A comparison of three total variation based texture extraction models
01 Jun 2007-Journal of Visual Communication and Image Representation (Academic Press, Inc.)-Vol. 18, Iss: 3, pp 240-252
TL;DR: This paper qualitatively compares three recently proposed models for signal/image texture extraction based on total variation minimization: the Meyer, Vese-Osher (VO), and TV-L^1[12,38,2-4,29-31] models.
About: This article is published in Journal of Visual Communication and Image Representation.The article was published on 2007-06-01 and is currently open access. It has received 68 citations till now. The article focuses on the topics: Image texture.
Summary (2 min read)
Jump to: [1 Introduction] – [1.1 The spaces BV and G] – [1.3 Second-order cone programming] – [2.2.3 The Vese-Osher (VO) model] – [3 Numerical results] – [Example 2:] and [4 Conclusion]
1 Introduction
- Let f be an observed image that contains texture and/or noise.
- Texture is characterized as repeated and meaningful structure of small patterns.
- Noise is characterized as uncorrelated random patterns.
- The rest of an image, which is called cartoon, contains object hues and sharp edges .
1.1 The spaces BV and G
- In image processing, the space BV and the total variation semi-norm were first used by Rudin, Osher, and Fatemi [33] to remove noise from images.
- The ROF model is the precursor to a large number of image processing models having a similar form.
1.3 Second-order cone programming
- Since a one-dimensional second-order cone corresponds to a semi-infinite ray, SOCPs can accommodate nonnegative variables.
- In fact if all cones are onedimensional, then the above SOCP is just a standard form linear program.
- As is the case for linear programs, SOCPs can be solved in polynomial time by interior point methods.
- This is the approach that the authors take to solve the TV-based cartoon-texture decomposition models in this paper.
2.2.3 The Vese-Osher (VO) model
- This is equivalent to solving the residual-free version (45) below.
- The authors chose to solve the latter in their numerical tests because using a large λ in (44) makes it difficult to numerically solve its SOCP accurately.
3 Numerical results
- Similar artifacts can also be found in the results Figures 2 (h )-(j) of the VO model, but the differences are that the VO model generated u's that have a block-like structure and thus v's with more complicated patterns.
- In Figure 2 (h), most of the signal in the second and third section was extracted from u, leaving very little signal near the boundary of these signal parts.
- In short, the VO model performed like an approximation of Meyer's model but with certain features closer to those of the TV-L 1 model.
Example 2:
- This fingerprint has slightly inhomogeneous brightness because the background near the center of the finger is whiter than the rest.
- The authors believe that the inhomogeneity like this is not helpful to the recognition and comparison of fingerprints so should better be corrected.
- The authors can observe in Figures 4 (a ) and (b) that their cartoon parts are close to each other, but slightly different from the cartoon in Figure 4 (c).
- The VO and the TV-L 1 models gave us more satisfactory results than Meyer's model.
- Compared to the parameters used in the three models for decomposing noiseless images in Example 3, the parameters used in the Meyer and VO models in this set of tests were changed due to the increase in the G-norm of the texture/noise part v that resulted from adding noise.
4 Conclusion
- The authors have computationally studied three total variation based models with discrete inputs: the Meyer, VO, and TV-L 1 models.
- The authors tested these models using a variety of 1D sig- nals and 2D images to reveal their differences in decomposing inputs into their cartoon and oscillating/small-scale/texture parts.
- The Meyer model tends to capture the pattern of the oscillations in the input, which makes it well-suited to applications such as fingerprint image processing.
- On the other hand, the TV-L 1 model decomposes the input into two parts according to the geometric scales of the components in the input, independent of the signal intensities, one part containing large-scale components and the other containing smallscale ones.
- These results agree with those in [9] , which compares the ROF, Meyer, and TV-L 1 models.
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Figures (6)
Fig. 5. Example 4 and 5, cartoon-texture decomposition: left halves - cartoon, right halves - texture/noise. 
Fig. 1. Example 1: 1D signal decomposition 
Table 1 Mosek termination measures 
Fig. 3. Inputs: (a) original 117×117 fingerprint, (b) original 512×512 Barbara, (c) a 256×256 part of original Barbara, (d) a 256×256 part of noisy Barbara (std.=20), (e) original 256× 256 4texture. 
Fig. 2. Example 1: 1D signal decomposition (continue) 
Fig. 4. Examples 2 and 3, cartoon-texture decomposition results: left halves - cartoon, right halves - texture.
Citations
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TL;DR: The purpose in this paper is to propose well-posed variational models in the continuous domain that can be naturally associated to exemplar-based algorithms and to investigate their ability to reconsider geometry.
