Modeling Textures with Total Variation Minimization and Oscillating Patterns in Image Processing
TL;DR: This paper decomposes a given (possible textured) image f into a sum of two functions u+v, where u∈BV is a function of bounded variation (a cartoon or sketchy approximation of f), while v is afunction representing the texture or noise.
Abstract: This paper is devoted to the modeling of real textured images by functional minimization and partial differential equations. Following the ideas of Yves Meyer in a total variation minimization framework of L. Rudin, S. Osher, and E. Fatemi, we decompose a given (possible textured) image f into a sum of two functions u+v, where u∈BV is a function of bounded variation (a cartoon or sketchy approximation of f), while v is a function representing the texture or noise. To model v we use the space of oscillating functions introduced by Yves Meyer, which is in some sense the dual of the BV space. The new algorithm is very simple, making use of differential equations and is easily solved in practice. Finally, we implement the method by finite differences, and we present various numerical results on real textured images, showing the obtained decomposition u+v, but we also show how the method can be used for texture discrimination and texture segmentation.
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TL;DR: A general mathematical and experimental methodology to compare and classify classical image denoising algorithms and a nonlocal means (NL-means) algorithm addressing the preservation of structure in a digital image are defined.
Abstract: The search for efficient image denoising methods is still a valid challenge at the crossing of functional analysis and statistics In spite of the sophistication of the recently proposed methods, m
4,153 citations
Cites methods from "Modeling Textures with Total Variat..."
...Because of the different and more ambitious scopes of the Meyer method [2, 36, 26], which makes it parameter- and implementation-dependent, we could not draw it into the discussion....
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TL;DR: It is shown that various inverse problems in signal recovery can be formulated as the generic problem of minimizing the sum of two convex functions with certain regularity properties, which makes it possible to derive existence, uniqueness, characterization, and stability results in a unified and standardized fashion for a large class of apparently disparate problems.
Abstract: We show that various inverse problems in signal recovery can be formulated as the generic problem of minimizing the sum of two convex functions with certain regularity properties. This formulation makes it possible to derive existence, uniqueness, characterization, and stability results in a unified and standardized fashion for a large class of apparently disparate problems. Recent results on monotone operator splitting methods are applied to establish the convergence of a forward-backward algorithm to solve the generic problem. In turn, we recover, extend, and provide a simplified analysis for a variety of existing iterative methods. Applications to geometry/texture image decomposition schemes are also discussed. A novelty of our framework is to use extensively the notion of a proximity operator, which was introduced by Moreau in the 1960s.
2,645 citations
Cites background from "Modeling Textures with Total Variat..."
...The variational formulations proposed in [5, 6, 7, 71, 72] to achieve this decomposition based on a noisy observation z ∈ H of the signal of interest are of the general form...
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...1 will be shown to cover a wide range of apparently unrelated signal recovery formulations, including constrained least-squares problems [35, 48, 63], multiresolution sparse regularization problems [10, 30, 31, 36], Fourier regularization problems [46, 50], geometry/texture image decomposition problems [5, 6, 7, 57, 71], hard-constrained inconsistent feasibility problems [26], alternating projection signal synthesis problems [38, 60], least square-distance problems [22], split feasibility problems [13, 15], total variation problems [19, 62], as well as certain maximum a posteriori problems [68, 69]....
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TL;DR: A new iterative regularization procedure for inverse problems based on the use of Bregman distances is introduced, with particular focus on problems arising in image processing.
Abstract: We introduce a new iterative regularization procedure for inverse problems based on the use of Bregman distances, with particular focus on problems arising in image processing. We are motivated by the problem of restoring noisy and blurry images via variational methods by using total variation regularization. We obtain rigorous convergence results and effective stopping criteria for the general procedure. The numerical results for denoising appear to give significant improvement over standard models, and preliminary results for deblurring/denoising are very encouraging.
1,858 citations
TL;DR: This topic can be viewed as an extension of spectral graph theory and the diffusion geometry framework to functional analysis and PDE-like evolutions to define new types of flows and functionals for image processing and elsewhere.
Abstract: We propose the use of nonlocal operators to define new types of flows and functionals for image processing and elsewhere. A main advantage over classical PDE-based algorithms is the ability to handle better textures and repetitive structures. This topic can be viewed as an extension of spectral graph theory and the diffusion geometry framework to functional analysis and PDE-like evolutions. Some possible applications and numerical examples are given, as is a general framework for approximating Hamilton–Jacobi equations on arbitrary grids in high demensions, e.g., for control theory.
