Augmented Lagrangian method for a mean curvature based image denoising model
Wei Zhu Xue-Cheng Tai Tony Chan
High order derivative information has been widely used in developing variational models in image processing to accomplish more advanced tasks. However, it is a nontrivial issue to construct efficient numerical algorithms to deal with the minimization of these variational models due to the associated high order Euler-Lagrange equations. In this paper, we propose an efficient numerical method for a mean curvature based image denoising model using the augmented Lagrangian method. A special technique is introduced to handle the mean curvature model for the augmented Lagrangian scheme. We detail the procedures of finding the related saddle-points of the functional. We present numerical experiments to illustrate the effectiveness and efficiency of the proposed numerical method, and show a few important features of the image denoising model such as keeping corners and image contrast. Moreover, a comparison with the gradient descent method further demonstrates the efficiency of the proposed augmented Lagrangian method.
keywords: mean curvature Augmented lagrangian method image denoising.
Wavelet inpainting by nonlocal total variation
Xiaoqun Zhang Tony F. Chan
Wavelet inpainting problem consists of filling in missed data in the wavelet domain. In [17], Chan, Shen, and Zhou proposed an efficient method to recover piecewise constant or smooth images by combining total variation regularization and wavelet representations. In this paper, we extend it to nonlocal total variation regularization in order to recover textures and local geometry structures simultaneously. Moreover, we apply an efficient algorithm framework for both local and nonlocal regularizers. Extensive experimental results on a variety of loss scenarios and natural images validate the performance of this approach.
keywords: nonlocal total variation. wavelet inpainting Inverse problems
Detection of shape deformities using Yamabe flow and Beltrami coefficients
Lok Ming Lui Tsz Wai Wong Wei Zeng Xianfeng Gu Paul M. Thompson Tony F. Chan Shing Tung Yau
We address the problem of detecting deformities on elastic surfaces. This is of great importance for shape analysis, with applications such as detecting abnormalities in biological shapes (e.g., brain structures). We propose an effective algorithm to detect abnormal deformations by generating quasi-conformal maps between the original and deformed surfaces. We firstly flatten the 3D surfaces conformally onto 2D rectangles using the discrete Yamabe flow and use them to compute a quasi-conformal map that matches internal features lying within the surfaces. The deformities on the elastic surface are formulated as non-conformal deformations, whereas normal deformations that preserve local geometry are formulated as conformal deformations. We then detect abnormalities by computing the Beltrami coefficient associated uniquely with the quasi-conformal map. The Beltrami coefficient is a complex-valued function defined on the surface. It describes the deviation of the deformation from conformality at each point. By considering the norm of the Beltrami coefficient, we can effectively segment the regions of abnormal changes, which are invariant under normal (non-rigid) deformations that preserve local geometry. Furthermore, by considering the argument of the Beltrami coefficient, we can capture abnormalities induced by local rotational changes. We tested the algorithm by detecting abnormalities on synthetic surfaces, 3D human face data and MRI-derived brain surfaces. Experimental results show that our algorithm can effectively detect abnormalities and capture local rotational alterations. Our method is also more effective than other existing methods, such as the isometric indicator, for locating abnormalities.
keywords: Conformal map Shape Analysis. Beltrami coefficient Surface registration Quasi-conformal map
Tony F. Chan Yunmei Chen Nikos Paragios
Life expectancy in the developed and developing countries is constantly increasing. Medicine has benefited from novel biomarkers for screening and diagnosis. At least for a number of diseases, biomedical imaging is one of the most promising means of early diagnosis. Medical hardware manufacturer's progress has led to a new generation of measurements to understand the human anatomical and functional states. These measurements go beyond simple means of anatomical visualization (e.g. X-ray images) and therefore their interpretation becomes a scientific challenge for humans mostly because of the volume and flow of information as well as their nature. Computer-aided diagnosis develops mathematical models and their computational solutions to assist data interpretation in a clinical setting. In simple words, one would like to be able to provide a formal answer to a clinical question using the available measurements. The development of mathematical models for automatic clinical interpretation of multi-modalities is a great challenge.

