds and performs well on noisy images. For initialization, we set 0 to be zero (a black image). Solving Eq. 13b yields a noisy 0 that also contains useful highfrequency image structures. In the alternating minimization process, the noise in 0 is gradually reduced, while the high-frequency image details are preserved. To show the effectiveness of our method, we compare it with other two recent non-blind deconvolution methods, i.e., Zoran and Weiss [23] and Cho et al. [5] in Fig. 5. 3.2.2 4. Experimental results Final noise-aware nonblind deconvolution We implemented our method in Matlab on an Intel Core i5 CPU with 8GB of RAM. We apply directional ﬁlters along 36 regularly sampled orientations, that is, one sample every 5◦ . The computation time is a few minutes for a one-megapixel image. For all the experiments, we set the Once an accurate k0 is estimated, we use it to estimate a good latent image 0 from the noisy input b0 . This is not a trivial task when b0 contains severe noise [21]. However, since k0 is ﬁxed at this stage, it is safe to apply existing 616 614 (a) Abbey(input, 5% noise) TaiandLin (b) Chalet(input, 5% noise) Ours (c) Aque(input,10% noise) TaiandLin (f) Abbey (result, 5% noise) TaiandLin Ours (i) Abbey (result, 10% noise) Ours (d) Kernel 1 TaiandLin i d i (g) Chalet (result, 5% noise) TaiandLin Ours (j) Chalet (result, 10% noise) (e) Kernel 2 Ours (h) Aque (result, 5% noise) TaiandLin i d i Ours (k) Aque (result, 10% noise) Figure 6. Comparing Tai and Lin’s method [17] and our method on synthetic data. Three input blurry image examples with different levels of noise are shown in (a),(b),(c). (d) and (e) are the ground truth blur kernels from Levin et. al. [14]. (d) is used for the examples “Abbey” and “Chalet”, and (e) is used for the example “Aque”. (f-k) show the estimated kernels and the latent images of Tai and Lin’s method and our method with 5% noise and 10% noise. Due to the space limit only the areas highlighted by the bounding boxes in (a-c) are shown. Full size images for comparison are in the supplementary material. We ﬁrst conducted experiments on images that we convolved with a known blur kernel and to which we added noise in a controlled fashion. This allows us to report quantitative measures in addition to visual results. mean and standard deviations of 0.05 and 0.1 for a [0,1] intensity range. Tai and Lin kindly provided the results for their method. The comparison shows that visually our estimated blur kernels are closer to the ground truth, and our estimated latent images contain more details and less ringing artifacts. We also evaluate the results quantitatively by computing the Peak Signal-to-Noise Ratio (PSNR) and Structural SIMilarity (SSIM) (Table 1). Comparisons with Tai and Lin’s method Tai and Lin’s method [17] is the most related work to ours since it also seeks to handle noisy images. This section focuses on comparing this method with our approach. We ﬁrst ran comparisons on synthetic images (Fig. 6), where the latent sharp images were blurred using two blur kernels provided by Levin et al. [14]. We then added Gaussian noise with zero Comparisons with other methods We also conducted experiments to explore how noise affects the performance of other state-of-the-art single-image blind deconvolution methods. Using the “Aque” image and the blur kernel shown in Fig. 6(e), we generated 10 input images with noise from 1% to 10%. We then applied different blind deconvolution methods to these test images, and measure the PSNR extent σf of the directional ﬁlter to 30 pixels. We also set w1 = 0.05 and w2 = 1 (Eq. 11), and w3 = 0.05 (Eq. 12). 