A Note on the Dual Treatment of Higher-Order Regularization Functionals

Steidl, Gabriele

doi:10.1007/s00607-005-0129-z

Cited by 75 publications

(51 citation statements)

References 29 publications

(27 reference statements)

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“…Based on this definition, we have (13) The Hessian spectral norm can be alternatively defined as (14) where are the eigenvalues of the Hessian matrix of at coordinates . These two eigenvalues are expressed as (15) where the associated differential operators are defined in Table I. The eigenvalues of the Hessian operator are called principal directions.…”

Section: Two-dimensional Regularizationmentioning

confidence: 99%

“…The resulting regularizers are therefore suitable extensions of TV for the secondorder case as they retain invariances while following the same principles. It is worthwhile to note that the Hessian Frobenius norm has been introduced as a regularization model in [9], [14], and [15] for image denoising. Meanwhile, the connection we establish between the Frobenius norm and the norm of justifies this regularizer as a valid extension of TV.…”

Section: Two-dimensional Regularizationmentioning

confidence: 99%

“…In this case, the proposed variational methods are either combining a second-order regularizer with the TV seminorm [8], [9], [11], [12] or employing a second-order regularizer in a standalone way [10], [13]- [15]. The regularizers that have already been used on this problem include the norm of the Laplacian operator [8], [13], [15] (1) where , with , the modified Laplacian [14], [15] (2) the Frobenius norm of the Hessian [9], [14], [15] (3) and the affine TV functional [10] (4) 1057-7149/$26.00 © 2011 IEEE For image denoising, these regularizers have been shown either to perform better than TV or to complement it favorably. This motivates us to investigate higher order regularization methods that constitute valid extensions of TV and that can be applied effectively to the more general problem of image restoration/deblurring.…”

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confidence: 99%

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Hessian-Based Norm Regularization for Image Restoration With Biomedical Applications

Lefkimmiatis

Bourquard

Unser

2012

IEEE Trans. on Image Process.

214

151

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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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Section: Two-dimensional Regularizationmentioning

confidence: 99%

Section: Two-dimensional Regularizationmentioning

confidence: 99%

mentioning

confidence: 99%

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Hessian-Based Norm Regularization for Image Restoration With Biomedical Applications

Lefkimmiatis

Bourquard

Unser

2012

IEEE Trans. on Image Process.

214

151

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“…of vectors are meant componentwise. By D ∈ R mn,n we denote either i) some discretization of the gradient operator as, e.g., those in [14,60] with m = 2, see (36), or ii) the NL-means operator with binary weights introduced in [37] with m associated to the number of permitted neighbors, see Section 5. Note that as in i) the rows of D contain exactly one entry −1 and one entry 1 or are zero rows.…”

Section: Discrete Denoising Modelmentioning

confidence: 99%

Removing Multiplicative Noise by Douglas-Rachford Splitting Methods

2009

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In this paper, we consider a variational restoration model consisting of the I-divergence as data fitting term and the total variation semi-norm or nonlocal means as regularizer for removing multiplicative Gamma noise. Although the I-divergence is the typical data fitting term when dealing with Poisson noise we substantiate why it is also appropriate for cleaning Gamma noise. We propose to compute the minimizers of our restoration functionals by applying Douglas-Rachford splitting techniques, resp. alternating direction methods of multipliers. For a particular splitting, we present a semi-implicit scheme to solve the involved nonlinear systems of equations and prove its Q-linear convergence. Finally, we demonstrate the performance of our methods by numerical examples.

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“…(L2) The operator associated with the Frobenius norm of the Hessian L := (∂ xx , ∂ yy , ∂ xy , ∂ yx ) T in [55] -here m = 4n, i.e. κ = 4 -and some of its relatives, see, e.g., [7,13,41,37].…”

Section: Restoration Of Images Corrupted By Gaussian Noisementioning

confidence: 99%

Homogeneous Penalizers and Constraints in Convex Image Restoration

2012

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Recently convex optimization models were successfully applied for solving various problems in image analysis and restoration. In this paper, we are interested in relations between convex constrained optimization problems of the form argmin{Φ(x) subject to Ψ(x) ≤ τ } and their penalized counterparts argmin{Φ(x) + λΨ(x)}. We recall general results on the topic by the help of an epigraphical projection. Then we deal with the special setting Ψ := L · with L ∈ R m,n and Φ := ϕ(H·), where H ∈ R n,n and ϕ : R n → R ∪ {+∞} meet certain requirements which are often fulfilled in image processing models. In this case we prove by incorporating the dual problems that there exists a bijective function such that the solutions of the constrained problem coincide with those of the penalized problem if and only if τ and λ are in the graph of this function. We illustrate the relation between τ and λ for various problems arising in image processing. In particular, we point out the relation to the Pareto frontier for joint sparsity problems. We demonstrate the performance of the constrained model in restoration tasks of images corrupted by Poisson noise with the I-divergence as data fitting term ϕ and in inpainting models with the constrained nuclear norm. Such models can be useful if we have a priori knowledge on the image rather than on the noise level.

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A Note on the Dual Treatment of Higher-Order Regularization Functionals

Cited by 75 publications

References 29 publications

Hessian-Based Norm Regularization for Image Restoration With Biomedical Applications

Hessian-Based Norm Regularization for Image Restoration With Biomedical Applications

Removing Multiplicative Noise by Douglas-Rachford Splitting Methods

Homogeneous Penalizers and Constraints in Convex Image Restoration

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