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Brain anatomical feature detection by solving partial differential equations on general manifolds
Lok Ming Lui Yalin Wang Tony F. Chan Paul M. Thompson
One important problem in human brain mapping research is to locate the important anatomical features. Anatomical features on the cortical surface are usually represented by landmark curves, called sulci/gyri curves. These landmark curves are important information for neuroscientists to study brain disease and to match different cortical surfaces. Manual labelling of these landmark curves is time-consuming, especially when large sets of data have to be analyzed. In this paper, we present algorithms to automatically detect and match landmark curves on cortical surfaces to get an optimized brain conformal parametrization. First, we propose an algorithm to obtain a hypothesized landmark region/curves using the Chan-Vese segmentation method, which solves a Partial Differential Equation (PDE) on a manifold with global conformal parameterization. This is done by segmentating the high mean curvature region. Second, we propose an automatic landmark curve tracing method based on the principal directions of the local Weingarten matrix. Based on the global conformal parametrization of a cortical surface, our method adjusts the landmark curves iteratively on the spherical or rectangular parameter domain of the cortical surface along its principal direction field, using umbilic points of the surface as anchors. The landmark curves can then be mapped back onto the cortical surface. Experimental results show that the landmark curves detected by our algorithm closely resemble these manually labeled curves. Next, we applied these automatically labeled landmark curves to generate an optimized conformal parametrization of the cortical surface, in the sense that homologous features across subjects are caused to lie at the same parameter locations in a conformal grid. Experimental results show that our method can effectively help in automatically matching cortical surfaces across subjects.
keywords: Conformal Parametrization Partial Differential Equations Brain Anatomical Feature Curvature. Variational problem
Fast dual minimization of the vectorial total variation norm and applications to color image processing
Xavier Bresson Tony F. Chan
We propose a regularization algorithm for color/vectorial images which is fast, easy to code and mathematically well-posed. More precisely, the regularization model is based on the dual formulation of the vectorial Total Variation (VTV) norm and it may be regarded as the vectorial extension of the dual approach defined by Chambolle in [13] for gray-scale/scalar images. The proposed model offers several advantages. First, it minimizes the exact VTV norm whereas standard approaches use a regularized norm. Then, the numerical scheme of minimization is straightforward to implement and finally, the number of iterations to reach the solution is low, which gives a fast regularization algorithm. Finally, and maybe more importantly, the proposed VTV minimization scheme can be easily extended to many standard applications. We apply this $L^1$ vectorial regularization algorithm to the following problems: color inverse scale space, color denoising with the chromaticity-brightness color representation, color image inpainting, color wavelet shrinkage, color image decomposition, color image deblurring, and color denoising on manifolds. Generally speaking, this VTV minimization scheme can be used in problems that required vector field (color, other feature vector) regularization while preserving discontinuities.
keywords: image inpainting image decomposition dual formulation ROF model BV space image deblurring wavelet shrinkage Vector-valued TV norm inverse scale space chromaticity-brightness color representation image denoising denoising on manifold
A multiphase logic framework for multichannel image segmentation
Matthew S. Keegan Berta Sandberg Tony F. Chan
We propose a novel framework for energy-based multiphase segmentation over multiple channels. The framework allows the user to combine the information from each channel as the user sees fit, and thus allows the user to define how the information from each channel should influence the result. The framework extends the two-phase Logic Framework [J. Vis. Commun. Image R. 16 (2005) 333-358] model. The logic operators of the Logic Framework are used to define objective functions for multiple phases and a condition is defined that prevents conflict between energy terms. This condition prevents local minima that may occur using ad hoc methods, such as summing the objective functions of each region.
keywords: Image segmentation.
Four color theorem and convex relaxation for image segmentation with any number of regions
Ruiliang Zhang Xavier Bresson Tony F. Chan Xue-Cheng Tai
Image segmentation is an essential problem in imaging science. One of the most successful segmentation models is the piecewise constant Mumford-Shah minimization model. This minimization problem is however difficult to carry out, mainly due to the non-convexity of the energy. Recent advances based on convex relaxation methods are capable of estimating almost perfectly the geometry of the regions to be segmented when the mean intensity and the number of segmented regions are known a priori. The next important challenge is to provide a tight approximation of the optimal geometry, mean intensity and the number of regions simultaneously while keeping the computational time and memory usage reasonable. In this work, we propose a new algorithm that combines convex relaxation methods with the four color theorem to deal with the unsupervised segmentation problem. More precisely, the proposed algorithm can segment any a priori unknown number of regions with only four intensity functions and four indicator (``labeling") functions. The number of regions in our segmentation model is decided by one parameter that controls the regularization strength of the geometry, i.e., the total length of the boundary of all the regions. The segmented image function can take as many constant values as needed.
keywords: four color theorem Mumford-Shah model Unsupervised segmentation convex relaxation method.

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