iccv iccv2013 iccv2013-35 knowledge-graph by maker-knowledge-mining
Source: pdf
Author: Netalee Efrat, Daniel Glasner, Alexander Apartsin, Boaz Nadler, Anat Levin
Abstract: Over the past decade, single image Super-Resolution (SR) research has focused on developing sophisticated image priors, leading to significant advances. Estimating and incorporating the blur model, that relates the high-res and low-res images, has received much less attention, however. In particular, the reconstruction constraint, namely that the blurred and downsampled high-res output should approximately equal the low-res input image, has been either ignored or applied with default fixed blur models. In this work, we examine the relative importance ofthe imageprior and the reconstruction constraint. First, we show that an accurate reconstruction constraint combined with a simple gradient regularization achieves SR results almost as good as those of state-of-the-art algorithms with sophisticated image priors. Second, we study both empirically and theoretically the sensitivity of SR algorithms to the blur model assumed in the reconstruction constraint. We find that an accurate blur model is more important than a sophisticated image prior. Finally, using real camera data, we demonstrate that the default blur models of various SR algorithms may differ from the camera blur, typically leading to over- smoothed results. Our findings highlight the importance of accurately estimating camera blur in reconstructing raw low- res images acquired by an actual camera.
Reference: text
sentIndex sentText sentNum sentScore
1 Estimating and incorporating the blur model, that relates the high-res and low-res images, has received much less attention, however. [sent-4, score-0.315]
2 In particular, the reconstruction constraint, namely that the blurred and downsampled high-res output should approximately equal the low-res input image, has been either ignored or applied with default fixed blur models. [sent-5, score-0.63]
3 In this work, we examine the relative importance ofthe imageprior and the reconstruction constraint. [sent-6, score-0.192]
4 First, we show that an accurate reconstruction constraint combined with a simple gradient regularization achieves SR results almost as good as those of state-of-the-art algorithms with sophisticated image priors. [sent-7, score-0.393]
5 Second, we study both empirically and theoretically the sensitivity of SR algorithms to the blur model assumed in the reconstruction constraint. [sent-8, score-0.591]
6 We find that an accurate blur model is more important than a sophisticated image prior. [sent-9, score-0.393]
7 Finally, using real camera data, we demonstrate that the default blur models of various SR algorithms may differ from the camera blur, typically leading to over- smoothed results. [sent-10, score-0.631]
8 Our findings highlight the importance of accurately estimating camera blur in reconstructing raw low- res images acquired by an actual camera. [sent-11, score-0.45]
9 The second challenge is to enforce the reconstruction constraint Ax ≈ y, which implies that up to imaging noise, the recovered HR image should be consistent with the LR input. [sent-18, score-0.231]
10 In contrast, the reconstruction constraint has received relatively little attention. [sent-21, score-0.226]
11 Those that do often assume a predefined blur kernel. [sent-23, score-0.292]
12 Examples include antialiasing with bicubic interpolation (Matlab’s default imresize function) [8, 27], Gaussian blur [3], Gaussian blur followed by bicubic interpolation [7], simple pixel averaging [5], and sampling without any pre-smoothing [15]. [sent-24, score-1.242]
13 A critical concern in applying such SR algorithms to real images is how well these synthetic forward models approximate real camera blur. [sent-25, score-0.181]
14 For example, the bicubic interpolation used by many algorithms is generally not physically feasible, as it involves negative weights. [sent-26, score-0.312]
15 Furthermore, in most SR algorithms, the blur kernel is not an input parameter: it is coupled to various internal components which are not easily adjusted. [sent-27, score-0.516]
16 A few single-image SR works which do attempt to estimate or take the unknown kernel into account include [3, 24, 17, 10, 11, 12]. [sent-28, score-0.224]