1,397 citations
Additional excerpts
...in [55, 2, 3]....
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TL;DR: A novel method for separating images into texture and piecewise smooth (cartoon) parts, exploiting both the variational and the sparsity mechanisms is presented, combining the basis pursuit denoising (BPDN) algorithm and the total-variation (TV) regularization scheme.
Abstract: The separation of image content into semantic parts plays a vital role in applications such as compression, enhancement, restoration, and more. In recent years, several pioneering works suggested such a separation be based on variational formulation and others using independent component analysis and sparsity. This paper presents a novel method for separating images into texture and piecewise smooth (cartoon) parts, exploiting both the variational and the sparsity mechanisms. The method combines the basis pursuit denoising (BPDN) algorithm and the total-variation (TV) regularization scheme. The basic idea presented in this paper is the use of two appropriate dictionaries, one for the representation of textures and the other for the natural scene parts assumed to be piecewise smooth. Both dictionaries are chosen such that they lead to sparse representations over one type of image-content (either texture or piecewise smooth). The use of the BPDN with the two amalgamed dictionaries leads to the desired separation, along with noise removal as a by-product. As the need to choose proper dictionaries is generally hard, a TV regularization is employed to better direct the separation process and reduce ringing artifacts. We present a highly efficient numerical scheme to solve the combined optimization problem posed by our model and to show several experimental results that validate the algorithm's performance.
1,032 citations
References
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TL;DR: In this article, a constrained optimization type of numerical algorithm for removing noise from images is presented, where the total variation of the image is minimized subject to constraints involving the statistics of the noise.
15,225 citations
"Modeling Textures with Total Variat..." refers background or methods in this paper
...The result u obtained with the ROF model is presented in Fig....
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...• In the standard Rudin–Osher–Fatemi model [22], the residual is given by:...
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...The proposed model combines the idea of the total variation minimization in image restoration of Rudin, Osher, and Fatemi [22] with the ideas introduced by Meyer [17] for the appropriate space to model texture or noise....
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...Following the ideas of Yves Meyer [17], we show in this paper how we can extract from f both components u and v, in a simple total variation minimization framework of Rudin, Osher, and Fatemi [22]....
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...In [17], Yves Meyer proves that the ROF model will remove the texture, if l is small enough....
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TL;DR: A new definition of scale-space is suggested, and a class of algorithms used to realize a diffusion process is introduced, chosen to vary spatially in such a way as to encourage intra Region smoothing rather than interregion smoothing.
Abstract: A new definition of scale-space is suggested, and a class of algorithms used to realize a diffusion process is introduced. The diffusion coefficient is chosen to vary spatially in such a way as to encourage intraregion smoothing rather than interregion smoothing. It is shown that the 'no new maxima should be generated at coarse scales' property of conventional scale space is preserved. As the region boundaries in the approach remain sharp, a high-quality edge detector which successfully exploits global information is obtained. Experimental results are shown on a number of images. Parallel hardware implementations are made feasible because the algorithm involves elementary, local operations replicated over the image. >
12,560 citations
TL;DR: A new model for active contours to detect objects in a given image, based on techniques of curve evolution, Mumford-Shah (1989) functional for segmentation and level sets is proposed, which can detect objects whose boundaries are not necessarily defined by the gradient.
Abstract: We propose a new model for active contours to detect objects in a given image, based on techniques of curve evolution, Mumford-Shah (1989) functional for segmentation and level sets. Our model can detect objects whose boundaries are not necessarily defined by the gradient. We minimize an energy which can be seen as a particular case of the minimal partition problem. In the level set formulation, the problem becomes a "mean-curvature flow"-like evolving the active contour, which will stop on the desired boundary. However, the stopping term does not depend on the gradient of the image, as in the classical active contour models, but is instead related to a particular segmentation of the image. We give a numerical algorithm using finite differences. Finally, we present various experimental results and in particular some examples for which the classical snakes methods based on the gradient are not applicable. Also, the initial curve can be anywhere in the image, and interior contours are automatically detected.
10,404 citations
"Modeling Textures with Total Variat..." refers methods in this paper
...Other methods for texture segmentation are using the so called ‘‘textons’’, as local averages of curvature of level lines and of the orientation of tangents to level lines (see for instance Koepfler, Lopez, and Morel [12] and Chan and Vese [9])....
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...The detected contour is obtained by applying the active contour model without edges from [8, 9] to |g1 |....
