Denoising an Image by Denoising Its Curvature Image

Bertalmío, Marcelo; Levine, Stacey

doi:10.1137/120901246

Cited by 30 publications

(26 citation statements)

References 27 publications

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“…In some sense, the denoising process preserves the first order local geometry of the image. Inspired by [3] where it is shown that curvature information is less corrupted than first order local geometry under additive Gaussian noise, we are currently investigating projections taking into the curvature information of Riemannian manifold. Finally, we would like to point out that the step 2 of our denoising method might be extended to any denoising method.…”

Section: Resultsmentioning

confidence: 99%

Generalized Gradient on Vector Bundle – Application to Image Denoising

Batard

Bertalmío

2013

Lecture Notes in Computer Science

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Abstract. We introduce a gradient operator that generalizes the Euclidean and Riemannian gradients. This operator acts on sections of vector bundles and is determined by three geometric data: a Riemannian metric on the base manifold, a Riemannian metric and a covariant derivative on the vector bundle. Under the assumption that the covariant derivative is compatible with the metric of the vector bundle, we consider the problems of minimizing the L2 and L1 norms of the gradient. In the L2 case, the gradient descent for reaching the solutions is a heat equation of a differential operator of order two called connection Laplacian. We present an application to color image denoising by replacing the regularizing term in the Rudin-Osher-Fatemi (ROF) denoising model by the L1 norm of a generalized gradient associated with a well-chosen covariant derivative. Experiments are validated by computations of the PSNR and Q-index.

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Section: Resultsmentioning

confidence: 99%

Generalized Gradient on Vector Bundle – Application to Image Denoising

Batard

Bertalmío

2013

Lecture Notes in Computer Science

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“…Such an approach motivated the Bregman iterations-based denoising algorithm of Osher et al [26], [36]. More recently, Bertalmío and Levine [9] adopted a similar approach but dealing with the curvature of the level-lines.…”

Section: A Short Overview On Variational Methods For Image Regularizamentioning

confidence: 99%

“…Dealing with grey-level images, we compare our method with three variational denoising methods whose fidelity term, like our method, is not spatially adapted, namely the curvature-based denoising method of Bertalmío and Levine applied to the ROF model [9], Bregman iterations [36], and Chambolle's algorithm [14]. We also compare with the method of Gilboa et al [23] that does contain a spatially adapted fidelity term dedicated to take into account the texture information of the image.…”

Section: 52mentioning

confidence: 99%

“…The results for the curvature-based and Bregman iterations methods included in the tables have already been represented in [9], Figure 4.2. We obtained the results for Chambolle's algorithm using the demo available in IPOL [20].…”

Section: 52mentioning

confidence: 99%

“…We also compare our method with a color extension of the Split Bregman algorithm [26] using the demo available in IPOL [22], and we test an extension of the spatially adapted method of Gilboa et al [23] by applying it on each component of the color images. Note that the curvature-based denoising method in [9] that we compared with in the case of grey-level images has not been designed, as the method of Gilboa et al, for multi-channel images. Results are available in Table 4.9 (increase of PSNR) and Table 4.10 (increase of Q-Index).…”

Section: 52mentioning

confidence: 99%

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On Covariant Derivatives and Their Applications to Image Regularization

Batard¹,

Bertalmío²

2014

SIAM J. Imaging Sci.

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Abstract. We present a generalization of the Euclidean and Riemannian gradient operators to a vector bundle, a geometric structure generalizing the concept of manifold. One of the key ideas is to replace the standard differentiation of a function by the covariant differentiation of a section. Dealing with covariant derivatives satisfying the property of compatibility with vector bundle metrics, we construct generalizations of existing mathematical models for image regularization that involve the Euclidean gradient operator, namely the linear scale-space and the Rudin-Osher-Fatemi denoising model. For well-chosen covariant derivatives, we show that our denoising model outperforms stateof-the-art variational denoising methods of the same type both in terms of PSNR and Q-index [45].

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Image denoising using the Gaussian curvature of the image surface

Brito‐Loeza

Chen

Uc-Cetina

2015

Numer. Methods Partial Differential Eq.

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A number of high-order variational models for image denoising have been proposed within the last few years. The main motivation behind these models is to fix problems such as the staircase effect and the loss of image contrast that the classical Rudin-Osher-Fatemi model [Leonid I. Rudin, Stanley Osher and Emad Fatemi, Nonlinear total variation based noise removal algorithms, Physica D 60 (1992), pp. 259-268] and others also based on the gradient of the image do have. In this work, we propose a new variational model for image denoising based on the Gaussian curvature of the image surface of a given image. We analytically study the proposed model to show why it preserves image contrast, recovers sharp edges, does not transform piecewise smooth functions into piecewise constant functions and is also able to preserve corners. In addition, we also provide two fast solvers for its numerical realization. Numerical experiments are shown to illustrate the good performance of the algorithms and test results.

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Denoising an Image by Denoising Its Curvature Image

Cited by 30 publications

References 27 publications

Generalized Gradient on Vector Bundle – Application to Image Denoising

Generalized Gradient on Vector Bundle – Application to Image Denoising

On Covariant Derivatives and Their Applications to Image Regularization

Image denoising using the Gaussian curvature of the image surface

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