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Anisotropic smoothness classes : from finite element approximation to image models

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Jean-Marie Mirebeau, Albert Cohen

Numerical Analysis · Submitted 6 Jan 2011

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim This motivates for using them in image processing as an alternative to the frequently used record variation semi-norm which does not account for the geometric smoothness of the edges.

how it is stated · asserted 0.60 read the record →

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arXiv ID1101.1321 (version 1)
Pith IDY5OS4DRR, Pith's permanent record of this paper and its review
CitationAlbert Cohen, Jean-Marie Mirebeau. "Anisotropic smoothness classes : from finite element approximation to image models." arXiv:1101.1321 (2011). https://pith.science/paper/Y5OS4DRR
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Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,319 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.88 · human participants 0.02 · controlled comparison 0.10 · randomized assignment 0.01 · states quantitative result 0.08 · compares with prior results 0.83 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

We propose and study quantitative measures of smoothness which are adapted to anisotropic features such as edges in images or shocks in PDE's. These quantities govern the rate of approximation by adaptive finite elements, when no constraint is imposed on the aspect ratio of the triangles, the simplest examples of such quantities are based on the determinant of the hessian of the function to be approximated. Since they are not semi-norms, these quantities cannot be used to define linear function spaces. We show that they can be well defined by mollification when the function to be approximated has jump discontinuities along piecewise smooth curves. This motivates for using them in image processing as an alternative to the frequently used record variation semi-norm which does not account for the geometric smoothness of the edges.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We propose and study quantitative measures of smoothness which are adapted to anisotropic features such as edges in images or shocks in PDE's. contribution 1.00 0.90 0.18 asserted 1.00
s1 These quantities govern the rate of approximation by adaptive finite elements, when no constraint is imposed on the aspect ratio of the triangles, the simplest examples of such quantities are based on the determinant of the hessian of the function to be approximated. result 0.55 0.43 0.17 asserted 1.00
s2 Since they are not semi-norms, these quantities cannot be used to define linear function spaces. limitation 0.52 0.40 0.08 asserted 1.00
s3 We show that they can be well defined by mollification when the function to be approximated has jump discontinuities along piecewise smooth curves. result 0.50 0.91 0.11 asserted 0.99
s4 This motivates for using them in image processing as an alternative to the frequently used record variation semi-norm which does not account for the geometric smoothness of the edges. interpretation 0.84 0.24 0.26 asserted 0.60

What kind of paper the classifier reports this is: theoretical result 0.61 · evidence from proof or derivation 0.98.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

We propose and study quantitative measures of smoothness which are adapted to anisotropic features such as edges in images or shocks in PDE's. These quantities govern the rate of approximation by adaptive finite elements, when no constraint is imposed on the aspect ratio of the triangles, the simplest examples of such quantities are based on the determinant of the hessian of the function to be approximated. Since they are not semi-norms, these quantities cannot be used to define linear function spaces. We show that they can be well defined by mollification when the function to be approximated has jump discontinuities along piecewise smooth curves. This motivates for using them in image processing as an alternative to the frequently used record variation semi-norm which does not account for the geometric smoothness of the edges.

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