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212 nips-2013-Non-Uniform Camera Shake Removal Using a Spatially-Adaptive Sparse Penalty

Source: pdf

Author: Haichao Zhang, David Wipf

Abstract: Typical blur from camera shake often deviates from the standard uniform convolutional assumption, in part because of problematic rotations which create greater blurring away from some unknown center point. Consequently, successful blind deconvolution for removing shake artifacts requires the estimation of a spatiallyvarying or non-uniform blur operator. Using ideas from Bayesian inference and convex analysis, this paper derives a simple non-uniform blind deblurring algorithm with a spatially-adaptive image penalty. Through an implicit normalization process, this penalty automatically adjust its shape based on the estimated degree of local blur and image structure such that regions with large blur or few prominent edges are discounted. Remaining regions with modest blur and revealing edges therefore dominate on average without explicitly incorporating structureselection heuristics. The algorithm can be implemented using an optimization strategy that is virtually tuning-parameter free and simpler than existing methods, and likely can be applied in other settings such as dictionary learning. Detailed theoretical analysis and empirical comparisons on real images serve as validation.

Reference: text

Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 com Abstract Typical blur from camera shake often deviates from the standard uniform convolutional assumption, in part because of problematic rotations which create greater blurring away from some unknown center point. [sent-3, score-1.056]

2 Consequently, successful blind deconvolution for removing shake artifacts requires the estimation of a spatiallyvarying or non-uniform blur operator. [sent-4, score-1.113]

3 Using ideas from Bayesian inference and convex analysis, this paper derives a simple non-uniform blind deblurring algorithm with a spatially-adaptive image penalty. [sent-5, score-0.88]

4 Through an implicit normalization process, this penalty automatically adjust its shape based on the estimated degree of local blur and image structure such that regions with large blur or few prominent edges are discounted. [sent-6, score-1.572]

5 Remaining regions with modest blur and revealing edges therefore dominate on average without explicitly incorporating structureselection heuristics. [sent-7, score-0.64]

6 1 Introduction Image blur is an undesirable degradation that often accompanies the image formation process and may arise, for example, because of camera shake during acquisition. [sent-10, score-1.122]

7 Blind image deblurring strategies aim to recover a sharp image from only a blurry, compromised observation. [sent-11, score-0.916]

8 Unfortunately, many real-world photographs contain blur effects that vary across the image plane, such as when unknown rotations are introduced by camera shake [17]. [sent-13, score-1.119]

9 Note that the original uniform blur model can be achieved equivalently when H is forced to adopt certain structure (e. [sent-15, score-0.557]

10 In general, nonuniform blur may arise under several different contexts. [sent-18, score-0.528]

11 This paper will focus on the blind removal of non-uniform blur caused by general camera shake (as opposed to blur from object motion) using only a single image, with no additional hardware assistance. [sent-19, score-1.737]

12 While existing algorithms for addressing non-uniform camera shake have displayed a measure of success, several important limitations remain. [sent-20, score-0.367]

13 First, some methods require either additional spe1 cialized hardware such as high-speed video capture [23] or inertial measurement sensors [13] for estimating motion, or else multiple images of the same scene [4]. [sent-21, score-0.121]

14 Secondly, even the algorithms that operate given only data from a single image typically rely on carefully engineered initializations, heuristics, and trade-off parameters for selecting salient image structure or edges, in part to avoid undesirable degenerate, no-blur solutions [7, 8, 9, 11]. [sent-22, score-0.469]

15 This transparency leads to a nearly tuning-parameter free algorithm based upon a sparsity penalty whose shape adapts to the estimated degree of local blur, and provides theoretical arguments regarding how to robustly handle non-uniform degradations. [sent-25, score-0.236]

16 Section 2 brieﬂy describes relevant existing work on non-uniform blind deblurring operators and implementation techniques. [sent-27, score-0.729]

17 Section 3 then introduces the proposed non-uniform blind deblurring model, while further theoretical justiﬁcation and analyses are provided in Section 4. [sent-28, score-0.692]

