The Experts below are selected from a list of 240 Experts worldwide ranked by ideXlab platform
Martin Welk - One of the best experts on this subject based on the ideXlab platform.
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Multivariate Median Filters and Partial Differential Equations
Journal of Mathematical Imaging and Vision, 2016Co-Authors: Martin WelkAbstract:Multivariate median filters have been proposed as generalizations of the well-established median filter for gray-value images to multichannel images. As multivariate median, most of the recent approaches use the $$L^1$$ L 1 median, i.e., the minimizer of an objective function that is the sum of distances to all input points. Many properties of univariate median filters generalize to such a filter. However, the famous result by Guichard and Morel about approximation of the mean Curvature Motion PDE by median filtering does not have a comparably simple counterpart for $$L^1$$ L 1 multivariate median filtering. We discuss the affine equivariant Oja median and the affine equivariant transformation–retransformation $$L^1$$ L 1 median as alternatives to $$L^1$$ L 1 median filtering. We analyze multivariate median filters in a space-continuous setting, including the formulation of a space-continuous version of the transformation–retransformation $$L^1$$ L 1 median, and derive PDEs approximated by these filters in the cases of bivariate planar images, three-channel volume images, and three-channel planar images. The PDEs for the affine equivariant filters can be interpreted geometrically as combinations of a diffusion and a principal-component-wise Curvature Motion contribution with a cross-effect term based on torsions of principal components. Numerical experiments are presented, which demonstrate the validity of the approximation results.
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multivariate median filters and partial differential equations
arXiv: Computer Vision and Pattern Recognition, 2015Co-Authors: Martin WelkAbstract:Multivariate median filters have been proposed as generalisations of the well-established median filter for grey-value images to multi-channel images. As multivariate median, most of the recent approaches use the $L^1$ median, i.e.\ the minimiser of an objective function that is the sum of distances to all input points. Many properties of univariate median filters generalise to such a filter. However, the famous result by Guichard and Morel about approximation of the mean Curvature Motion PDE by median filtering does not have a comparably simple counterpart for $L^1$ multivariate median filtering. We discuss the affine equivariant Oja median and the affine equivariant transformation--retransformation $L^1$ median as alternatives to $L^1$ median filtering. We analyse multivariate median filters in a space-continuous setting, including the formulation of a space-continuous version of the transformation--retransformation $L^1$ median, and derive PDEs approximated by these filters in the cases of bivariate planar images, three-channel volume images and three-channel planar images. The PDEs for the affine equivariant filters can be interpreted geometrically as combinations of a diffusion and a principal-component-wise Curvature Motion contribution with a cross-effect term based on torsions of principal components. Numerical experiments are presented that demonstrate the validity of the approximation results.
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SSVM - Partial Differential Equations of Bivariate Median Filters
Lecture Notes in Computer Science, 2015Co-Authors: Martin WelkAbstract:Multivariate median filters have been proposed as generalisations of the well-established median filter for grey-value images to multi-channel images. As multivariate median, most of the recent approaches use the \(L^1\) median, i.e. the minimiser of an objective function that is the sum of distances to all input points. Many properties of univariate median filters generalise to such a filter. However, the famous result by Guichard and Morel about approximation of the mean Curvature Motion PDE by median filtering does not have a comparably simple counterpart for \(L^1\) multivariate median filtering. We discuss the affine equivariant Oja median as an alternative to \(L^1\) median filtering. We derive the PDE approximated by Oja median filtering in the bivariate case, and demonstrate its validity by a numerical experiment.
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Curvature-Driven PDE Methods for Matrix-Valued Images
International Journal of Computer Vision, 2006Co-Authors: Christian Feddern, Joachim Weickert, Bernhard Burgeth, Martin WelkAbstract:Matrix-valued data sets arise in a number of applications including diffusion tensor magnetic resonance imaging (DT-MRI) and physical measurements of anisotropic behaviour. Consequently, there arises the need to filter and segment such tensor fields. In order to detect edge-like structures in tensor fields, we first generalise Di Zenzo’s concept of a structure tensor for vector-valued images to tensor-valued data. This structure tensor allows us to extend scalar-valued mean Curvature Motion and self-snakes to the tensor setting. We present both two-dimensional and three-dimensional formulations, and we prove that these filters maintain positive semidefiniteness if the initial matrix data are positive semidefinite. We give an interpretation of tensorial mean Curvature Motion as a process for which the corresponding curve evolution of each generalised level line is the gradient descent of its total length. Moreover, we propose a geodesic active contour model for segmenting tensor fields and interpret it as a minimiser of a suitable energy functional with a metric induced by the tensor image. Since tensorial active contours incorporate information from all channels, they give a contour representation that is highly robust under noise. Experiments on three-dimensional DT-MRI data and an indefinite tensor field from fluid dynamics show that the proposed methods inherit the essential properties of their scalar-valued counterparts.