17 This state of affairs naturally raises the following questions, which are the focus of our paper: i) what is the effect of an incorrect blur model on SR algorithms? [sent-29, score-0.328]
18 ii) what is the importance of the reconstruction constraint compared to that of the image prior? [sent-30, score-0.241]
19 First, we argue that the reconstruction constraint is at least as important as the image prior. [sent-32, score-0.203]
20 In particular, we demonstrate that combining a simple prior, an L2 penalty on image gradients, with an accurate reconstruction constraint, provides SR results almost as good as those produced by × state-of-the-art SR algorithms with sophisticated priors. [sent-33, score-0.293]
21 Second, we empirically examine the sensitivity of several SR algorithms to the accuracy of the estimated blur ker2832 nel. [sent-34, score-0.426]
22 We show that incorporating an accurate estimate into these algorithms improves their output, by allowing them to take full advantage of the reconstruction constraint. [sent-35, score-0.227]
23 In contrast, when the SR algorithms utilize an inaccurate blur kernel, the resulting images are either too blurred or contain over-sharpening artifacts. [sent-36, score-0.4]
24 For the L2 prior case, we also present a theoretical analysis explaining these phenomena, via a frequency analysis of the kernel mismatch. [sent-38, score-0.372]
25 Finally, we demonstrate the importance of accurately estimating camera blur when applying SR to raw images captured with a real camera. [sent-39, score-0.447]
26 We show that the default kernels used by many algorithms are not sufficiently close to the camera blur, and produce over-smoothed results. [sent-40, score-0.275]
27 Moreover, we show that incorporating a more accurate estimate of the camera blur improves the results. [sent-41, score-0.394]
28 The relation between x and y is typically expressed as y=k ∗ x ↓s +n (2) where k denotes a blur kernel (low pass filter), ↓s denotes subsampling by factor s, and n is imaging noise. [sent-45, score-0.564]
29 First, we need to know A, that is, to have an accurate estimate of the blur k. [sent-49, score-0.327]
30 Moreover, some methods are patch-based and either do not enforce the global reconstruction constraint Ax ≈ y at all, or apply it only in a separate post-processing step, with a default blur kernel. [sent-61, score-0.612]
31 In this paper we study, both qualitatively and quantitatively, the importance of an accurate blur model and the corresponding reconstruction constraint in SR algorithms. [sent-62, score-0.568]
32 Since visual plausibility is somewhat subjective, we focus on the reconstruction task, which can be evaluated numerically against ground truth. [sent-69, score-0.156]
33 (4) with both a Gaussian 2833 to the kernel the algorithm assumes. [sent-87, score-0.224]
34 The rightmost column presents results of the unmodified algorithm with its default bi-cubic kernel. [sent-88, score-0.149]
35 On the diagonal, the On the upper right off-diagonal im- ages, the assumed kernel is smoother than the true one, leading to over sharpening artifacts. [sent-96, score-0.463]
36 For the lower left, the assumed kernel is sharper than the correct one, leading to over-smoothed results. [sent-97, score-0.332]
37 We denote by kT the true kernel used to synthesize a test LR image and by kA the kernel assumed by a SR algorithm. [sent-100, score-0.546]
38 pixels; 4) a real camera blur kernel, estimated by capturing a known calibration target with a Canon 5D Mark II camera, shown in Fig. [sent-103, score-0.412]
39 The algorithms of Yang, Kim, and Glasner use as default the bicubic kernel, kA = b, whereas Freeman uses kA = b ∗ g1. [sent-114, score-0.384]
40 Unfortunately, these algorithms do not accept a kernel as one of their input parameters and adjusting them to use a different kernel is not straightforward. [sent-115, score-0.553]
41 upsample the image in gradual steps, and defining a kernel for each intermediate step is non-trivial. [sent-117, score-0.224]
42 Second, following [27], we introduced a reconstruction constraint with the desired kernel in post-processing. [sent-120, score-0.427]
43 (6) is not needed for the simple regularization algorithms which already optimize the reconstruction constraint in Eq. [sent-128, score-0.292]
44 SR was applied to the 4 test sets (prepared with different kernels kT), each time adjusting the algorithm to use a different kernel kA in reconstruction. [sent-134, score-0.296]
