Sparse Wavelet Representations of Spatially Varying Blurring Operators

Escande, Paul; Weiss, Pierre

doi:10.1137/151003465

Cited by 16 publications

(27 citation statements)

References 44 publications

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“…Finally, we implement the iterative algorithm on a GPU. The first two ideas, which constitute the main contribution of this paper, are motivated by our recent observation that spatially varying blur operators are compressible and have a well characterized structure in the wavelet domain [15]. We showed that matrix Θ = Ψ * HΨ,…”

Section: The Proposed Ideas In a Nutshellmentioning

confidence: 99%

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Accelerating ℓ1−ℓ2 deblurring using wavelet expansions of operators

Escande

Weiss

2018

Journal of Computational and Applied Mathematics

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Image deblurring is a fundamental problem in imaging, usually solved with computationally intensive optimization procedures. We show that the minimization can be significantly accelerated by leveraging the fact that images and blur operators are compressible in the same orthogonal wavelet basis. The proposed methodology consists of three ingredients: i) a sparse approximation of the blur operator in wavelet bases, ii) a diagonal preconditioner and iii) an implementation on massively parallel architectures. Combing the three ingredients leads to acceleration factors ranging from 30 to 250 on a typical workstation. For instance, a 1024 × 1024 image can be deblurred in 0.15 seconds, which corresponds to real-time.

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Section: The Proposed Ideas In a Nutshellmentioning

confidence: 99%

“…Our experience is that diagonal approximations are too crude to provide sufficiently good approximations. More recently the authors of [35,15] proposed independently to compress operators in the wavelet domain. However they did not explore its implications for the fast resolution of inverse problems.…”

Section: Related Workmentioning

confidence: 99%

Accelerating ℓ1−ℓ2 deblurring using wavelet expansions of operators

Escande

Weiss

2018

Journal of Computational and Applied Mathematics

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“…We began investigating a much narrower version of the problem in a preliminary version of [EW15]. An anonymous reviewer however suggested that it would be more interesting to make a general analysis and we therefore discarded the aspects related to convolution-product expansions from [EW15]. We thank the reviewer for motivating us to initiate this research.…”

Section: Acknowledgmentsmentioning

confidence: 99%

Approximation of Integral Operators Using Product-Convolution Expansions

Escande

Weiss

2017

J Math Imaging Vis

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We consider a class of linear integral operators with impulse responses varying regularly in time or space. These operators appear in a large number of applications ranging from signal/image processing to biology. Evaluating their action on functions is a computationally intensive problem necessary for many practical problems. We analyze a technique called convolution-product expansion: the operator is locally approximated by a convolution, allowing to design fast numerical algorithms based on the fast Fourier transform. We design various types of expansions, provide their explicit rates of approximation and their complexity depending on the time varying impulse response smoothness. This analysis suggests novel wavelet based implementations of the method with numerous assets such as optimal approximation rates, low complexity and storage requirements as well as adaptivity to the kernels regularity. The proposed methods are an alternative to more standard procedures such as panel clustering, cross approximations, wavelet expansions or hierarchical matrices.

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“…The multiplication by the matrix A or A T can be implemented implicitly by filtering with spatially varying filter coefficients [52,50,29]. Using the piecewise constant convolution function and induced DFT implementation, multiplication by A or A T can be approximated efficiently [39,36,22,25,48,40]. However, the approximation of A and A T introduces mismatches of blur models in the acquisition and restoration processes.…”

Section: Tv Restorationmentioning

confidence: 99%

“…The KADMM algorithm can be applied by identifying (25). The KADMM for the optimization problem in (26) is given in Algorithm 4.…”

Section: Application Of Kadmmmentioning

confidence: 99%

ADMM in Krylov Subspace and Its Application to Total Variation Restoration of Spatially Variant Blur

Yun¹,

Yang²

2017

SIAM J. Imaging Sci.

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Abstract. In this paper we propose an efficient method for a convex optimization problem which involves a large nonsymmetric and non-Toeplitz matrix. The proposed method is an instantiation of the alternating direction method of multipliers applied in Krylov subspace. Our method offers significant advantages in computational speed for the convex optimization problems involved with general matrices of large size. We apply the proposed method to the restoration of spatially variant blur. The matrix representing spatially variant blur is not block circulant with circulant blocks (BCCB). Efficient implementation based on diagonalization of BCCB matrices by the discrete Fourier transform is not applicable for spatially variant blur. Since the proposed method can efficiently work with general matrices, the restoration of spatially variant blur is a good application of our method. Experimental results for total variation restoration of spatially variant blur show that the proposed method provides meaningful solutions in a short time.Key words. ADMM, Krylov subspace methods, image restoration, space variant blur, total variation AMS subject classifications. 65F10, 65F22, 68U10, 90C25DOI. 10.1137/16M10835051. Introduction. Many interesting problems in image processing involve finding a solution of a system of linear equations given by g = Af , where A ∈ R d×d and g, f ∈ R d . When the problem is ill-posed, a regularized solution can be found by reducing the solution space with constraints that reflect our prior knowledge about the solution. With convex constraints that reflect our prior knowledge, the problem can be written as a convex optimization. The alternating direction method of multipliers (ADMM) [11,26,27] is a convex optimization tool, which recently received considerable attention for its ease of incorporating diverse convex constraints to the problem, ease of implementation, and fast computational speed.The Krylov subspace method [44,45] is a projection method which restricts the solution space of the problem g = Af to the subspace spanned by the nth order Krylov sequence x, Ax, A 2 x, . . . , A n−1 x constructed from a vector x ∈ R d . By reducing the solution space to a subspace of small dimensionality, a solution can be found efficiently. The orthonormal basis of the Krylov subspace can be found by the Lanczos bidiagonalization when A is symmetric and by the Arnoldi process when A is nonsymmetric. Examples of the Krylov subspace method are minimum residual (MINRES) [41] and generalized minimum residual (GMRES)

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Sparse Wavelet Representations of Spatially Varying Blurring Operators

Cited by 16 publications

References 44 publications

Accelerating ℓ1−ℓ2 deblurring using wavelet expansions of operators

Accelerating ℓ1−ℓ2 deblurring using wavelet expansions of operators

Approximation of Integral Operators Using Product-Convolution Expansions

ADMM in Krylov Subspace and Its Application to Total Variation Restoration of Spatially Variant Blur

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