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...Here, we also show the contour between the two textures, extracted by applying the active contour model without gradient based segmentation from [8, 9] to the image |g1 |....
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...The detected contour is obtained by applying the active contour model without edges from [8, 9] to |g2 |....
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...A recent work for segmentation of textured images using segmentation based active contour models in a Gabor transform framework, is proposed by Sandberg, Chan, and Vese [23]....
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Book•
01 Jan 1992
TL;DR: In this article, the authors define and define elementary properties of BV functions, including the following: Sobolev Inequalities Compactness Capacity Quasicontinuity Precise Representations of Soboleve Functions Differentiability on Lines BV Function Differentiability and Structure Theorem Approximation and Compactness Traces Extensions Coarea Formula for BV Functions isoperimetric inequalities The Reduced Boundary The Measure Theoretic Boundary Gauss-Green Theorem Pointwise Properties this article.
Abstract: GENERAL MEASURE THEORY Measures and Measurable Functions Lusin's and Egoroff's Theorems Integrals and Limit Theorems Product Measures, Fubini's Theorem, Lebesgue Measure Covering Theorems Differentiation of Radon Measures Lebesgue Points Approximate continuity Riesz Representation Theorem Weak Convergence and Compactness for Radon Measures HAUSDORFF MEASURE Definitions and Elementary Properties Hausdorff Dimension Isodiametric Inequality Densities Hausdorff Measure and Elementary Properties of Functions AREA AND COAREA FORMULAS Lipschitz Functions, Rademacher's Theorem Linear Maps and Jacobians The Area Formula The Coarea Formula SOBOLEV FUNCTIONS Definitions And Elementary Properties Approximation Traces Extensions Sobolev Inequalities Compactness Capacity Quasicontinuity Precise Representations of Sobolev Functions Differentiability on Lines BV FUNCTIONS AND SETS OF FINITE PERIMETER Definitions and Structure Theorem Approximation and Compactness Traces Extensions Coarea Formula for BV Functions Isoperimetric Inequalities The Reduced Boundary The Measure Theoretic Boundary Gauss-Green Theorem Pointwise Properties of BV Functions Essential Variation on Lines A Criterion for Finite Perimeter DIFFERENTIABILITY AND APPROXIMATION BY C1 FUNCTIONS Lp Differentiability ae Approximate Differentiability Differentiability AE for W1,P (P > N) Convex Functions Second Derivatives ae for convex functions Whitney's Extension Theorem Approximation by C1 Functions NOTATION REFERENCES
5,769 citations
"Modeling Textures with Total Variat..." refers background in this paper
...This problem has minimizers in the space BV(R(2)) of functions of bounded variation, which is defined by [11]: u ¥ BV(R(2)) iff u ¥ L(1)(R(2)) and...
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TL;DR: In this article, the authors introduce and study the most basic properties of three new variational problems which are suggested by applications to computer vision, and study their application in computer vision.
Abstract: : This reprint will introduce and study the most basic properties of three new variational problems which are suggested by applications to computer vision. In computer vision, a fundamental problem is to appropriately decompose the domain R of a function g (x,y) of two variables. This problem starts by describing the physical situation which produces images: assume that a three-dimensional world is observed by an eye or camera from some point P and that g1(rho) represents the intensity of the light in this world approaching the point sub 1 from a direction rho. If one has a lens at P focusing this light on a retina or a film-in both cases a plane domain R in which we may introduce coordinates x, y then let g(x,y) be the strength of the light signal striking R at a point with coordinates (x,y); g(x,y) is essentially the same as sub 1 (rho) -possibly after a simple transformation given by the geometry of the imaging syste. The function g(x,y) defined on the plane domain R will be called an image. What sort of function is g? The light reflected off the surfaces Si of various solid objects O sub i visible from P will strike the domain R in various open subsets R sub i. When one object O1 is partially in front of another object O2 as seen from P, but some of object O2 appears as the background to the sides of O1, then the open sets R1 and R2 will have a common boundary (the 'edge' of object O1 in the image defined on R) and one usually expects the image g(x,y) to be discontinuous along this boundary. (JHD)
5,516 citations
"Modeling Textures with Total Variat..." refers methods in this paper
...In this category, we mention Rudin, Osher, and Fatemi [22], Mumford and Shah [18], Perona and Malik [20], Alvarez et al. [2], Chambolle and Lions [10], Aubert and Vese [5], among many others....
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...In this category, we mention Rudin, Osher, and Fatemi [22], Mumford and Shah [18], Perona and...
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