18 The downside with this type of model is that geometric relationships between the blur kernels of different regions derived from the the physical motion path of the camera are ignored. [sent-32, score-0.86]

19 In this regard, (1) represents a more general model that has been used in many recent non-uniform deblurring efforts [23, 25, 7, 11, 4]. [sent-35, score-0.452]

20 PMP also retains the bilinear property of uniform convolution, meaning that y = Hx + n = Dw + n, where H = j (2) wj Pj and D = [P1 x, P2 x, · · · , Pj x, · · · ] is a matrix of transformed sharp images. [sent-36, score-0.193]

21 However, EFF can be combined with the PMP model by introducing a set of basis images efﬁciently generated by transforming a grid of delta peak images [9]. [sent-38, score-0.142]

22 3 A New Non-Uniform Blind Deblurring Model Following previous work [6, 16], we will work in the derivative domain of images for ease of modeling and better performance, meaning that x ∈ R m and y ∈ Rn will denote the lexicographically ordered sharp and blurry image derivatives respectively. [sent-40, score-0.504]

23 While presently γ is unknown, if we ﬁrst marginalize over the unknown x, we can estimate it jointly along with the blur parameters w and the unknown noise variance λ. [sent-52, score-0.582]

24 The ﬁnal sharp image can then be recovered using the estimated kernel and noise level along with standard non-blind deblurring algorithms (e. [sent-54, score-0.839]

25 For example, consider the simpliﬁed non-uniform deblurring situation where the true x has a single non-zero element and H is deﬁned such that each column indexed by i is independently parameterized with ﬁnite support symmetric around pixel i. [sent-61, score-0.486]

26 Moreover, assume this support matches the true support of the unknown blur operator. [sent-62, score-0.528]

27 Then we have the following: Lemma 1 Given the idealized non-uniform deblurring problem described above, the cost function (4) will be characterized by a unique minimizing solution that correctly locates the nonzero element in x and the corresponding true blur kernel at this location. [sent-63, score-1.092]

28 Speciﬁcally, using Hadamard’s inequality we have = n log λ + log |Γ| + log λ−1 HT H + Γ−1 ≤ log HΓHT + λI n log λ + log |Γ| + log λ−1 diag HT H + Γ−1 ¯ log λ + γi wi = 2 2 + (n − m) log λ, (6) i ¯ where wi denotes the i-th column of H. [sent-71, score-0.534]

29 We then have log |λ−1 HT H + Γ−1 | = 2 log |V| ≤ 2 log ( i vi 2 ) = log diag λ−1 HT H + Γ−1 , leading to the stated result. [sent-76, score-0.104]

30 3 ¯ Also, the quantity wi 2 which appears in (6) can be viewed as a measure of the degree of local blur at location i. [sent-77, score-0.733]

31 Given the feasible region w ≥ 0 and without loss of generality the constraint ¯ 2 wi 2 ≤ 1, where i wi = 1 for normalization purposes, it can easily be shown that 1/L ≤ ¯ L is the maximum number of elements in any local blur kernel wi or column of H. [sent-78, score-1.099]

32 The upper bound is achieved when the local kernel is a delta solution, meaning only one nonzero element ¯ 2 and therefore minimal blur. [sent-79, score-0.165]

33 In contrast, the lower bound on wi 2 occurs when every element of ¯ wi has an equal value, constituting the maximal possible blur. [sent-80, score-0.266]

34 This metric, which will inﬂuence ¯ 2 our analysis in the next section, can be computing using wi 2 = wT (BT Bi )w, where Bi i [P1 ei , P2 ei , · · · , Pj ei , · · · ] and ei denotes an all-zero image with a one at site i. [sent-81, score-0.441]

35 In the uniform ¯ deblurring case, B T Bi = I ignoring edge effects, and therefore wi 2 = w 2 for all i. [sent-82, score-0.614]

36 However, it is still well-equipped for estimating sparse image gradients and avoiding degenerate no-blur solutions. [sent-84, score-0.293]