Abstract: Among all methods for reconstructing missing regions in a digital image, the so-called exemplar-based algorithms are very efficient and often produce striking results. They are based on the simple idea—initially used for texture synthesis—that the unknown part of an image can be reconstructed by simply pasting samples extracted from the known part. Beyond heuristic considerations, there have been very few contributions in the literature to explain from a mathematical point of view the performances of these purely algorithmic and discrete methods. More precisely, a recent paper by Levina and Bickel [Ann. Statist., 34 (2006), pp. 1751–1773] provides a theoretical explanation of their ability to recover very well the texture, but nothing equivalent has been done so far for the recovery of geometry. Our purpose in this paper is twofold: (1) to propose well-posed variational models in the continuous domain that can be naturally associated to exemplar-based algorithms; (2) to investigate their ability to recons...
104 citations
Cites methods from "A comparison of three total variati..."
...In this paper we will use a TV-L1 model originally introduced in the context of image denoising [1, 72] and studied explicitly as a decomposition model in [11, 27, 34, 87] for instance....
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TL;DR: A novel approach based on total variation regularization and principal component pursuit (TV-PCP) is presented to deal with the detection of infrared dim targets and shows superior detection ability under various backgrounds.
94 citations
Cites background from "A comparison of three total variati..."
...It is alsowidely used in image decomposition [25], which can decompose an image into two parts: one is uncorrelated random patterns, while the other is sharp edges and piecewise-smooth components [26]....
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TL;DR: It is shown that, in the convex case, exact solutions of the TVL1 problem are given by an opening followed by a simple test over the ratio perimeter/area, and a new and efficient numerical scheme to apply the model to digital images is suggested.
Abstract: The aim of this paper is to investigate the geometrical behavior of the TVL1 model used in image processing, by making use of the notion of Cheeger sets. This mathematical concept was recently related to the celebrated Rudin–Osher–Fatemi image restoration model, yielding important advances in both fields. We provide the reader with a geometrical characterization of the TVL1 model. We show that, in the convex case, exact solutions of the TVL1 problem are given by an opening followed by a simple test over the ratio perimeter/area. Shapes remain or suddenly vanish depending on this test. As a result of our theoretical study, we suggest a new and efficient numerical scheme to apply the model to digital images. As a by-product, we justify the use of the TVL1 model for image decomposition, by establishing a connection between the model and morphological granulometry. Eventually, we propose an extension of TVL1 into an adaptive framework, in which we derive some theoretical results.
92 citations
TL;DR: A variational regularisation model endowed with a Kantorovich-Rubinstein discrepancy term and total variation regularization in the context of image denoising and cartoon-texture decomposition is discussed and connections to several other recently proposed methods are pointed out.
Abstract: We propose the use of the Kantorovich--Rubinstein norm from optimal transport in imaging problems. In particular, we discuss a variational regularization model endowed with a Kantorovich--Rubinstein discrepancy term and total variation regularization in the context of image denoising and cartoon-texture decomposition. We point out connections of this approach to several other recently proposed methods such as total generalized variation and norms capturing oscillating patterns. We also show that the respective optimization problem can be turned into a convex-concave saddle point problem with simple constraints and hence can be solved by standard tools. Numerical examples exhibit interesting features and favorable performance for denoising and cartoon-texture decomposition.
85 citations
TL;DR: In this article, a variational model for image denoising and/or texture identification is presented, which involves a L 2 -data fitting term and a Tychonov-like regularization term.
Abstract: We present a variational model for image denoising and/or texture identification. Noise and textures may be modelled as oscillating components of images. The model involves a L
2-data fitting term and a Tychonov-like regularization term. We choose the BV
2 norm instead of the classical BV norm. Here BV
2 is the bounded hessian function space that we define and describe. The main improvement is that we do not observe staircasing effects any longer, during denoising process. Moreover, texture extraction can be performed with the same method. We give existence results and present a discretized problem. An algorithm close to the one set by Chambolle (J Math Imaging Vis 20:89–97, 2004) is used: we prove convergence and present numerical tests.
80 citations
Cites background from "A comparison of three total variati..."
...5 3 The variational model We now assume that the image we want to recover from the data ud can be decomposed as f = u + v where u and v are functions that characterize different parts of f (see [6,23,26] for example)....
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...We shall assume as well that the image we want to recover from the data ud can be decomposed as f = u+ v where u and v are functions that characterize different parts of f (see [6,23,26] for example)....
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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
"A comparison of three total variati..." refers methods in this paper
...Moreover, in [19] it is shown that each interior-point iteration takes O(n(3)) time and O(n(2)logn) bytes for solving an SOCP formulation of the Rudin–Osher–Fatemi model [33]....
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...Moreover, in [19] it is shown that each interior-point iteration takes O(n3) time and O(n2logn) bytes for solving an SOCP formulation of the Rudin–Osher–Fatemi model [33]....
[...]
...In image processing, the space BV and the total variation semi-norm were first used by Rudin, Osher, and Fatemi [33] to remove noise from images....
