Polyakov Action Minimization for Efficient Color Image Processing

Rosman, Guy; Tai, Xue–Cheng; Dascal, Lorina; Kimmel, Ron

doi:10.1007/978-3-642-35740-4_5

Cited by 18 publications

(15 citation statements)

References 29 publications

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“…To overcome this issue, we use the iterative re-weighted least squares (IRLS) technique, which has been successfully used in the context of Beltrami in [36]. IRLS iteratively minimizes the square root term c .…”

Section: Algorithm 1 Lagrangian Methods For Minimization Of Hac Methodsmentioning

confidence: 99%

Harmonic Active Contours

Estellers

Zosso

Bresson

et al. 2014

IEEE Trans. on Image Process.

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Abstract-We propose a segmentation method based on the geometric representation of images as 2-D manifolds embedded in a higher dimensional space. The segmentation is formulated as a minimization problem, where the contours are described by a level set function and the objective functional corresponds to the surface of the image manifold. In this geometric framework, both data-fidelity and regularity terms of the segmentation are represented by a single functional that intrinsically aligns the gradients of the level set function with the gradients of the image and results in a segmentation criterion that exploits the directional information of image gradients to overcome image inhomogeneities and fragmented contours. The proposed formulation combines this robust alignment of gradients with attractive properties of previous methods developed in the same geometric framework: 1) the natural coupling of image channels proposed for anisotropic diffusion and 2) the ability of subjective surfaces to detect weak edges and close fragmented boundaries. The potential of such a geometric approach lies in the general definition of Riemannian manifolds, which naturally generalizes existing segmentation methods (the geodesic active contours, the active contours without edges, and the robust edge integrator) to higher dimensional spaces, non-flat images, and feature spaces. Our experiments show that the proposed technique improves the segmentation of multi-channel images, images subject to inhomogeneities, and images characterized by geometric structures like ridges or valleys.

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Section: Algorithm 1 Lagrangian Methods For Minimization Of Hac Methodsmentioning

confidence: 99%

Harmonic Active Contours

Estellers

Zosso

Bresson

et al. 2014

IEEE Trans. on Image Process.

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“…Some of these techniques make use of the structure tensor of the image (see for instance [40], [42], [44], [46] for anisotropic diffusion and [33], [38] for denoising). In particular, the approaches in [38], [42] are based on a generalization of the gradient operator to Riemannian manifolds. Let us also cite the recent work of Lenzen et al [32] who introduced spatial adaptivity in the TGV for denoising purposes.…”

Section: A Short Overview On Variational Methods For Image Regularizamentioning

confidence: 99%

“…where the regularizing term is the L 1 norm of the gradient associated to this geometric triplet is the denoising model of Rosman et al [38].…”

Section: Geometric Triplets That Lead To Existing Regularization Methmentioning

confidence: 99%

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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“…Since then, this model has been extended in several ways (see e.g. [4], [10], [15], [16], [17], [18], [24] for local methods based on a modification of the regularizing term, and [9], [11] for nonlocal methods). In this paper, we construct a new regularizing term by the introduction of a generalization of the gradient operator.…”

Section: Introductionmentioning

confidence: 99%

“…The ROF denoising model based of this new gradient operator generalizes the Euclidean approach of [19] and its multidimensional extension [4], as well as the Riemannian ROF denoising model in [18]. The key idea is to treat the term ∇I as a vector-valued differential 1-form ∇ E I, that we call connection gradient of I, where the operator ∇ E is a covariant derivative (also called connection).…”

Section: Introductionmentioning

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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Polyakov Action Minimization for Efficient Color Image Processing

Cited by 18 publications

References 29 publications

Harmonic Active Contours

Harmonic Active Contours

On Covariant Derivatives and Their Applications to Image Regularization

Generalized Gradient on Vector Bundle – Application to Image Denoising

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