37 For example, consider the case of an asymptotically large image with iid distributed sparse image gradients, with some constant fraction exactly equal to zero and the remaining nonzero elements drawn from any continuous distribution. [sent-85, score-0.42]

38 Now suppose that this image is corrupted with a non-uniform blur operator of the form H = j wj Pj , where the cardinality of the summation is ﬁnite and H satisﬁes minimal regularity conditions. [sent-86, score-0.793]

39 Then it can be shown that any global minimum of (4), with or without the bound from (6), will produce the true blur operator. [sent-87, score-0.528]

40 Related intuition applies when noise is present or when the image gradients are not exactly sparse (we will defer more detailed analysis to a future publication). [sent-88, score-0.287]

41 Regardless, the simpliﬁed γ-dependent cost function is still far less intuitive than the penalized regression models dependent on x such as (5) that are typically employed for non-uniform blind deblurring. [sent-89, score-0.297]

42 However, using the framework from [26], it can be shown that the kernel estimate obtained by this process is formally equivalent to the one obtained via 1 ¯ min ψ(|xi | wi 2 , λ) + (n − m) log λ, with (7) y − Hx 2 + 2 x;w≥0,λ≥0 λ i 2u √ u ≥ 0. [sent-90, score-0.218]

43 ψ(u, λ) + log 2λ + u2 + u 4λ + u2 u + 4λ + u2 The optimization from (7) closely resembles a standard penalized regression (or equivalently MAP) problem used for blind deblurring. [sent-91, score-0.296]

44 The primary distinction is the penalty term ψ, which jointly regularizes x, w, and λ as discussed Section 4. [sent-92, score-0.106]

45 The underlying procedure is related to variational Bayesian (VB) models from [1, 16, 20]; however, these models are based on a completely different mean-ﬁeld approximation and a uniform blur assumption, and they do not learn the noise parameter. [sent-94, score-0.583]

46 4 Model Properties The proposed blind deblurring strategy involves simply minimizing (7); no additional steps for tradeoff parameter selection or structure/salient-edge detection are required unlike other state-of-the-art approaches. [sent-96, score-0.692]

47 First, we will demonstrate a form of intrinsic column normalization that facilitates the balanced sparse estimation of the unknown latent image and implicitly de-emphasizes regions with large blur and few dominate edges. [sent-98, score-0.952]

48 1 Column-Normalized Sparse Estimation ¯ Using the simple reparameterization z i xi wi 1 2 y − Hz 2 + min z;w≥0,λ≥0 λ 2 it follows that (7) is exactly equivalent to solving ψ(|zi |, λ) + (n − m) log λ, i 4 (8) where z = [z1 , . [sent-102, score-0.213]

49 Moreover, it can be shown that this ψ is a concave, non-decreasing function of |z|, and hence represents a canonical sparsity-promoting penalty function with respect to z [26]. [sent-106, score-0.106]

50 Moreover, no additional heuristic kernel penalty terms such as in (5) are required since H is in some sense self-regularized by the normalization. [sent-110, score-0.165]

51 While this will indeed result in normalized columns and a properly balanced data-ﬁt term, these raw norms will now appear in the penalty function g, giving the equivalent objective min z;w≥0 y − Hz 2 2 ¯ g z i wi +α −1 2 +β i h(wj ). [sent-114, score-0.283]

52 ¯ Simply put, the problem (9) will favor solutions where the ratio z i / wi 2 is sparse or nearly so, ¯ which can be achieved by either making many z i zero or many wi 2 big. [sent-116, score-0.31]

53 If some z i is estimated to be zero (and many z i will provably be exactly zero at any local minima if g(x) is a concave, ¯ non-decreasing function of |x|), then the corresponding wi 2 will be unconstrained. [sent-117, score-0.244]

54 In contrast, ¯ if a given zi is non-zero, there will be a stronger push for the associated wi 2 to be large, i. [sent-118, score-0.133]

55 Thus, the relative penalization of the kernel norms will depend on the estimated local image gradients, and no-blur delta solutions may be arbitrarily favored in parts of the image plane dominated by edges, the very place where blur estimation information is paramount. [sent-121, score-1.204]

