An image decomposition model using the total variation and the infinity Laplacian (original) (raw)
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Image Decomposition into a Bounded Variation Component and an Oscillating Component
Journal of Mathematical Imaging and Vision, 2005
We construct an algorithm to split an image into a sum u + v of a bounded variation component and a component containing the textures and the noise. This decomposition is inspired from a recent work of Y. Meyer. We find this decomposition by minimizing a convex functional which depends on the two variables u and v, alternately in each variable. Each minimization is based on a projection algorithm to minimize the total variation. We carry out the mathematical study of our method. We present some numerical results. In particular, we show how the u component can be used in nontextured SAR image restoration.
Cartoon + Texture Image Decomposition by the TV-L1 Model
Image Processing On Line, 2014
We consider the problem of decomposing an image into a cartoon part and a textural part. The geometric and smoothly-varying component, referred to as cartoon, is composed of object hues and boundaries. The texture is an oscillatory component capturing details and noise. Variational models form a general framework to obtain u + v image decompositions, where cartoon and texture are forced into different functional spaces. The TV-L1 model consists in a L 1 data fidelity term and a Total Variation (TV) regularization term. The L 1 norm is particularly well suited for the cartoon+texture decomposition since it better preserves geometric features than the L 2 norm. The TV regularization has become famous in inverse problems because it enables to recover sharp variations. However, the nondifferentiability of TV makes the underlying problems challenging to solve. There exists a wide literature of variants and numerical attempts to solve these optimization problems. In this paper, we present an implementation of a primal dual algorithm proposed by Antonin Chambolle and Thomas Pock applied to this image decomposition problem with the TV-L1 model. A thorough experimental comparison is performed with a recent filter pair proposed in IPOL for the cartoon+texture decomposition. Source Code The source code and the online demonstration are accessible at the IPOL web part of this article 1 .
Total variation based image cartoon-texture decomposition
2005
Abstract. This paper studies algorithms for decomposing a real image into the sum of cartoon and texture based on total variation minimization and second-order cone programming (SOCP). The cartoon is represented as a function of bounded variation while texture (and ...
Image decompositions using bounded variation and generalized homogeneous Besov spaces
Applied and Computational Harmonic Analysis, 2007
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. Y. Meyer [Y. Meyer, Oscillating Patterns in Image Processing and Nonlinear Evolution Equations, University Lecture Series, vol. 22, Amer. Math. Soc., Providence, RI, 2001] proposed refinements of the total variation model [L. Rudin, S. Osher, E. Fatemi, Nonlinear total variation based noise removal algorithms, Phys. D 60 (1992) 259-268] that better represent the oscillatory part v: the weaker spaces of generalized functions G = div(L ∞ ), F = div(BMO), and E =Ḃ −1 ∞,∞ have been proposed to model v, instead of the standard L 2 space, while keeping u ∈ BV, a function of bounded variation. Such new models separate better geometric structures from oscillatory structures, but it is difficult to realize them in practice. D. Mumford and B. Gidas [D. Mumford, B. Gidas, Stochastic models for generic images, Quart. Appl. Math. 59 (1) (2001) 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.
Weighted and extended total variation for image restoration and decomposition
Pattern Recognition, 2010
In various information processing tasks obtaining regularized versions of a noisy or corrupted image data is often a prerequisite for successful use of classical image analysis algorithms. Image restoration and decomposition methods need to be robust if they are to be useful in practice. In particular, this property has to be verified in engineering and scientific applications. By robustness, we mean that the performance of an algorithm should not be affected significantly by small deviations from the assumed model. In image processing, total variation (TV) is a powerful tool to increase robustness. In this paper, we define several concepts that are useful in robust restoration and robust decomposition. We propose two extended total variation models, weighted total variation (WTV) and extended total variation (ETV). We state generic approaches. The idea is to replace the TV penalty term with more general terms. The motivation is to increase the robustness of ROF (Rudin, Osher, Fatemi) model and to prevent the staircasing effect due to this method. Moreover, rewriting the non-convex sublinear regularizing terms as WTV, we provide a new approach to perform minimization via the well-known Chambolle's algorithm. The implementation is then more straightforward than the half-quadratic algorithm. The behavior of image decomposition methods is also a challenging problem, which is closely related to anisotropic diffusion. ETV leads to an anisotropic decomposition close to edges improving the robustness. It allows to respect desired geometric properties during the restoration, and to control more precisely the regularization process. We also discuss why compression algorithms can be an objective method to evaluate the image decomposition quality.
