Pré-Publication, Document De Travail Année : 2024

Image = Cartoon+Texture: How Yves Meyer's "Oscillating patterns in image processing and in some nonlinear evolution equations" ended up in a computer vision model

Résumé

While analyzing variational restoration methods in imaging with the eye of an harmonic analyst, Yves Meyer proposed a seminal reformulation of image decomposition methods, which aim at separating the "cartoon" and "texture" parts of an image. In this note, we describe Meyer's analysis of the image denoising problem. This analysis occupied an important part of his book "Oscillating patterns in image processing and in some nonlinear evolution equations". Starting from the famous Rudin, Osher and Fatemi (ROF) Total Variationbased model for natural image denoising, Meyer tries to bracket the space BV , of functions with bounded variation, and its dual between spaces that can be characterized by the decay of their wavelet coefficients. In that way, he first attempts to build a bridge between the Donoho and Johnstone wavelet shrinkage model for image denoising and the ROF model. This discussion leads him to explore the consistency of the ROF model on examples. He proves that the ROF model sometimes finds the right decomposition, namely BV + an oscillatory component, and sometimes not. This leads him to pay attention to the dual of BV and eventually to the discovery that small norms in this space characterize oscillatory signals very well. This exploration yields a reinterpretation and extension of the ROF model, with the proposal of decomposing a distribution f = u + v into two components, the first one u in BV and another one in a dual space where oscillating components have a small norm, in the spirit of interpolation theory. We illustrate the resulting cartoon+texture model and its developments with striking experiments which show its impact on medicine and society.
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Dates et versions

hal-04893423 , version 1 (20-01-2025)

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  • HAL Id : hal-04893423 , version 1

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Antonin Chambolle, Jean-Michel Morel. Image = Cartoon+Texture: How Yves Meyer's "Oscillating patterns in image processing and in some nonlinear evolution equations" ended up in a computer vision model. 2025. ⟨hal-04893423⟩
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