56 ¯ In reality, the local kernel norms wi 2 , which quantify the degree of local blur as mentioned previously, should be completely independent of the sparsity of the image gradients in the same location. [sent-122, score-1.119]

57 This is of course because the different blurring effects from camera shake are independent of the locations of strong edges in a given scene, since the blur operator is only a function of camera motion (at least to ﬁrst order approximation). [sent-123, score-1.293]

58 One way to compensate for this independence would be ¯ to simply optimize (9) with wi 2 removed from g. [sent-124, score-0.133]

59 In contrast, our algorithm handles these complications seamlessly without any additional penalty terms. [sent-127, score-0.106]

60 2 Noise-Dependent, Parameter-Free Homotopy Continuation Column normalization can be viewed as a principled ﬁrst step towards solving challenging sparse estimation problems. [sent-129, score-0.118]

61 However, when non-convex sparse regularizers are used for the image penalty, e. [sent-130, score-0.232]

62 , p norms with p < 1, then local minima can be a signiﬁcant problem. [sent-132, score-0.129]

63 When applied to a sharp image, any blur operator will necessarily contribute two opposing effects: (i) It reduces a measure of the image sparsity, which normally increases the penalty i |yi |p , and p (ii) It broadly reduces the overall image variance, which actually reduces i |yi | . [sent-134, score-1.127]

64 Note that we can always apply greater and greater blur to any sharp image x such that the variance of the resulting blurry y is arbitrarily small. [sent-136, score-1.027]

65 This then produces an arbitrarily small p norm, which implies that p p i |yi | < i |xi | , meaning that the penalty actually favors the blurry image over the sharp one. [sent-137, score-0.561]

66 In a practical sense though, the amount of blur that can be tolerated before this undesirable preference for y over x occurs is much larger as p approaches zero. [sent-138, score-0.567]

67 This is because the more concave the image penalty becomes (as a function of coefﬁcient magnitudes), the less sensitive it is to image variance and the more sensitive it is to image sparsity. [sent-139, score-0.752]

68 2 We may therefore expect such a highly concave, sparsity promoting penalty to favor the sharp image over the blurry one in a broader range of blur conditions. [sent-141, score-1.12]

69 Even with other families of penalty functions the same basic notion holds: greater concavity means greater sparsity preference and less sensitivity to variance changes that favor no-blur degenerate solutions. [sent-142, score-0.345]

70 From an implementational standpoint, homotopy continuation methods provide one attractive means of dealing with difﬁcult non-convex penalty functions and the associated constellation of local minima [3]. [sent-143, score-0.375]

71 The basic idea is to use a parameterized family of sparsity-promoting functions g(x; θ), where different values of θ determine the relative degree of concavity allowing a transition from something convex such as the 1 norm (with θ large) to something concave such as the 0 norm (with θ small). [sent-144, score-0.25]

72 While potentially effective in practice, homotopy continuation methods require both a trade-off parameter for g(x; θ) and a pre-deﬁned schedule or heuristic for adjusting θ, both of which could potentially be image dependent. [sent-147, score-0.335]

73 The proposed deblurring algorithm automatically implements a form of noise-dependent, parameterfree homotopy continuation with several attractive auxiliary properties [26]. [sent-148, score-0.599]

74 To make this claim precise and facilitate subsequent analysis, we ﬁrst introduce the deﬁnition of relative concavity [19]: Deﬁnition 1 Let u be a strictly increasing function on [a, b]. [sent-149, score-0.13]

75 The function ν is concave relative to ν u on the interval [a, b] if and only if ν(y) ≤ ν(x) + u (x) [u(y) − u(x)] holds ∀x, y ∈ [a, b]. [sent-150, score-0.109]

76 (x) We will use ν ≺ u to denote that ν is concave relative to u on [0, ∞). [sent-151, score-0.109]

77 This can be understood as a natural generalization of the traditional notion of a concavity, in that a concave function is equivalently concave relative to a linear function per Deﬁnition 1. [sent-152, score-0.191]

