Infimal convolution regularizations with discrete ℓ1-type functionals

Abstract-We present nonquadratic Hessian-based regularization methods that can be effectively used for image restoration problems in a variational framework. Motivated by the great success of the total-variation (TV) functional, we extend it to also include second-order differential operators. Specifically, we derive second-order regularizers that involve matrix norms of the Hessian operator. The definition of these functionals is based on an alternative interpretation of TV that relies on mixed norms of directional derivatives. We show that the resulting regularizers retain some of the most favorable properties of TV, i.e., convexity, homogeneity, rotation, and translation invariance, while dealing effectively with the staircase effect. We further develop an efficient minimization scheme for the corresponding objective functions. The proposed algorithm is of the iteratively reweighted least-square type and results from a majorization-minimization approach. It relies on a problem-specific preconditioned conjugate gradient method, which makes the overall minimization scheme very attractive since it can be applied effectively to large images in a reasonable computational time. We validate the overall proposed regularization framework through deblurring experiments under additive Gaussian noise on standard and biomedical images.

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“…For alternative higher order extensions of TV based on tensor algebra, we refer the interested reader to two recent works in [17] and [18].…”

Section: Two-dimensional Regularizationmentioning

confidence: 99%

Hessian-Based Norm Regularization for Image Restoration With Biomedical Applications

Lefkimmiatis

Bourquard

Unser

2012

IEEE Trans. on Image Process.

210

151

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“…As suggested in [24], we use three iterations and as in [16] decrease σ d and σ A by the square root of the number of iterations. In addition to bilateral filtering, we apply infimal convolution (IC) [4,21] (in combination with a weighted data term) and the non-local (NL) means algorithm [3] (using the publicly available implementation by Peyre 1 ). In Fig.…”

Section: Methodsmentioning

confidence: 99%

“…For the seminal work we refer to [18]; adaptive TV variants are described e.g. in [22,9,14]; higher order TV approaches are considered in [2,21,19,4]. As an alternative, nonlocal techniques have been proposed, e.g.…”

Section: Introductionmentioning

confidence: 99%

Denoising Time-Of-Flight Data with Adaptive Total Variation

Lenzen

Schäfer

Garbe

2011

Advances in Visual Computing

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Abstract. For denoising depth maps from time-of-flight (ToF) cameras we propose an adaptive total variation based approach of first and second order. This approach allows us to take into account the geometric properties of the depth data, such as edges and slopes. To steer adaptivity we utilize a special kind of structure tensor based on both the amplitude and phase of the recorded ToF signal. A comparison to state-of-the-art denoising methods shows the advantages of our approach.

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“…Some forms of generalized TV are based on infimal convolution (related to the Moreau envelope) [5], [7], [11], [49]. But these works propose convex penalties suitable for non-piecewise-constant signals, while we propose non-convex penalties suitable for piecewise-constant signals.…”

Section: A Relation To Prior Workmentioning

confidence: 99%

Total Variation Denoising Via the Moreau Envelope

Selesnick

2017

IEEE Signal Process. Lett.

104

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Abstract-Total variation denoising is a nonlinear filtering method well suited for the estimation of piecewise-constant signals observed in additive white Gaussian noise. The method is defined by the minimization of a particular non-differentiable convex cost function. This paper describes a generalization of this cost function that can yield more accurate estimation of piecewise constant signals. The new cost function involves a nonconvex penalty (regularizer) designed to maintain the convexity of the cost function. The new penalty is based on the Moreau envelope. The proposed total variation denoising method can be implemented using forward-backward splitting.

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Infimal convolution regularizations with discrete ℓ1-type functionals

Cited by 106 publications

References 38 publications

Hessian-Based Norm Regularization for Image Restoration With Biomedical Applications

Hessian-Based Norm Regularization for Image Restoration With Biomedical Applications

Denoising Time-Of-Flight Data with Adaptive Total Variation

Total Variation Denoising Via the Moreau Envelope

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