[...]
3,377 citations
TL;DR: SOCP formulations are given for four examples: the convex quadratically constrained quadratic programming (QCQP) problem, problems involving fractional quadRatic functions, and many of the problems presented in the survey paper of Vandenberghe and Boyd as examples of SDPs can in fact be formulated as SOCPs and should be solved as such.
Abstract: Second-order cone programming (SOCP) problems are convex optimization problems in which a linear function is minimized over the intersection of an affine linear manifold with the Cartesian product of second-order (Lorentz) cones. Linear programs, convex quadratic programs and quadratically constrained convex quadratic programs can all be formulated as SOCP problems, as can many other problems that do not fall into these three categories. These latter problems model applications from a broad range of fields from engineering, control and finance to robust optimization and combinatorial optimization. On the other hand semidefinite programming (SDP)—that is the optimization problem over the intersection of an affine set and the cone of positive semidefinite matrices—includes SOCP as a special case. Therefore, SOCP falls between linear (LP) and quadratic (QP) programming and SDP. Like LP, QP and SDP problems, SOCP problems can be solved in polynomial time by interior point methods. The computational effort per iteration required by these methods to solve SOCP problems is greater than that required to solve LP and QP problems but less than that required to solve SDP’s of similar size and structure. Because the set of feasible solutions for an SOCP problem is not polyhedral as it is for LP and QP problems, it is not readily apparent how to develop a simplex or simplex-like method for SOCP. While SOCP problems can be solved as SDP problems, doing so is not advisable both on numerical grounds and computational complexity concerns. For instance, many of the problems presented in the survey paper of Vandenberghe and Boyd [VB96] as examples of SDPs can in fact be formulated as SOCPs and should be solved as such. In §2, 3 below we give SOCP formulations for four of these examples: the convex quadratically constrained quadratic programming (QCQP) problem, problems involving fractional quadratic functions ∗RUTCOR, Rutgers University, e-mail:alizadeh@rutcor.rutgers.edu. Research supported in part by the U.S. National Science Foundation grant CCR-9901991 †IEOR, Columbia University, e-mail: gold@ieor.columbia.edu. Research supported in part by the Department of Energy grant DE-FG02-92ER25126, National Science Foundation grants DMS-94-14438, CDA-97-26385 and DMS-01-04282.
1,535 citations
"A comparison of three total variati..." refers background or methods in this paper
...When 1 < p <1, we use second-order cone formulations presented in [1]....
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...With these definitions an SOCP can be written in the following form [1]:...
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01 Sep 2001
TL;DR: It turns out that this mathematics involves new properties of various Besov-type function spaces and leads to many deep results, including new generalizations of famous Gagliardo-Nirenberg and Poincare inequalities.
Abstract: From the Publisher:
"Image compression, the Navier-Stokes equations, and detection of gravitational waves are three seemingly unrelated scientific problems that, remarkably, can be studied from one perspective. The notion that unifies the three problems is that of "oscillating patterns", which are present in many natural images, help to explain nonlinear equations. and are pivotal in studying chirps and frequency-modulated signals." "In the book, the author describes both what the oscillating patterns are and the mathematics necessary for their analysis. It turns out that this mathematics involves new properties of various Besov-type function spaces and leads to many deep results, including new generalizations of famous Gagliardo-Nirenberg and Poincare inequalities." This book can be used either as a textbook in studying applications of wavelets to image processing or as a supplementary resource for studying nonlinear evolution equations or frequency-modulated signals. Most of the material in the book did not appear previously in monograph literature.
1,147 citations
"A comparison of three total variati..." refers background or methods in this paper
...G is the dual of the closed subspace BV of BV, where BV :1⁄4 fu 2 BV : jrf j 2 L(1)g [27]....
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...Meyer’s model To extract cartoon u in the space BV and texture and/or noise v as an oscillating function, Meyer [27] proposed the following model:...
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...Among the recent total variation-based cartoon-texture decomposition models, Meyer [27] and Haddad and Meyer [20] proposed using the G-norm defined above, Vese and Osher [35] approximated the G-norm by the div(L)-norm, Osher, Sole and Vese [32] proposed using the H (1)-norm, Lieu and Vese [26] proposed using the more general H -norm, and Le and Vese [24] and Garnett, Le, Meyer and Vese [18] proposed using the homogeneous Besov space _ Bp;q, 2 < s < 0, 1 6 p, q 61, extending Meyer’s _ B 1 1;1, to model the oscillation component of an image....
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...This paper qualitatively compares three recently proposed models for signal/image texture extraction based on total variation minimization: the Meyer [27], Vese–Osher (VO) [35], and TV-L(1) [12,38,2–4,29–31] models....
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...Meyer gave a few examples in [27], including the one shown at the end of next paragraph, illustrating the appropriateness of modeling oscillating patterns by functions in G....
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