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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Cites background from "A comparison of three total variati..."
...It is proved that the recovery is perfect, i.e., the solution uopt = ū, for any ū whenever k, m, n, and A satisfy certain conditions (e.g., see [13, 30, 37, 42, 78, 95, 96] )....
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TL;DR: In this letter, an enhanced pixel domain JND model with a new algorithm for CM estimation is proposed, and the proposed one shows its advantages brought by the better EM and TM estimation.
Abstract: In just noticeable difference (JND) models, evaluation of contrast masking (CM) is a crucial step. More specifically, CM due to edge masking (EM) and texture masking (TM) needs to be distinguished due to the entropy masking property of the human visual system. However, TM is not estimated accurately in the existing JND models since they fail to distinguish TM from EM. In this letter, we propose an enhanced pixel domain JND model with a new algorithm for CM estimation. In our model, total-variation based image decomposition is used to decompose an image into structural image (i.e., cartoon like, piecewise smooth regions with sharp edges) and textural image for estimation of EM and TM, respectively. Compared with the existing models, the proposed one shows its advantages brought by the better EM and TM estimation. It has been also applied to noise shaping and visual distortion gauge, and favorable results are demonstrated by experiments on different images.
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...In [9], different TV-based image decomposition models are considered and the model of minimizing TV with an L1-norm fidelity term is shown to achieve better results; we adopt this (TV-L1) model in our work for image decomposition, and then (1) becomes as follows:...
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...2 to 2 [8], [9] for most natural images....
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TL;DR: New fast algorithms to minimize total variation and more generally $l^1$-norms under a general convex constraint and a recent advance in convex optimization proposed by Yurii Nesterov are presented.
Abstract: This paper presents new fast algorithms to minimize total variation and more generally $l^1$-norms under a general convex constraint. Such problems are standards of image processing. The algorithms are based on a recent advance in convex optimization proposed by Yurii Nesterov. Depending on the regularity of the data fidelity term, we solve either a primal problem or a dual problem. First we show that standard first-order schemes allow one to get solutions of precision $\epsilon$ in $O(\frac{1}{\epsilon^2})$ iterations at worst. We propose a scheme that allows one to obtain a solution of precision $\epsilon$ in $O(\frac{1}{\epsilon})$ iterations for a general convex constraint. For a strongly convex constraint, we solve a dual problem with a scheme that requires $O(\frac{1}{\sqrt{\epsilon}})$ iterations to get a solution of precision $\epsilon$. Finally we perform some numerical experiments which confirm the theoretical results on various problems of image processing.
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TL;DR: This paper converts the linear model, which reduces to a low-pass/high-pass filter pair, into a nonlinear filter pair involving the total variation, which retains both the essential features of Meyer's models and the simplicity and rapidity of thelinear model.
Abstract: 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.
203 citations
TL;DR: It is shown that the images produced by this model can be formed from the minimizers of a sequence of decoupled geometry sub-problems, and that the TV-L1 model is able to separate image features according to their scales.
Abstract: This paper studies the total variation regularization with an $L^1$ fidelity term (TV‐$L^1$) model for decomposing an image into features of different scales. We first show that the images produced by this model can be formed from the minimizers of a sequence of decoupled geometry subproblems. Using this result we show that the TV‐$L^1$ model is able to separate image features according to their scales, where the scale is analytically defined by the G‐value. A number of other properties including the geometric and morphological invariance of the TV‐$L^1$ model are also proved and their applications discussed.
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...decomposition can also be used to fllter 1D signal [3], to remove impulsive (salt-npepper) noise [35], to extract textures from natural images [ 45 ], to remove varying illumination in face images for face recognition [22, 21], to decompose 2D/3D images for multiscale MR image registration [20], to assess damage from satellite imagery [19], and to remove inhomogeneous background from cDNA microarray and digital microscopic images [44]....
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References
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TL;DR: A definition of the G-norm as norm of linear functionals on BV, which seems to be more feasible for numerical computation, is used and the equivalence between Meyer’s original definition and the authors' is established and it is shown that computing the norm can be expressed as an interface problem.
Abstract: In this paper we apply Meyer's G-norm for image processing problems. We use a definition of the G-norm as norm of linear functionals on BV, which seems to be more feasible for numerical computation. We establish the equivalence between Meyer's original definition and ours and show that computing the norm can be expressed as an interface problem. This allows us to define an algorithm based on the level set method for its solution. Alternatively we propose a fixed point method based on mean curvature type equations. A computation of the G-norm according to our definition additionally gives functions which can be used for denoising of simple structures in images under a high level of noise. We present some numerical computations of this denoising method which support this claim.
20 citations
"A comparison of three total variati..." refers methods in this paper
...To calculate the G-norm of a function f alone, one can choose to solve an SOCP or use the dual method by Kindermann, Osher and Xu [22]....
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Abstract: Registration, that is, the alignment of multiple images, has been one of the most challenging problems in the field of computer vision. It also serves as an important role in biomedical image analysis and its applications. Although various methods have been proposed for solving different kinds of registration problems in computer vision, the results are still far from ideal when it comes to real world biomedical image applications. For instance, in order to register 3D brain MR images, current state of the art registration methods use a multi-resolution coarse-to-fine algorithm, which typically involves starting with low resolution images and working progressively through to higher resolutions, with the aim to avoid the local maximum "traps". However, these methods do not always successfully avoid the local maximum. Consequently, various rather sophisticated optimization methods are developed to attack this problem. In this paper, we propose a novel viewpoint on the coarse-to-fine registration, in which coarse and fine images are distinguished by different scales of the objects instead of different resolutions of the images. Based on this new perspective, we develop a new image registration framework by combining the multi-resolution method with novel multi-scale algorithm, which could achieve higher accuracy and robustness on 3D brain MR images. We believe this work has great contribution to biomedical image analysis and related applications.
19 citations
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...These properties are important in various applications in biomedical engineering and computer vision such as background correction [36], face recognition [14,15], and brain MR image registration [13]....
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TL;DR: This work partitions an image into piecewise-constant regions using energy minimization and curve evolution approaches and decomposes a natural image into a cartoon or geometric component and an oscillatory or texture component using a variational approach and dual functionals.
Abstract: This work is devoted to new computational models for image segmentation, image restoration and image decomposition. In particular, we partition an image into piecewise-constant regions using energy minimization and curve evolution approaches. Applications of denoising-segmentation in polar coordinates (motivated by impedance tomography) and of segmentation of brain images will be presented. Also, we decompose a natural image into a cartoon or geometric component and an oscillatory or texture component using a variational approach and dual functionals. Thus, new computational methods will be presented for denoising, deblurring and texture modeling.
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"A comparison of three total variati..." refers methods in this paper
...Other numerical approaches based on the dual representation of the G-norm are introduced in [16] by Chung, Le, Lieu, Tanushev, and Vese, [25] by Lieu, and [23] by Le, Lieu, and Vese....
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