78 In the context of homotopy continuation, an ideal candidate penalty would be one for which g(x; θ 1 ) ≺ g(x; θ2 ) whenever θ1 ≤ θ2 . [sent-156, score-0.181]

79 This would ensure that greater sparsity-inducing concavity is introduced as θ is reduced. [sent-157, score-0.139]

80 Additionally, because λ must be relatively large to arrive at this 1 approximation, the estimation need only focus on reproducing the largest elements in z since the sparse penalty will dominate the data ﬁt term. [sent-168, score-0.212]

81 Furthermore, these larger elements are on average more likely to be in regions of relatively lower blurring or high ¯ ¯ wi 2 value by virtue of the reparameterization z i = xi wi 2 . [sent-169, score-0.431]

82 Consequently, the less concave ¯ initial estimation can proceed successfully by de-emphasizing regions with high blur or low wi 2 , and focusing on coarsely approximating regions with relatively less blur. [sent-170, score-0.874]

83 2 Note that even if the true sharp image is not exactly sparse, as long as it can be reasonably wellapproximated by some exactly sparse image in an 2 norm sense, then the analysis here still holds [27]. [sent-171, score-0.508]

84 Later as the estimation proceeds and w and z are reﬁned, λ will be reduced which in turn necessarily increases the relative concavity of the penalty ψ per Theorem 1. [sent-174, score-0.265]

85 Eventually the penalty can even approach the 0 norm (although images are generally not exactly sparse, and other noise factors and unmodeled artifacts are usually present such that λ will never go all the way to zero). [sent-176, score-0.218]

86 Figure 1 displays results of this procedure both with and without the spatially-varying column normalizations and the implicit adaptive penalization that help compensate for locally varying image blur. [sent-178, score-0.259]

87 The supplementary ﬁle contains a number of additional comparisons, including assessments with a benchmark uniform blind deblurring dataset where ground truth is available. [sent-180, score-0.721]

88 Figure 2 displays deblurring comparisons based on the Butchershop and Vintage-car images. [sent-189, score-0.478]

89 Note that with these images, ground truth blur kernels were independently estimated using a special capturing process [8]. [sent-191, score-0.554]

90 As shown in the supplementary ﬁle, the estimated blur kernel patterns obtained from our algorithm better resemble the ground truth relative to the other methods, a performance result that compensates for any differences in the non-blind step. [sent-192, score-0.64]

91 [25]: Results on the Pantheon test image from [25] are shown in Figure 3 (top row), where we observe that the deblurred image from Whyte et al. [sent-194, score-0.413]

92 [7]: We next experiment using the test image Building from [7], which contains large rotational blurring that can be challenging for blind deblurring algorithms. [sent-198, score-0.94]

93 presents a deblurring algorithm that relies upon additional hardware for estimating camera motion [13]. [sent-202, score-0.768]

94 However, even without this additional in7 Butchershop Vintage-car B LURRY H ARMELING H IRSCH O UR B LURRY H ARMELING H IRSCH O UR Pantheon Figure 2: Non-uniform deblurring results. [sent-203, score-0.452]

95 (better viewed electronically with zooming) W HYTE O UR B LURRY G UPTA O UR B LURRY J OSHI O UR Sculpture Building B LURRY Figure 3: Non-uniform deblurring results. [sent-205, score-0.489]

96 (better viewed electronically with zooming) formation, our algorithm produces a better sharp estimate of the Sculpture image from [13], with fewer ringing artifacts and higher resolution details. [sent-207, score-0.406]

97 6 Conclusion This paper presents a strikingly simple yet effective method for non-uniform camera shake removal based upon a principled, transparent cost function that is open to analysis and further extensions/reﬁnements. [sent-209, score-0.472]

98 One √ ¯ such simple example is a penalty of the form i log( λ + |xi | wi 2 ); many others are possible. [sent-213, score-0.239]

99 Richardson-Lucy deblurring for scenes under a projective motion path. [sent-386, score-0.551]

100 Multi-image blind deblurring using a coupled adaptive sparse prior. [sent-430, score-0.736]

similar papers computed by tfidf model

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