45 The fifth column shows the original authors’ results, unmodified, using the default bicubic kernel. [sent-135, score-0.346]
46 cross section primal Fourier primal Fourier The camera blur attenuates high frequencies more than the bicubic one. [sent-148, score-0.759]
47 First, incorporating the reconstruction constraint with the true kernel improves accuracy. [sent-153, score-0.466]
48 One can see this by comparing the fifth column (the original algorithm with its default kernel), with the diagonal entries, for which the reconstruction constraint uses kA = kT. [sent-154, score-0.371]
49 Moreover, SR using an incorrect kernel drastically increases reconstruction error. [sent-156, score-0.354]
50 In particular, when the assumed kernel is smoother than the true kernel, the recovered image is blurred. [sent-158, score-0.388]
51 On the other hand, when the assumed kernel is sharper than the true kernel, high frequency ringing artifacts appear, as illustrated in Fig. [sent-159, score-0.493]
52 Real images: Given the sensitivity of SR algorithms to the assumed blur kernel, it is interesting to assess their performance on raw LR images acquired by an actual camera. [sent-164, score-0.514]
53 To this end, we captured images with a Canon 5D Mark II camera, and estimated its blur using a known calibration target (calibration details can be found in [4]). [sent-165, score-0.319]
54 2 compares our estimated camera blur with the bicubic kernel. [sent-167, score-0.564]
55 As seen in the Fourier domain, the camera kernel attenuates high frequencies more than the bicubic one. [sent-169, score-0.609]
56 4) predicts that using the sharper bicubic kernel will result in over-smoothed SR images. [sent-171, score-0.478]
57 The default implementation of various algorithms that assume a bicubic kernel indeed yields over-smoothedresults, while adjusting the algorithms to incorporate the camera kernel sharpens them. [sent-174, score-1.004]
58 reconstruction constraint: Next, we consider the relative importance of the assumed image prior vs. [sent-178, score-0.269]
59 To this end, Table 2 compares all algorithms2 and their modified versions, which incorporate the reconstruction constraint, on two test sets - blurred with kT = b and with kT = b ∗ g1. [sent-180, score-0.176]
60 All LR images were corrupted by noise at can see the of a more sophisticated prior by comparing rows 2-3 to rows 4-6 in the left columns. [sent-192, score-0.228]
61 Comparing the = kT images are marked in same rows in the right columns shows the effect of using the correct blur kernel instead of the default one. [sent-193, score-0.74]
62 The effect of using the exact blur kernel is more dominant than that of the prior. [sent-195, score-0.552]
63 While visual comparison is somewhat subjective and one may argue in favor of one algorithm or the other, overall the results of all algorithms (except the baseline bicubic interpolation) are not significantly different. [sent-200, score-0.315]
64 Second, the influence of different image priors is much smaller than the effect of kernel mismatch. [sent-201, score-0.3]
65 5dB), which correspond to different image priors, to the difference of almost 2dB between the third and fourth rows, which capture the effect of using a correct kernel instead of the default bicubic one. [sent-205, score-0.582]
66 In the two left columns all algorithms use the true kernel and hence produce comparable results. [sent-208, score-0.351]
67 In contrast, in the two right columns, only the sparse and L2 algorithms use the correct kernel, others use their default one. [sent-209, score-0.179]
68 This emphasizes that an accurate reconstruction constraint can be more important than a sophisticated prior. [sent-211, score-0.304]
69 Since this is not a trivial task most algorithms only impose the global reconstruction constraint in post-processing. [sent-216, score-0.265]
70 In contrast, simple gradient regularization methods explicitly optimize a functional which jointly accounts for the global reconstruction constraint and the prior. [sent-217, score-0.23]
71 [87d6 ]efault kernel is compared with a modified one which accepts a kernel as a parameter, when applicable learns a dictionary with it, and also enforces the reconstruction constraint (RC). [sent-231, score-0.651]