Fabio Toninelli - One of the best experts on this subject based on the ideXlab platform.
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Mixing Times of Monotone Surfaces and SOS Interfaces: A Mean Curvature Approach
Communications in Mathematical Physics, 2012Co-Authors: Pietro Caputo, Fabio Martinelli, Fabio ToninelliAbstract:We consider stochastic spin-flip dynamics for: (i) monotone discrete surfaces in $${\mathbb {Z}^3}$$ with planar boundary height and (ii) the one-dimensional discrete Solid-on-Solid (SOS) model confined to a box. In both cases we show almost optimal bounds O ( L ^2polylog( L )) for the mixing time of the chain, where L is the natural size of the system. The dynamics at a macroscopic scale should be described by a deterministic mean Curvature Motion such that each point of the surface feels a drift which tends to minimize the local surface tension (Spohn in J Stat Phys 71:1081–1132, 1993 ). Inspired by this heuristics, our approach consists in bounding the dynamics with an auxiliary one which, with very high probability, follows quite closely the deterministic mean Curvature evolution. Key technical ingredients are monotonicity, coupling and an argument due to Wilson (Ann Appl Probab 14:274–325, 2004 ) in the framework of lozenge tiling Markov Chains. Our approach works equally well for both models despite the fact that their equilibrium maximal height fluctuations occur on very different scales (log L for monotone surfaces and $${\sqrt L}$$ for the SOS model). Finally, combining techniques from kinetically constrained spin systems (Cancrini et al. in Probab Th Rel Fields 140:459–504, 2008 ) together with the above mixing time result, we prove an almost diffusive lower bound of order 1/ L ^2polylog( L ) for the spectral gap of the SOS model with horizontal size L and unbounded heights.
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Mixing times of monotone surfaces and SOS interfaces: a mean Curvature approach
Communications in Mathematical Physics, 2012Co-Authors: Pietro Caputo, Fabio Martinelli, Fabio ToninelliAbstract:We consider stochastic spin-flip dynamics for: (i) monotone discrete surfaces in Z^3 with planar boundary height and (ii) the one-dimensional discrete Solid-on-Solid (SOS) model confined to a box. In both cases we show almost optimal bounds O(L^2polylog(L)) for the mixing time of the chain, where L is the natural size of the system. The dynamics at a macroscopic scale should be described by a deterministic mean Curvature Motion such that each point of the surface feels a drift which tends to minimize the local surface tension. Inspired by this heuristics, our approach consists in bounding the dynamics with an auxiliary one which, with very high probability, follows quite closely the deterministic mean Curvature evolution. Key technical ingredients are monotonicity, coupling and an argument due to D.B. Wilson in the framework of lozenge tiling Markov Chains. Our approach works equally well for both models despite the fact that their equilibrium maximal height fluctuations occur on very different scales (logarithmic for monotone surfaces and L^{1/2} for the SOS model). Finally, combining techniques from kinetically constrained spin systems together with the above mixing time result, we prove an almost diffusive lower bound of order 1/L^2 up to logarithmic corrections for the spectral gap of the SOS model with horizontal size L and unbounded heights.
