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Can images be decomposed into the sum of a geometric part and a textural part? In a theoretical breakthrough, [Y. Meyer, Oscillating Patterns in Image Processing and Nonlinear Evolution Equations. Providence, RI: American Mathematical Society, 2001] proposed variational models that force the geometric part into the space of functions with bounded variation, and the textural part into a space of oscillatory distributions. Meyer's models are simple minimization problems extending the famous total variation model. However, their numerical solution has proved challenging. It is the object of a literature rich in variants and numerical attempts. This paper starts with the linear model, which reduces to a low-pass/high-pass filter pair. A simple conversion of the linear filter pair into a nonlinear filter pair involving the total variation is introduced. This new-proposed nonlinear filter pair retains both the essential features of Meyer's models and the simplicity and rapidity of the linear model. It depends upon only one transparent parameter: the texture scale, measured in pixel mesh. Comparative experiments show a better and faster separation of cartoon from texture. One application is illustrated: edge detection.

This paper is devoted to the decomposition of an image f into u + v, with u a piecewise-smooth or "cartoon" component, and v an oscillatory component (texture or noise), in a variational approach. Meyer [Oscillating Patterns in Image Processing and Nonlinear Evolution Equations, Univ. Lecture Ser. 22, AMS, Providence, RI, 2001] proposed refinements of the total variation model (Rudin, Osher, and Fatemi [Phys. D, 60 (1992), pp. 259-268]) that better represent the oscillatory part v: the spaces of generalized functions G = div(L ∞) and F = div(BM O) (this last space arises in the study of Navier-Stokes equations; see Koch and Tataru [Adv. Math., 157 (2001), pp. 22-35]) have been proposed to model v, instead of the standard L 2 space, while keeping u a function of bounded variation. Mumford and Gidas [Quart. Appl. Math., 59 (2001), pp. 85-111] also show that natural images can be seen as samples of scale-invariant probability distributions that are supported on distributions only and not on sets of functions. However, there is no simple solution to obtain in practice such decompositions f = u + v when working with G or F. In earlier works [L.

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We propose a new model for segmenting piecewise constant images with irregular object boundaries: a variant of the Chan-Vese model [10], where the length penalization of the boundaries is replaced by the area of their neighborhood of thickness ε. Our aim is to keep fine details and irregularities of the boundaries while denoising additive Gaussian noise. For the numerical computation we revisit the classical BV level set formulation [24] considering suitable Lipschitz level set functions instead of BV ones.