72 Most algorithms use kA = b as default, thus in the 3rd row algorithms process images with kA kT and in the 4th row adjust to the correct kernel kA = kT. [sent-234, score-0.348]
73 Theoretical Analysis To gain further insight into the detrimental effect of kernel mismatch, let us examine SR with a simple L2 gradient regularization (Gaussian prior) as presented in Eq. [sent-237, score-0.337]
74 SR algorithms that assume a kernel KA in fact assume that Y and X are related via Yω = KA,ωXω + KA,ω? [sent-253, score-0.286]
75 The following lemma characterizes the resulting estimator and the relation between the estimated signal Xˆ and the true signal X in case of kernel mismatch (KA KT). [sent-258, score-0.54]
76 The MAP SR estimate Xˆ, assuming a Gaussian prior on X and a kernel KA as in, Eq. [sent-264, score-0.266]
77 (13) In other words, an L2 prior on image gradients is a diagonal Gaussian prior in the Fourier domain, whose variance at each frequency ω is the power ofthe derivative filter. [sent-299, score-0.239]
78 In the presence of noise with variance η2, at frequencies where σω2 ? [sent-348, score-0.136]
79 contributes to the low frequency reconstruction at Xˆω, and visa versa. [sent-357, score-0.207]
80 Kernel Mismatch: Next, we study the implication of the lemma when the signal was blurred with KT while the MAP = estimator assumed a different kernel KA KT. [sent-358, score-0.442]
81 At high frequencies, where signal variance is lower than noise level, HA,ω ≈ 0, and kernel mismatch has little effect on the output. [sent-361, score-0.458]
82 In contrast, at other frequencies, an incorrect kernel may have strong detrimental effects. [sent-362, score-0.25]
83 In the first case KT,ω/KA,ω acts as a blurring filter and in the second as a sharpening filter (see Fig. [sent-365, score-0.244]
84 βT > βA yields a Gaussian blur filter (left); βT < βA gives a sharpening filter (right). [sent-378, score-0.515]
85 For βT > βA we obtain a Gaussian blur filter, and for βT < βA a sharpening filter (the exponent is positive). [sent-379, score-0.455]
86 High frequencies whose expected power is below the noise variance are not amplified by the reconstruction filter. [sent-384, score-0.266]
87 An assumed wider kernel results in ringing, and a narrow kernel in over-blurring. [sent-387, score-0.507]
88 Kernel Uncertainty: In practice, SR algorithms may be applied to real images whose blur kernel is not precisely known. [sent-389, score-0.604]
89 One approach, taken by several SR algorithms, × is to ignore the true blur kernel, and utilize some default, such as bicubic. [sent-390, score-0.331]
90 As noted previously, when the assumed default kernel is narrower than the true kernel, as occurs with our actual camera (see Fig. [sent-391, score-0.535]
91 A second approach taken by various SR algorithms to cope with imprecise knowledge of the true kernel, is to give more weight to the image prior and reduce the weight of the reconstruction constraint (e. [sent-394, score-0.346]
92 This makes such algorithms less sensitive to kernel mismatch, but also reduces the quality of their results. [sent-398, score-0.286]
93 We now derive a principled estimation strategy to take into account kernel uncertainty. [sent-399, score-0.224]
94 The effect of kernel uncertainty is similar to that of noise: the kernel covariance ΣK adds to the noise covariance part in Eq. [sent-423, score-0.591]
95 If an accurate model of k can be found (a smaller-norm certainty covariance ΣK), the reconstruction accuracy is improved. [sent-427, score-0.189]
96 Discussion In this paper, we examined the effect of two components of SR: the natural image prior and the reconstruction constraint. [sent-429, score-0.208]
97 We showed that an accurate blur model and its corresponding reconstruction constraint are crucial to the success of SR algorithms. [sent-430, score-0.53]
98 The influence of an accurate estimate of the blur kernel is significantly larger than that of a sophisticated prior. [sent-431, score-0.617]
99 While existing blind motion deblurring methods can be adapted to this task, we note that SR kernel recovery is a simpler task, since the blur is a property of the sensor and is fixed for all images captured by the same camera under similar imaging conditions. [sent-433, score-0.624]
100 As described in [4], this camera blur can be calibrated using two images of the same calibration target. [sent-434, score-0.386]
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