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Zero'' temperature stochastic 3D Ising model and dimer covering fluctuations: a first step towards interface mean Curvature Motion
Communications on Pure and Applied Mathematics, 2011Co-Authors: Pietro Caputo, Fabio Martinelli, François Simenhaus, Fabio ToninelliAbstract:We consider the Glauber dynamics for the Ising model with "+" boundary conditions, at zero temperature or at a temperature that goes to zero with the system size (hence the quotation marks in the title). In dimension d = 3 we prove that an initial domain of linear size L of "−" spins disappears within a time τ+, which is at most L2(log L)c and at least L2/(c log L) for some c > 0. The proof of the upper bound proceeds via comparison with an auxiliary dynamics which mimics the Motion by mean Curvature that is expected to describe, on large time scales, the evolution of the interface between "+" and "−" domains. The analysis of the auxiliary dynamics requires recent results on the fluctuations of the height function associated to dimmer coverings of the infinite honeycomb lattice. Our result, apart from the spurious logarithmic factors, is the first rigorous confirmation of the Lifshitz law τ+ ≃ const × L2, conjectured on heuristic grounds [8, 13]. In dimension d = 2, τ+ can be shown to be of order L2 without logarithmic corrections: the upper bound was proven in [6], and here we provide the lower bound. For d = 2, we also prove that the spectral gap of the generator behaves like equation image for L large, as conjectured in [2].
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zero temperature stochastic 3d ising model and dimer covering fluctuations a first step towards interface mean Curvature Motion
arXiv: Mathematical Physics, 2010Co-Authors: Pietro Caputo, Fabio Martinelli, François Simenhaus, Fabio ToninelliAbstract:We consider the Glauber dynamics for the Ising model with "+" boundary conditions, at zero temperature or at temperature which goes to zero with the system size (hence the quotation marks in the title). In dimension d=3 we prove that an initial domain of linear size L of "-" spins disappears within a time \tau_+ which is at most L^2(\log L)^c and at least L^2/(c\log L), for some c>0. The proof of the upper bound proceeds via comparison with an auxiliary dynamics which mimics the Motion by mean Curvature that is expected to describe, on large time-scales, the evolution of the interface between "+" and "-" domains. The analysis of the auxiliary dynamics requires recent results on the fluctuations of the height function associated to dimer coverings of the infinite honeycomb lattice. Our result, apart from the spurious logarithmic factor, is the first rigorous confirmation of the expected behavior \tau_+\simeq const\times L^2, conjectured on heuristic grounds. In dimension d=2, \tau_+ can be shown to be of order L^2 without logarithmic corrections: the upper bound was proven in [Fontes, Schonmann, Sidoravicius, 2002] and here we provide the lower bound. For d=2, we also prove that the spectral gap of the generator behaves like c/L for L large, as conjectured in [Bodineau-Martinelli, 2002].
Pietro Caputo - One of the best experts on this subject based on the ideXlab platform.
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Mixing Times of Monotone Surfaces and SOS Interfaces: A Mean Curvature Approach
Communications in Mathematical Physics, 2012Co-Authors: Pietro Caputo, Fabio Martinelli, Fabio ToninelliAbstract:We consider stochastic spin-flip dynamics for: (i) monotone discrete surfaces in $${\mathbb {Z}^3}$$ with planar boundary height and (ii) the one-dimensional discrete Solid-on-Solid (SOS) model confined to a box. In both cases we show almost optimal bounds O ( L ^2polylog( L )) for the mixing time of the chain, where L is the natural size of the system. The dynamics at a macroscopic scale should be described by a deterministic mean Curvature Motion such that each point of the surface feels a drift which tends to minimize the local surface tension (Spohn in J Stat Phys 71:1081–1132, 1993 ). Inspired by this heuristics, our approach consists in bounding the dynamics with an auxiliary one which, with very high probability, follows quite closely the deterministic mean Curvature evolution. Key technical ingredients are monotonicity, coupling and an argument due to Wilson (Ann Appl Probab 14:274–325, 2004 ) in the framework of lozenge tiling Markov Chains. Our approach works equally well for both models despite the fact that their equilibrium maximal height fluctuations occur on very different scales (log L for monotone surfaces and $${\sqrt L}$$ for the SOS model). Finally, combining techniques from kinetically constrained spin systems (Cancrini et al. in Probab Th Rel Fields 140:459–504, 2008 ) together with the above mixing time result, we prove an almost diffusive lower bound of order 1/ L ^2polylog( L ) for the spectral gap of the SOS model with horizontal size L and unbounded heights.
