Edge adaptive methods and machine learning for high-resolution image reconstruction
Agustin Somacal  1@  , Albert Cohen  2  , Olga Mula  3  , Matthieu Dolbeault  1  
1 : Laboratoire Jacques-Louis Lions
Sorbonne Université, Centre National de la Recherche Scientifique : UMR_7598, Université de Paris : UMR_7598
2 : LJLL
Sorbonne Université UPMC Paris VI
3 : Paris-Dauphine University, PSL Research University, CNRS, UMR 7534, CEREMADE, France.
Paris-Dauphine University

Edge-adapted methods have been introduced in the context of image processing [1][2] to reconstruct highresolution images from coarser cell averages. In particular, when images consist of piece-wise smooth functions, the interfaces can be approximated by a pre-specified functional class (lines, circle arcs, etc) through optimization (LVIRA [2]) or specific preprocessing (ENO-EA [1]). In this work, we extend the ENO-EA approach to polynomials of degree higher than 1 and compare this algebraic approach to that introduced in [2] as well as to learning-based methods [3] in which an artificial neural network (NN) (or in principle any other non linear sufficiently rich function family) is used to attain the same goal.

[1] F. Arandiga, A. Cohen, R. Donat, N. Dyn, B. Matei. Approximation of piecewise smooth func-tions and images by edge-adapted (ENO-EA) nonlinear multiresolution techniques. Applied andComputational Harmonic Analysis,24(2), 225–250, 2008. doi :10.1016/j.acha.2007.06.009.
[2] J. E. Pilliod, E. G. Puckett.Second-order accurate volume-of-fluid algorithms for tra-cking material interfaces. Journal of Computational Physics,199(2), 465–502, 2004. doi :10.1016/j.jcp.2003.12.023.
[3] B. Després, H. Jourdren.Machine Learning design of Volume of Fluid schemes for compressibleflows. Journal of Computational Physics,408, 2020. doi :10.1016/j.jcp.2020.109275.

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