Anisotropic Total Variation Filtering

Grasmair, Markus; Lenzen, Frank

doi:10.1007/s00245-010-9105-x

Cited by 100 publications

(103 citation statements)

References 14 publications

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“…Furthermore, a basic shortcoming of both TV and VTV is that the gradient magnitude, employed to penalize the image variation at every point x, is too simple as an image descriptor; it relies only on x without taking into account the available information from its neighborhood. In fact, most of the existing extensions of TV [30,33,55,58] as well as related regularizers [52] share the same drawback: they integrate a penalty of image variation that is completely localized.…”

Section: Tv Regularizationmentioning

confidence: 99%

“…However, these methods extract the local information either from the input image in a preprocessing step or as an additional unknown function of the optimization problem, not directly depending on the underlying image. On the contrary, the so-called anisotropic TV (ATV) [33,38] adapts the penalization of image variation by introducing a "diffusion" tensor that depends on the structure tensor of the unknown image itself. Nevertheless, in this case the adaptivity on image structures is heuristically designed, similarly to the design of the coherence-enhancing diffusion [57].…”

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confidence: 99%

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Structure Tensor Total Variation

Lefkimmiatis¹,

Roussos²,

Maragos³

et al. 2015

SIAM J. Imaging Sci.

119

129

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Abstract. We introduce a novel generic energy functional that we employ to solve inverse imaging problems within a variational framework. The proposed regularization family, termed as structure tensor total variation (STV), penalizes the eigenvalues of the structure tensor and is suitable for both grayscale and vector-valued images. It generalizes several existing variational penalties, including the total variation seminorm and vectorial extensions of it. Meanwhile, thanks to the structure tensor's ability to capture first-order information around a local neighborhood, the STV functionals can provide more robust measures of image variation. Further, we prove that the STV regularizers are convex while they also satisfy several invariance properties w.r.t. image transformations. These properties qualify them as ideal candidates for imaging applications. In addition, for the discrete version of the STV functionals we derive an equivalent definition that is based on the patch-based Jacobian operator, a novel linear operator which extends the Jacobian matrix. This alternative definition allow us to derive a dual problem formulation. The duality of the problem paves the way for employing robust tools from convex optimization and enables us to design an efficient and parallelizable optimization algorithm. Finally, we present extensive experiments on various inverse imaging problems, where we compare our regularizers with other competing regularization approaches. Our results are shown to be systematically superior, both quantitatively and visually.

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Section: Tv Regularizationmentioning

confidence: 99%

mentioning

confidence: 99%

Structure Tensor Total Variation

Lefkimmiatis¹,

Roussos²,

Maragos³

et al. 2015

SIAM J. Imaging Sci.

119

129

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“…The extension of TV to variational tensor-based formulations was investigated by Roussos and Maragos [49], Lefkimmiatis et al [43] and Grasmair and Lenzen [33]. These approaches consider the structure tensor [10,29] and model the objective functions in terms of the tensor eigenvalues.…”

Section: Variational Methodsmentioning

confidence: 99%

Mapping-Based Image Diffusion

2016

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In this work, we introduce a novel tensorbased functional for targeted image enhancement and denoising. Via explicit regularization, our formulation incorporates application dependent and contextual information using first principles. Few works in literature treat variational models that describe both application dependent information and contextual knowledge of the denoising problem. We prove the existence of a minimizer and present results on tensor symmetry constraints, convexity, and geometric interpretation of the proposed functional. We show that our framework excels in applications where nonlinear functions are present such as in gamma correction and targeted value range filtering. We also study general denoising performance where we show comparable results to dedicated PDE-based state of the art methods.

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“…Anisotropic diffusion tensor can be used to describe the local geometry at an image pixel, thus making it appealing for various image processing tasks [6][7][8][9][10][11][12][13]. Variational methods allow easy integration of constraints and use of powerful modern optimisation techniques such as primal-dual [14][15][16], fast iterative shrinkagethresholding algorithm [17,18], and alternating direction method of multipliers [2][3][4][19][20][21][22][23][24].…”

Section: Introductionmentioning

confidence: 99%

Introducing diffusion tensor to high order variational model for image reconstruction

Duan

Ward

Sibbett

et al. 2017

Digital Signal Processing

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The version presented here may differ from the published version or from the version of record. If you wish to cite this item you are advised to consult the publisher's version. Please see the repository url above for details on accessing the published version and note that access may require a subscription.For more information, please contact eprints@nottingham.ac.uk JID:YDSPR AID:2153 /FLA [m5G; v1.219; Prn:17/07/2017; 11:02] Second order total variation (SOTV) models have advantages for image restoration over their first order counterparts including their ability to remove the staircase artefact in the restored image. However, such models tend to blur the reconstructed image when discretised for numerical solution [1][2][3][4][5]. To overcome this drawback, we introduce a new tensor weighted second order (TWSO) model for image restoration. Specifically, we develop a novel regulariser for the SOTV model that uses the Frobenius norm of the product of the isotropic SOTV Hessian matrix and an anisotropic tensor. We then adapt the alternating direction method of multipliers (ADMM) to solve the proposed model by breaking down the original problem into several subproblems. All the subproblems have closed-forms and can be solved efficiently. The proposed method is compared with state-of-the-art approaches such as tensor-based anisotropic diffusion, total generalised variation, and Euler's elastica. We validate the proposed TWSO model using extensive experimental results on a large number of images from the Berkeley BSDS500. We also demonstrate that our method effectively reduces both the staircase and blurring effects and outperforms existing approaches for image inpainting and denoising applications.

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Anisotropic Total Variation Filtering

Cited by 100 publications

References 14 publications

Structure Tensor Total Variation

Structure Tensor Total Variation

Mapping-Based Image Diffusion

Introducing diffusion tensor to high order variational model for image reconstruction

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