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Mixing times of monotone surfaces and SOS interfaces: a mean Curvature approach
Communications in Mathematical Physics, 2012Co-Authors: Pietro Caputo, Fabio Martinelli, Fabio ToninelliAbstract:We consider stochastic spin-flip dynamics for: (i) monotone discrete surfaces in Z^3 with planar boundary height and (ii) the one-dimensional discrete Solid-on-Solid (SOS) model confined to a box. In both cases we show almost optimal bounds O(L^2polylog(L)) for the mixing time of the chain, where L is the natural size of the system. The dynamics at a macroscopic scale should be described by a deterministic mean Curvature Motion such that each point of the surface feels a drift which tends to minimize the local surface tension. Inspired by this heuristics, our approach consists in bounding the dynamics with an auxiliary one which, with very high probability, follows quite closely the deterministic mean Curvature evolution. Key technical ingredients are monotonicity, coupling and an argument due to D.B. Wilson in the framework of lozenge tiling Markov Chains. Our approach works equally well for both models despite the fact that their equilibrium maximal height fluctuations occur on very different scales (logarithmic for monotone surfaces and L^{1/2} for the SOS model). Finally, combining techniques from kinetically constrained spin systems together with the above mixing time result, we prove an almost diffusive lower bound of order 1/L^2 up to logarithmic corrections for the spectral gap of the SOS model with horizontal size L and unbounded heights.
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Zero'' temperature stochastic 3D Ising model and dimer covering fluctuations: a first step towards interface mean Curvature Motion
Communications on Pure and Applied Mathematics, 2011Co-Authors: Pietro Caputo, Fabio Martinelli, François Simenhaus, Fabio ToninelliAbstract:We consider the Glauber dynamics for the Ising model with "+" boundary conditions, at zero temperature or at a temperature that goes to zero with the system size (hence the quotation marks in the title). In dimension d = 3 we prove that an initial domain of linear size L of "−" spins disappears within a time τ+, which is at most L2(log L)c and at least L2/(c log L) for some c > 0. The proof of the upper bound proceeds via comparison with an auxiliary dynamics which mimics the Motion by mean Curvature that is expected to describe, on large time scales, the evolution of the interface between "+" and "−" domains. The analysis of the auxiliary dynamics requires recent results on the fluctuations of the height function associated to dimmer coverings of the infinite honeycomb lattice. Our result, apart from the spurious logarithmic factors, is the first rigorous confirmation of the Lifshitz law τ+ ≃ const × L2, conjectured on heuristic grounds [8, 13]. In dimension d = 2, τ+ can be shown to be of order L2 without logarithmic corrections: the upper bound was proven in [6], and here we provide the lower bound. For d = 2, we also prove that the spectral gap of the generator behaves like equation image for L large, as conjectured in [2].
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zero temperature stochastic 3d ising model and dimer covering fluctuations a first step towards interface mean Curvature Motion
arXiv: Mathematical Physics, 2010Co-Authors: Pietro Caputo, Fabio Martinelli, François Simenhaus, Fabio ToninelliAbstract:We consider the Glauber dynamics for the Ising model with "+" boundary conditions, at zero temperature or at temperature which goes to zero with the system size (hence the quotation marks in the title). In dimension d=3 we prove that an initial domain of linear size L of "-" spins disappears within a time \tau_+ which is at most L^2(\log L)^c and at least L^2/(c\log L), for some c>0. The proof of the upper bound proceeds via comparison with an auxiliary dynamics which mimics the Motion by mean Curvature that is expected to describe, on large time-scales, the evolution of the interface between "+" and "-" domains. The analysis of the auxiliary dynamics requires recent results on the fluctuations of the height function associated to dimer coverings of the infinite honeycomb lattice. Our result, apart from the spurious logarithmic factor, is the first rigorous confirmation of the expected behavior \tau_+\simeq const\times L^2, conjectured on heuristic grounds. In dimension d=2, \tau_+ can be shown to be of order L^2 without logarithmic corrections: the upper bound was proven in [Fontes, Schonmann, Sidoravicius, 2002] and here we provide the lower bound. For d=2, we also prove that the spectral gap of the generator behaves like c/L for L large, as conjectured in [Bodineau-Martinelli, 2002].
