Nonlocal PDEs on Graphs: From Tug-of-War Games to Unified Interpolation on Images and Point Clouds

Elmoataz, Abderrahim; Lozes, François; Toutain, Matthieu

doi:10.1007/s10851-016-0683-3

Cited by 19 publications

(12 citation statements)

References 61 publications

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“…Analysis on graphs is an important mathematical topic and has has many applications in other fields, such as imagine processing, data analysis, neural network, etc. [6,9,10,[12][13][14][15]. Recently, many researchers have pay attention to the differential equations on graphs.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Convergence of ground state solutions for nonlinear Schrödinger equations on graphs

Zhang

Zhao

2018

Sci. China Math.

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We consider the nonlinear Schrödinger equation −∆u + (λa(x) + 1)u = |u| p−1 u on a locally finite graph G = (V, E). We prove via the Nehari method that if a(x) satisfies certain assumptions, for any λ > 1, the equation admits a ground state solution u λ . Moreover, as λ → ∞, the solution u λ converges to a solution of the Dirichlet problem −∆u + u = |u| p−1 u which is defined on the potential well Ω. We also provide a numerical experiment which solves the equation on a finite graph to illustrate our results.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Convergence of ground state solutions for nonlinear Schrödinger equations on graphs

Zhang

Zhao

2018

Sci. China Math.

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“…IV. Color processing on point clouds: There are several works studying the process of colors on point clouds [44]- [49]. These methods utilize the morphological partial differential equation (PDE) operators on graphs, and treat the color operation on point clouds as an iterative diffusion process.…”

Section: Related Workmentioning

confidence: 99%

Sparse Graph Regularized Mesh Color Edit Propagation

Ли

Lai

Rosin

2020

IEEE Trans. on Image Process.

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Mesh color edit propagation aims to propagate the color from a few color strokes to the whole mesh, which is useful for mesh colorization, color enhancement and color editing, etc. Compared with image edit propagation, luminance information is not available for 3D mesh data, so the color edit propagation is more difficult on 3D meshes than images, with far less research carried out. This paper proposes a novel solution based on sparse graph regularization. Firstly, a few color strokes are interactively drawn by the user, and then the color will be propagated to the whole mesh by minimizing a sparse graph regularized nonlinear energy function. The proposed method effectively measures geometric similarity over shapes by using a set of complementary multiscale feature descriptors, and effectively controls color bleeding via a sparse 1 optimization rather than quadratic minimization used in existing work. The proposed framework can be applied for the task of interactive mesh colorization, mesh color enhancement and mesh color editing. Extensive qualitative and quantitative experiments show that the proposed method outperforms the state-of-the-art methods.

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“…First, we construct a local or non‐local graph. Then, we use the framework of PDE method that defined discrete difference operators on graph [30–32]; we use the definition of the mean curvature flows on graphs and the morphological scheme, described in [29], that it can be used to perform regularisation. Then, we discuss the connection of the Eikonal equation [28] to weighted graphs.…”

Section: Proposed Algorithm For Saliency Detectionmentioning

confidence: 99%