Uniqueness of Viscosity Mean Curvature Flow Solution in Two Sub-Riemannian Structures

Baspinar, Emre; Citti, Giovanna

doi:10.1137/17m1150797

Cited by 7 publications

(13 citation statements)

References 41 publications

(94 reference statements)

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“…4. We use the Gabor filter banks obtained from (6) and (8) with scale value of 2 pixels (the total filter size is 24 pixels) to lift the test images (see Fig. 5 for [64] We see in the middle and right columns those two images now transformed and then inverse transformed with different numbers of orientation samples.…”

Section: Gabor Transformmentioning

confidence: 99%

“…The lifting procedure is performed by the Gabor filters of the type given by (6) and (8) with scale = 2 pixels (the filter size is 12 × scale = 24 pixels) and time step t = 0.1 in the experiments.…”

Section: Enhancementmentioning

confidence: 99%

“…10, we provide the results of our algorithm applied with different numbers of iterations on a 256 × 256 grayscale image taken from [47]. We use the Gabor filter banks obtained from (6) and (8) with scale value of 2 pixels (total filter size is 48 pixels) to lift the test images. We sample the space at frequencies ω ∈ {1.45, 1.51, 1.58, 1.66, 1.74, 1.82, 1.91, 2, 2.09, 2.19} and orientations θ ∈ { 2π 16 , 4π 16 , .…”

Section: Enhancementmentioning

confidence: 99%

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A sub-Riemannian model of the visual cortex with frequency and phase

2020

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In this paper, we present a novel model of the primary visual cortex (V1) based on orientation, frequency, and phase selective behavior of V1 simple cells. We start from the first-level mechanisms of visual perception, receptive profiles. The model interprets V1 as a fiber bundle over the two-dimensional retinal plane by introducing orientation, frequency, and phase as intrinsic variables. Each receptive profile on the fiber is mathematically interpreted as rotated, frequency modulated, and phase shifted Gabor function. We start from the Gabor function and show that it induces in a natural way the model geometry and the associated horizontal connectivity modeling of the neural connectivity patterns in V1. We provide an image enhancement algorithm employing the model framework. The algorithm is capable of exploiting not only orientation but also frequency and phase information existing intrinsically in a two-dimensional input image. We provide the experimental results corresponding to the enhancement algorithm.

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Section: Gabor Transformmentioning

confidence: 99%

Section: Enhancementmentioning

confidence: 99%

Section: Enhancementmentioning

confidence: 99%

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A sub-Riemannian model of the visual cortex with frequency and phase

2020

Self Cite

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“…There might be additional difficulties due to the fact that Carnot groups with step higher than two differ in some important geometric properties. Nonetheless step two groups already have important applications that make their study quite interesting as for instance in models of the visual cortex, see [6] and the references therein for details.…”

Section: Viscosity Solutionsmentioning

confidence: 99%

“…Remark 3.6. In [6] the authors require a subsolution u of the horizontal mean curvature flow equation to satisfy…”

Section: Viscosity Solutionsmentioning

confidence: 99%

On Viscosity and Equivalent Notions of Solutions for Anisotropic Geometric Equations

Zan

Soravia

2020

Abstract and Applied Analysis

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We prove that viscosity solutions of geometric equations in step two Carnot groups can be equivalently reformulated by restricting the set of test functions at the singular points. These are characteristic points for the level sets of the solutions and are usually difficult to deal with. A similar property is known in the euclidian space, and in Carnot groups is based on appropriate properties of a suitable homogeneous norm. We also use this idea to extend to Carnot groups the definition of generalised flow, and it works similarly to the euclidian setting. These results simplify the handling of the singularities of the equation, for instance to study the asymptotic behaviour of singular limits of reaction diffusion equations. We provide examples of using the simplified definition, showing for instance that boundaries of strictly convex subsets in the Carnot group structure become extinct in finite time when subject to the horizontal mean curvature flow even if characteristic points are present.

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Horizontal convex envelope in the Heisenberg group and applications to sub-elliptic equations

Liu¹,

Zhou²

2021

Annali Scuola Normale Superiore - Classe Di Scienze

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This paper introduces in a natural way a notion of horizontal convex envelopes of continuous functions in the Heisenberg group. We provide a convexification process to find the envelope in a constructive manner. We also apply the convexification process to show h-convexity of viscosity solutions to a class of fully nonlinear elliptic equations in the Heisenberg group satisfying a certain symmetry condition. Our examples show that in general one cannot expect h-convexity of solutions without the symmetry condition.

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Uniqueness of Viscosity Mean Curvature Flow Solution in Two Sub-Riemannian Structures

Cited by 7 publications

References 41 publications

A sub-Riemannian model of the visual cortex with frequency and phase

A sub-Riemannian model of the visual cortex with frequency and phase

On Viscosity and Equivalent Notions of Solutions for Anisotropic Geometric Equations

Horizontal convex envelope in the Heisenberg group and applications to sub-elliptic equations

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