Fabio Martinelli - One of the best experts on this subject based on the ideXlab platform.
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Mixing Times of Monotone Surfaces and SOS Interfaces: A Mean Curvature Approach
Communications in Mathematical Physics, 2012Co-Authors: Pietro Caputo, Fabio Martinelli, Fabio ToninelliAbstract:We consider stochastic spin-flip dynamics for: (i) monotone discrete surfaces in $${\mathbb {Z}^3}$$ with planar boundary height and (ii) the one-dimensional discrete Solid-on-Solid (SOS) model confined to a box. In both cases we show almost optimal bounds O ( L ^2polylog( L )) for the mixing time of the chain, where L is the natural size of the system. The dynamics at a macroscopic scale should be described by a deterministic mean Curvature Motion such that each point of the surface feels a drift which tends to minimize the local surface tension (Spohn in J Stat Phys 71:1081–1132, 1993 ). Inspired by this heuristics, our approach consists in bounding the dynamics with an auxiliary one which, with very high probability, follows quite closely the deterministic mean Curvature evolution. Key technical ingredients are monotonicity, coupling and an argument due to Wilson (Ann Appl Probab 14:274–325, 2004 ) in the framework of lozenge tiling Markov Chains. Our approach works equally well for both models despite the fact that their equilibrium maximal height fluctuations occur on very different scales (log L for monotone surfaces and $${\sqrt L}$$ for the SOS model). Finally, combining techniques from kinetically constrained spin systems (Cancrini et al. in Probab Th Rel Fields 140:459–504, 2008 ) together with the above mixing time result, we prove an almost diffusive lower bound of order 1/ L ^2polylog( L ) for the spectral gap of the SOS model with horizontal size L and unbounded heights.
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Mixing times of monotone surfaces and SOS interfaces: a mean Curvature approach
Communications in Mathematical Physics, 2012Co-Authors: Pietro Caputo, Fabio Martinelli, Fabio ToninelliAbstract:We consider stochastic spin-flip dynamics for: (i) monotone discrete surfaces in Z^3 with planar boundary height and (ii) the one-dimensional discrete Solid-on-Solid (SOS) model confined to a box. In both cases we show almost optimal bounds O(L^2polylog(L)) for the mixing time of the chain, where L is the natural size of the system. The dynamics at a macroscopic scale should be described by a deterministic mean Curvature Motion such that each point of the surface feels a drift which tends to minimize the local surface tension. Inspired by this heuristics, our approach consists in bounding the dynamics with an auxiliary one which, with very high probability, follows quite closely the deterministic mean Curvature evolution. Key technical ingredients are monotonicity, coupling and an argument due to D.B. Wilson in the framework of lozenge tiling Markov Chains. Our approach works equally well for both models despite the fact that their equilibrium maximal height fluctuations occur on very different scales (logarithmic for monotone surfaces and L^{1/2} for the SOS model). Finally, combining techniques from kinetically constrained spin systems together with the above mixing time result, we prove an almost diffusive lower bound of order 1/L^2 up to logarithmic corrections for the spectral gap of the SOS model with horizontal size L and unbounded heights.
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Zero'' temperature stochastic 3D Ising model and dimer covering fluctuations: a first step towards interface mean Curvature Motion
Communications on Pure and Applied Mathematics, 2011Co-Authors: Pietro Caputo, Fabio Martinelli, François Simenhaus, Fabio ToninelliAbstract:We consider the Glauber dynamics for the Ising model with "+" boundary conditions, at zero temperature or at a temperature that goes to zero with the system size (hence the quotation marks in the title). In dimension d = 3 we prove that an initial domain of linear size L of "−" spins disappears within a time τ+, which is at most L2(log L)c and at least L2/(c log L) for some c > 0. The proof of the upper bound proceeds via comparison with an auxiliary dynamics which mimics the Motion by mean Curvature that is expected to describe, on large time scales, the evolution of the interface between "+" and "−" domains. The analysis of the auxiliary dynamics requires recent results on the fluctuations of the height function associated to dimmer coverings of the infinite honeycomb lattice. Our result, apart from the spurious logarithmic factors, is the first rigorous confirmation of the Lifshitz law τ+ ≃ const × L2, conjectured on heuristic grounds [8, 13]. In dimension d = 2, τ+ can be shown to be of order L2 without logarithmic corrections: the upper bound was proven in [6], and here we provide the lower bound. For d = 2, we also prove that the spectral gap of the generator behaves like equation image for L large, as conjectured in [2].