4.1. Synthetic data 617 615 Noise Tai Ours Tai Ours Tai Ours Abbey Chalet Aque PSNR 5% 10% 22.43 21.05 22.73 21.61 19.79 18.95 22.80 19.35 26.58 24.53 28.46 25.58 ing artifacts. Overall, our approach produces visually more satisfying results. We further show our results on real-world photographs that were captured under common low-light conditions with a Nikon D90 DLSR camera and a 18 − 105mm lens. We compare our results with those of other state-of-theart methods, including Goldstein and Fattal [9], Cho and Lee [4], Cho et al. [6], Levin et al. [15]. The results (Fig. 9) show that our recovered latent images exhibit less artifacts, such as noise and ringing, and contain more high-frequency details at the same time. These observations are consistent across all test images. We provide additional examples in supplemental material. SSIM 5% 10% .8122 .7242 .8150 .7270 .8244 .7162 .8273 .7200 .8206 .7415 .8512 .7469 Table 1. The comparison experiments of our method and Tai and Lin [17] on synthetic blurry images with different amount of noises. The performances are evaluated by PSNR and SSIM, comparing the generated latent images with the ground truth. Goldstein and Fattal Cho and Lee Cho et al. Levin et al. Tai and Lin Ours 40 35 PSNR 30 5. Conclusion We have shown that most state-of-the-art image deblurring techniques are sensitive to image noise. In this paper, we propose a new single image blind deconvolution method that is more robust to noise than previous approaches. Our method uses directional ﬁlters to reduce the noise while keeping the blur information in their orthogonal direction intact. By applying a series of such directional ﬁlters, we showed how to recover correct 1D projections of the kernel in all directions, which we use to estimate an accurate blur kernel using the inverse Radon transform. We also introduced a noise-tolerant non-blind deconvolution technique that generates high-quality ﬁnal results. The effectiveness of the proposed approach is demonstrated on several comparisons on synthetic and real data. 25 20 15 10 1 2 3 4 5 6 Noise level (%) 7 8 9 10 Figure 7. The PSNR curves of various blind deconvolution algorithms, including Goldstein and Fattal [9], Cho and Lee [4], Cho et al. [6], Levin et al. [15] and our method, on the 10 synthetic test images with noise level from 1% to 10%, generated by the “Aque” image and the kernel shown in Fig. 6(e). The two data points of Tai and Lin’s method [17] are shown as black diamonds, which are provided by the authors. While the PSNR values are closer to ours, the visual difference is still signiﬁcant; our approach produces cleaner images (Fig. 8). All images are included in the supplementary material. Acknowledgements We would like to thank the anonymous reviewers for their helpful feedback. This research is partially supported by the following grants to Dimitris Metaxasa: NSF1069258 and Adobe Systems. References curve of each method (Fig. 7). The accuracies of previous methods degrade rapidly when the noise level increases. On the contrary, our method is more robust, i.e., it works more reliably in the presence of noise, and achieves satisfactory results even when the input noise level is high. This ﬁgure also includes two data points of the Tai and Lin’s method [17] provided by the authors themselves. [1] Neatimage. http://www.neatimage.com/. 1, 3 [2] A. Buades, C. B., and J.-M. Morel. 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The close-ups show that our method recovers more high-frequency details. For the boundaries of objects, our results have less noticeable ring- 618 616 Tai and Lin TaiandLin Ours TaiandLin Ours (a) Santorini (b) Books Figure 8. Comparisons of Tai and Lin’s method and our method on real-world images from [17]. Our results contain more high-frequency details and less ringing artifacts. Zoom-in regions are shown in bounding boxes. Input Goldstein & Fattal Goldstein&Fattal Cho et al Choetal. Cho & Lee Cho&Lee Levin et al Levinetal. Ours Figure 9. Comparisons on real-world examples, where we compare our results with the results of Goldstein and Fattal [9], Cho and Lee [4], Cho et al. [6], Levin et al. [15]. More results are in the supplementary material. tion. CVPR, 2011. 2, 7, 8 [16] Q. Shan, J. Jia, and A. Agarwala. High-quality motion deblurring from a single image. SIGGRAPH, 2008. 1 [17] Y. Tai and S. Lin. Motion-aware noise ﬁltering for deblurring of noisy and blurry images. 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