A primal-dual method for the Meyer model of cartoon and texture decomposition
Numerical Linear Algebra with Applications, 2018
In this paper, we study the original Meyer model of cartoon and texture decomposition in image processing. The model, which is a minimization problem, contains an l 1-based TV-norm and an l ∞-based G-norm. The main idea of this paper is to use the dual formulation to represent both TV-norm and G-norm. The resulting minimization problem of the Meyer model can be given as a minimax problem. A first-order primal-dual algorithm can be developed to compute the saddle point of the minimax problem. The convergence of the proposed algorithm is theoretically shown. Numerical results are presented to show that the original Meyer model can decompose better cartoon and texture components than the other testing methods.
A variational model for image fusion with simultaneous cartoon and texture decomposition
Proceedings of the 5th Eccomas Thematic Conference on Computational Vision and Medical Image Processing (VipIMAGE 2015, Tenerife, Spain, October 19-21, 2015), 2015
Image fusion is a technique that merges the information of multiple images, representing the same scene, to produce a single image that should gather the major and meaningful information contained in the different images. On the other hand, cartoon+texture image decomposition is another image processing technique, that decomposes the image into the sum of a cartoon image, containing the major geometric information, i.e., piece-wise smooth regions, and a textural image, containing the small details and oscillating patterns. We propose a model to perform the fusion of multiple images, relying on gradient information, that provides as well a cartoon and texture decomposition of the fused image. The model is formulated as a variational minimization problem and solved with the split Bregman method. The suitability of the proposed model is illustrated with some earth observation images and also medical images.
Leveling cartoons, texture energy markers, and image decomposition
8th Int. Symp. on Mathematical Morphology (ISMM '07), 2007
The variational u + v model for image decomposition aims at separating the image into a 'cartoon component' u, which consists of relatively at plateaus for the object regions surrounded by abrupt edges, and a 'texture component' v, which contains smaller-scale oscillations plus possibly noise. Exploiting this model leads to improved performance in several image analysis and computer vision problems. In this paper we propose alternative approaches for u+v decomposition based on levelings and texture energy. First, we propose an efficient method for obtaining a multiscale cartoon component using hierarchies of levelings based on Gaussian scale-space markers. We show that this corresponds to a constrained minimization driven by PDEs and link the leveling cartoons with total variation minimization. Second, we extract the texture component from levelings of the residuals between the image and its multiscale levelings. Further, we employ instantaneous nonlinear operators to estimate the spatial modulation energy in the most active texture frequency bands and use this as a new type of texture markers that yield an improved texture component from the leveling residuals. Finally, we provide experimental results that demonstrate the efficacy of the proposed image decomposition methods.
Lecture Notes in Computer Science, 2008
Dynamic, or temporal, texture is a spatially repetitive, timevarying visual pattern that forms an image sequence with a certain temporal stationarity. Important tasks are thus the detection, segmentation and perceptual characterization of Dynamic Texture (DT). Following recent work, color image decomposition appears as a good way to reach these different aims, however, to our best knowledge, no proposed model is currently able to deal with temporal aspect, inherent to color image sequences. The major contribution of this paper is to adapt static decomposition model to time aspect in order to deal with videos and color image sequences. In this paper we propose an extended decomposition model which splits a video into two components, a first one containing geometrical information, the structure of the sequence and a second one dynamic color texture and noise. Examples for color video decomposition and characterization of real dynamic present in texture component will be presented.