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zero temperature stochastic 3d ising model and dimer covering fluctuations a first step towards interface mean Curvature Motion
arXiv: Mathematical Physics, 2010Co-Authors: Pietro Caputo, Fabio Martinelli, François Simenhaus, Fabio ToninelliAbstract:We consider the Glauber dynamics for the Ising model with "+" boundary conditions, at zero temperature or at temperature which goes to zero with the system size (hence the quotation marks in the title). In dimension d=3 we prove that an initial domain of linear size L of "-" spins disappears within a time \tau_+ which is at most L^2(\log L)^c and at least L^2/(c\log L), for some c>0. The proof of the upper bound proceeds via comparison with an auxiliary dynamics which mimics the Motion by mean Curvature that is expected to describe, on large time-scales, the evolution of the interface between "+" and "-" domains. The analysis of the auxiliary dynamics requires recent results on the fluctuations of the height function associated to dimer coverings of the infinite honeycomb lattice. Our result, apart from the spurious logarithmic factor, is the first rigorous confirmation of the expected behavior \tau_+\simeq const\times L^2, conjectured on heuristic grounds. In dimension d=2, \tau_+ can be shown to be of order L^2 without logarithmic corrections: the upper bound was proven in [Fontes, Schonmann, Sidoravicius, 2002] and here we provide the lower bound. For d=2, we also prove that the spectral gap of the generator behaves like c/L for L large, as conjectured in [Bodineau-Martinelli, 2002].
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Some New Results on the Kinetic Ising Model in a Pure Phase
Journal of Statistical Physics, 2002Co-Authors: T. Bodineau, Fabio MartinelliAbstract:We consider a general class of Glauber dynamics reversible with respect to the standard Ising model in ℤ^ d with zero external field and inverse temperature β strictly larger than the critical value β _ c in dimension 2 or the so called “slab threshold” β in dimension d ≥ 3. We first prove that the inverse spectral gap in a large cube of side N with plus boundary conditions is, apart from logarithmic corrections, larger than N in d = 2 while the logarithmic Sobolev constant is instead larger than N ^2 in any dimension. Such a result substantially improves over all the previous existing bounds and agrees with a similar computations obtained in the framework of a one dimensional toy model based on mean Curvature Motion. The proof, based on a suggestion made by H. T. Yau some years ago, explicitly constructs a subtle test function which forces a large droplet of the minus phase inside the plus phase. The relevant bounds for general d ≥ 2 are then obtained via a careful use of the recent $$\mathbb{L}^{1}$$ –approach to the Wulff construction. Finally we prove that in d = 2 the probability that two independent initial configurations, distributed according to the infinite volume plus phase and evolving under any coupling, agree at the origin at time t is bounded from below by a stretched exponential $$\exp ( - \sqrt t )$$ , again apart from logarithmic corrections. Such a result should be considered as a first step toward a rigorous proof that, as conjectured by Fisher and Huse some years ago, the equilibrium time auto-correlation of the spin at the origin decays as a stretched exponential in d = 2.
Jean-michel Morel - One of the best experts on this subject based on the ideXlab platform.
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A proof of equivalence between level lines shortening and Curvature Motion in image processing
SIAM Journal on Mathematical Analysis, 2013Co-Authors: Adina Ciomaga, Jean-michel MorelAbstract:In this paper we define the continuous Level Lines Shortening evolution of a two-dimensional image as the Curve Shortening operator acting simultaneously and independently on all the level lines of the initial data, and show that it computes a viscosity solution for the mean Curvature Motion. This provides an exact analytical framework for its numerical implementation, which runs online on any image at http://www.ipol.im/. Analogous results hold for its affine variant version, the Level Lines Affine Shortening.
