From high energy physics to low level vision

Kimmel, Ron; Sochen, Nir; Malladi, R.

doi:10.1007/3-540-63167-4_54

Cited by 86 publications

(48 citation statements)

References 16 publications

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“…Finally, we review the variational formulation of image diffusion given in [40,64,39,63], where the authors consider images defined on Riemannian manifolds with a metric that depends on the image and reflects the anisotropy of the underlying problem (designed for edge preservation, for color image restoration, for texture analysis, etc.). The basic energy functional is the Polyakov action, which is the extension of the Dirichlet integral to maps between Riemannian manifolds [40,64].…”

Section: Variational Models For Image Diffusionmentioning

confidence: 99%

“…Then in subsection 6.2 we compute some structure tensors for video. We end these theoretical developments by some discussion in section 7 on diffusion operators defined by variational models written in terms of the Dirichlet integral (Polyakov action) on the Riemannian manifold (following the formulation in [40,64,39,63]). This reflects on one hand the unifying underlying principle reflected by the Dirichlet integral and on the other the main differences with multiscale analyses.…”

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confidence: 99%

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Multiscale Analysis for Images on Riemannian Manifolds

Calderero¹,

Caselles²

2014

SIAM J. Imaging Sci.

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Abstract. In this paper we study multiscale analyses for images defined on Riemannian manifolds and extend the axiomatic approach proposed byÁlvarez, Guichard, Lions, and Morel to this general case. This covers the case of two-and three-dimensional images and video sequences. After obtaining the general classification, we consider the case of morphological scale spaces, which are given in terms of geometric equations, and the linear case given by the Laplace-Beltrami flow. We consider in some detail the case of image metrics given in terms of the structure tensor and compute some cases of such a tensor for video. Then we comment on the connections with variational formulations of image diffusion comparing the anisotropies that appear. Finally, we include numerical experiments illustrating some of the models. Namely, we compare some examples for still images using the Laplace-Beltrami flow and some variational models. We also consider several examples in video: the mean curvature motion and the extension of the morphological and Galilean invariant scale spaces to the video manifold case, and the Laplace-Beltrami flow. We point out that the number of models that appear is huge, and we have restricted ourselves to such cases for the sake of brevity and illustration.

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Section: Variational Models For Image Diffusionmentioning

confidence: 99%

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confidence: 99%

Multiscale Analysis for Images on Riemannian Manifolds

Calderero¹,

Caselles²

2014

SIAM J. Imaging Sci.

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“…Sochen, Kimmel and Malladi introduced in [19] and [9] a general geometrical framework for low-level vision, based on an energy functional defined by Polyakov in [10]. In this framework which is widely used for image restoration, anisotropic smoothing and scale-spaces, images are seen as surfaces or hypersurfaces embedded in higher dimensional spaces.…”

Section: Weighted Polyakov Energymentioning

confidence: 99%

“…The spatial derivative within such a scale-space is now obtained as c∇, whereas the scale derivative is given by ρc∂ σ . The natural heat equation, that defines the scale-space, is: [9], [19]. The linear and the Beltrami scalespace are illustrated at the example of the fractal image of a Von Koch snowflake, and a single slice of a T1-weighted brain MR image in Fig.…”

Section: A Motivationmentioning

confidence: 99%

Geodesic Active Fields—A Geometric Framework for Image Registration

Zosso

Bresson

Thiran

2011

IEEE Trans. on Image Process.

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Abstract-In this paper we present a novel geometric framework called geodesic active fields for general image registration. In image registration, one looks for the underlying deformation field that best maps one image onto another. This is a classic ill-posed inverse problem, which is usually solved by adding a regularization term. Here, we propose a multiplicative coupling between the registration term and the regularization term, which turns out to be equivalent to embed the deformation field in a weighted minimal surface problem. Then, the deformation field is driven by a minimization flow toward a harmonic map corresponding to the solution of the registration problem. This proposed approach for registration shares close similarities with the well-known geodesic active contours model in image segmentation, where the segmentation term (the edge detector function) is coupled with the regularization term (the length functional) via multiplication as well. As a matter of fact, our proposed geometric model is actually the exact mathematical generalization to vector fields of the weighted length problem for curves and surfaces introduced by Caselles-Kimmel-Sapiro [1]. The energy of the deformation field is measured with the Polyakov energy weighted by a suitable image distance, borrowed from standard registration models. We investigate three different weighting functions, the squared error and the approximated absolute error for monomodal images, and the local joint entropy for multimodal images. As compared to specialized state-of-theart methods tailored for specific applications, our geometric framework involves important contributions. Firstly, our general formulation for registration works on any parametrizable, smooth and differentiable surface, including non-flat and multiscale images. In the latter case, multiscale images are registered at all scales simultaneously, and the relations between space and scale are intrinsically being accounted for. Secondly, this method is, to the best of our knowledge, the first re-parametrization invariant registration method introduced in the literature. Thirdly, the multiplicative coupling between the registration term, i.e. local image discrepancy, and the regularization term naturally results in a data-dependent tuning of the regularization strength. Finally, by choosing the metric on the deformation field one can freely interpolate between classic Gaussian and more interesting anisotropic, TV-like regularization.

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“…We represent an image as a two-dimensional Riemannian surface embedded in a higher dimensional spatial-feature Riemannian manifold [11,10,3,4,5,13,12]. Let σ µ , µ = 1, 2, be the local coordinates on the image surface and let X i , i = 1, 2, .…”

Section: A Geometric Measure On Embedded Mapsmentioning

confidence: 99%