A volume-based method for denoising on curved surfaces

Biddle, Harry; Glehn, Ingrid von; Macdonald, Colin B.; März, Thomas

doi:10.1109/icip.2013.6738109

Cited by 6 publications

(10 citation statements)

References 19 publications

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“…All the notations have same meaning for the traditional PINNs introduced in the previous subsections. Notice that we use the same training set for first and third term as they come from one equation N [u(x)] = f (x) in (4). By training this PINN, we can get the prediction of u.…”

Section: Pinns For Pdes On 3d Surfacesmentioning

confidence: 99%

“…Partial differential equations (PDEs) on manifold in R d arise in mathematical models for many nature phenomena. Image processing applications include the mapping an image on a given surface [1], the recovery of lost information on a surface [2] and the segmentation and deciphering of images on surfaces [3], [4]. It also widely used in biological and medical science.…”

Section: Introductionmentioning

confidence: 99%

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A Physics-Informed Neural Network Framework for PDEs on 3D Surfaces: Time Independent Problems

Fang

Zhan

2020

IEEE Access

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Partial differential equations (PDEs) on surfaces are ubiquitous in all the nature science. Many traditional mathematical methods has been developed to solve surfaces PDEs. However, almost all of these methods have obvious drawbacks and complicate in general problems. As the fast growth of machine learning area, we show an algorithm by using the physics-informed neural networks (PINNs) to solve surface PDEs. To deal with the surfaces, our algorithm only need a set of points and their corresponding normal, while the traditional methods need a partition or a grid on the surface. This is a big advantage for real computation. A variety of numerical experiments have been shown to verify our algorithm.

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Section: Pinns For Pdes On 3d Surfacesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A Physics-Informed Neural Network Framework for PDEs on 3D Surfaces: Time Independent Problems

Fang

Zhan

2020

IEEE Access

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“…Our next example concerns image denoising for a textured image on the unit sphere. Following [4], we apply the Perona-Malik model to denoise surface images. The equation has the form…”

Section: Image Denoising On a Spherementioning

confidence: 99%

“…Many applications in the natural and applied sciences require the solution of partial differential equations (PDEs) on surfaces. Image processing applications include the placement of an image on a surface [1], the restoration of a damaged pattern on a surface [2] and the segmentation and the denoising of images on surfaces [3,4]. In biology, applications include the formation of patterns on animal coats [5] and the wound healing process [6].…”

Section: Introductionmentioning

confidence: 99%

An RBF-FD closest point method for solving PDEs on surfaces

Petras

Ling

Ruuth

2018

Journal of Computational Physics

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Partial differential equations (PDEs) on surfaces appear in many applications throughout the natural and applied sciences. The classical closest point method (Ruuth and Merriman, J. Comput. Phys. 227(3):1943-1961) is an embedding method for solving PDEs on surfaces using standard finite difference schemes. In this paper, we formulate an explicit closest point method using finite difference schemes derived from radial basis functions (RBF-FD). Unlike the orthogonal gradients method (Piret, J. Comput. Phys. 231 (14):4662-4675, [2012]), our proposed method uses RBF centers on regular grid nodes. This formulation not only reduces the computational cost but also avoids the ill-conditioning from point clustering on the surface and is more natural to couple with a grid based manifold evolution algorithm (Leung and Zhao, J. Comput. Phys. 228(8):2993-3024, [2009]). When compared to the standard finite difference discretization of the closest point method, the proposed method requires a smaller computational domain surrounding the surface, resulting in a decrease in the number of sampling points on the surface. In addition, higher-order schemes can easily be constructed by increasing the number of points in the RBF-FD stencil. Applications to a variety of examples are provided to illustrate the numerical convergence of the method.

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“…Many applications in the natural and applied sciences involve the solution of partial differential equations (PDEs) on surfaces. Application areas for PDEs on static surfaces include image processing [1,2,3], biology [4,5] and computer graphics [6]. Applications for PDEs on moving surfaces also occur frequently.…”

Section: Introductionmentioning

confidence: 99%

A least-squares implicit RBF-FD closest point method and applications to PDEs on moving surfaces

Petras

Ling

Piret

et al. 2019

Journal of Computational Physics

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The closest point method (Ruuth and Merriman, J. Comput. Phys. 227(3):1943-1961) is an embedding method developed to solve a variety of partial differential equations (PDEs) on smooth surfaces, using a closest point representation of the surface and standard Cartesian grid methods in the embedding space. Recently, a closest point method with explicit time-stepping was proposed that uses finite differences derived from radial basis functions (RBF-FD). Here, we propose a least-squares implicit formulation of the closest point method to impose the constant-along-normal extension of the solution on the surface into the embedding space. Our proposed method is particularly flexible with respect to the choice of the computational grid in the embedding space. In particular, we may compute over a computational tube that contains problematic nodes. This fact enables us to combine the proposed method with the grid based particle method (Leung and Zhao, J. Comput. Phys.228 (8): 2993-3024, [2009]) to obtain a numerical method for approximating PDEs on moving surfaces. We present a number of examples to illustrate the numerical convergence properties of our proposed method. Experiments for advection-diffusion equations and Cahn-Hilliard equations that are strongly coupled to the velocity of the surface are also presented.

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A volume-based method for denoising on curved surfaces

Cited by 6 publications

References 19 publications

A Physics-Informed Neural Network Framework for PDEs on 3D Surfaces: Time Independent Problems

A Physics-Informed Neural Network Framework for PDEs on 3D Surfaces: Time Independent Problems

An RBF-FD closest point method for solving PDEs on surfaces

A least-squares implicit RBF-FD closest point method and applications to PDEs on moving surfaces

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