2015 # On the $p$-Laplacian and $\infty$-Laplacian on Graphs with Applications in Image and Data Processing

**Abstract:** International audienceIn this paper we introduce a new family of partial difference operators on graphs and study equations involving these operators. This family covers local variational $p$-Laplacian, $\infty$-Laplacian, nonlocal $p$-Laplacian and $\infty$-Laplacian, $p$-Laplacian with gradient terms, and gradient operators used in morphology based on the partial differential equation. We analyze a corresponding parabolic equation involving these operators which enables us to interpolate adaptively between $…

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“…In this section, we derive the determining equations of the generators of the symmetry group for both the ∞-Polylaplacian (6) and its reduction (8).…”

confidence: 99%

“…In this section, we derive the determining equations of the generators of the symmetry group for both the ∞-Polylaplacian (6) and its reduction (8).…”

confidence: 99%

“…In what follows, we will see that this is not the case. In this way, we obtain the Lie algebra for the symmetry generators for both Equations (6) and (8) and thus, using the Lie series, derive the full groups of Lie point symmetries for both equations. At the end of this section, we discuss about the discrete symmetries of Equations (6) and (8).…”

confidence: 99%

“…PDEs of the Laplacian or p-Laplacian type have been used successfully in many applications, such as image denoising, restoration, segmentation and inpainting; see [1][2][3][4] and the references therein. PDEs of the Hamilton-Jacobi type play a central role in continuous mathematical morphology.…”

confidence: 99%