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Image Visualization and Restoration by Curvature Motions
SIAM Multiscale Model. Simul., 2011Co-Authors: Adina Ciomaga, Pascal Monasse, Jean-michel MorelAbstract:This paper presents a review, analysis, and comparison of numerical methods implementing the Curvature Motion and the affine Curvature Motion for two-dimensional (2D) images, shapes, and curves. These Curvature scale spaces allow, in principle, one to compute an accurate multiscale Curvature in digital images. The fastest and most invariant of them can be used in a complete image processing chain. This numerical chain simulates the accurate subpixel evolution of an image by mean Curvature Motion or by affine invariant Curvature Motion. To do so, it lets all the level lines of the image evolve by Curvature shortening (of affine shortening), computes the image Curvature directly on the smoothed level lines, and reconstructs the evolved image and its Curvatures in an intrinsic, grid independent representation. The paper describes a careful implementation of this chain and analyzes its effects on many examples. The microscopic visualization of an image Curvature map reveals after processing many image details. This image process improves graphic images and gets rid of compression and aliasing effects. It also gives an accurate tool to explore the validity of Attneave's and Julesz's theories on shape perception and texture discrimination. The "Curvature microscope'' runs online for any image at http://www.ipol.im/pub/algo/cmmm_image_Curvature_microscope/.
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feature extraction from high density point clouds toward automation of an intelligent 3d contactless digitizing strategy
Computer-aided Design and Applications, 2010Co-Authors: Charyar Mehdisouzani, Nicolas Audfray, Julie Digne, Claire Lartigue, Jean-michel MorelAbstract:AbstractThis paper deals with a global intelligent 3D digitizing algorithm, which allows increasing the quality of the resulting cloud of points. Built from quality analysis and characteristic line extraction, the algorithm computes a new path belonging to an admissible space with the objective of increasing the quality of the new resulting points cloud. The extraction work is performed thanks to a scale space algorithm, based on an iterative projection algorithm and the concept of mean Curvature Motion (MCM). The scale space framework allows us to perform the detection at a coarse scale without any noise or digitizing error interference and to project the result back onto the original point cloud. Thus all the details of the real object’s shape can be identified. Several applications illustrate geometrical feature extraction and global intelligent 3D digitizing within the context of RE. An application is also proposed to compute the distance between the real object shape and an existent CAD-model for con...
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Neighborhood filters and PDE’s
Numerische Mathematik, 2006Co-Authors: Antoni Buades, Bartomeu Coll, Jean-michel MorelAbstract:Denoising images can be achieved by a spatial averaging of nearby pixels. However, although this method removes noise it creates blur. Hence, neighborhood filters are usually preferred. These filters perform an average of neighboring pixels, but only under the condition that their grey level is close enough to the one of the pixel in restoration. This very popular method unfortunately creates shocks and staircasing effects. In this paper, we perform an asymptotic analysis of neighborhood filters as the size of the neighborhood shrinks to zero. We prove that these filters are asymptotically equivalent to the Perona–Malik equation, one of the first nonlinear PDE’s proposed for image restoration. As a solution, we propose an extremely simple variant of the neighborhood filter using a linear regression instead of an average. By analyzing its subjacent PDE, we prove that this variant does not create shocks: it is actually related to the mean Curvature Motion. We extend the study to more general local polynomial estimates of the image in a grey level neighborhood and introduce two new fourth order evolution equations.
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Neighborhood filters and PDE's
Numerische Mathematik, 2006Co-Authors: Antoni Buades, Bartomeu Coll, Jean-michel MorelAbstract:Denoising images can be achieved by a spatial averaging of nearby pixels. However, although this method removes noise it creates blur. Hence, neighborhood filters are usually preferred. These filters perform an average of neighboring pixels, but only under the condition that their grey level is close enough to the one of the pixel in restoration. This very popular method unfortunately creates shocks and staircasing effects. In this paper, we perform an asymptotic analysis of neighborhood filters as the size of the neighborhood shrinks to zero. We prove that these filters are asymptotically equivalent to the Perona-Malik equation, one of the first nonlinear PDE's proposed for image restoration. As a solution, we propose an extremely simple variant of the neighborhood filter using a linear regression instead of an average. By analyzing its subjacent PDE, we prove that this variant does not create shocks: it is actually related to the mean Curvature Motion. We extend the study to more general local polynomial estimates of the image in a grey level neighborhood and introduce two new fourth order evolution equations.
