Accelerated bregman operator splitting with backtracking
Yunmei Chen Xianqi Li Yuyuan Ouyang Eduardo Pasiliao
Inverse Problems & Imaging 2017, 11(6): 1047-1070 doi: 10.3934/ipi.2017048

This paper develops two accelerated Bregman Operator Splitting (BOS) algorithms with backtracking for solving regularized large-scale linear inverse problems, where the regularization term may not be smooth. The first algorithm improves the rate of convergence for BOSVS [5] in terms of the smooth component in the objective function by incorporating Nesterov's multi-step acceleration scheme under the assumption that the feasible set is bounded. The second algorithm is capable of dealing with the case where the feasible set is unbounded. Moreover, it allows more aggressive stepsize than that in the first scheme by properly selecting the penalty parameter and jointly updating the acceleration parameter and stepsize. Both algorithms exhibit better practical performance than BOSVS and AADMM [21], while preserve the same accelerated rate of convergence as that for AADMM. The numerical results on total-variation based image reconstruction problems indicate the effectiveness of the proposed algorithms.

keywords: Bregman operator splitting accelerated ADMM convex optimization Barzilai-Borwein stepsize backtracking total variation image reconstruction
Variational denoising of diffusion weighted MRI
Tim McGraw Baba Vemuri Evren Özarslan Yunmei Chen Thomas Mareci
Inverse Problems & Imaging 2009, 3(4): 625-648 doi: 10.3934/ipi.2009.3.625
In this paper, we present a novel variational formulation for restoring high angular resolution diffusion imaging (HARDI) data. The restoration formulation involves smoothing signal measurements over the spherical domain and across the 3D image lattice. The regularization across the lattice is achieved using a total variation (TV) norm based scheme, while the finite element method (FEM) was employed to smooth the data on the sphere at each lattice point using first and second order smoothness constraints. Examples are presented to show the performance of the HARDI data restoration scheme and its effect on fiber direction computation on synthetic data, as well as on real data sets collected from excised rat brain and spinal cord.
keywords: denoising. Diffusion MRI
Deformable multi-modal image registration by maximizing Rényi's statistical dependence measure
Yunmei Chen Jiangli Shi Murali Rao Jin-Seop Lee
Inverse Problems & Imaging 2015, 9(1): 79-103 doi: 10.3934/ipi.2015.9.79
A novel variational model for deformable multi-modal image registration is presented in this work. As an alternative to the models based on maximizing mutual information, the Rényi's statistical dependence measure of two random variables is proposed as a measure of the goodness of matching in our objective functional. The proposed model does not require an estimation of the continuous joint probability density function. Instead, it only needs observed independent instances. Moreover, the theory of reproducing kernel Hilbert space is used to simplify the computation. Experimental results and comparisons with several existing methods are provided to show the effectiveness of the model.
keywords: mutual information. statistical dependence image registration deformable Multi-modal
A fractional-order derivative based variational framework for image denoising
Fangfang Dong Yunmei Chen
Inverse Problems & Imaging 2016, 10(1): 27-50 doi: 10.3934/ipi.2016.10.27
In this paper, we propose a unified variational framework for noise removal, which uses a combination of different orders of fractional derivatives in the regularization term of the objective function. The principle of the combination is taking the order two or higher derivatives for smoothing the homogeneous regions, and a fractional order less than or equal to one to smooth the locations near the edges. We also introduce a novel edge detector to better detect edges and textures. A main advantage of this framework is the superiority in dealing with textures and repetitive structures as well as eliminating the staircase effect. To effectively solve the proposed model, we extend the first-order primal dual algorithm to minimize a functional involving fractional-order derivatives. A set of experiments demonstrates that the proposed method is able to avoid the staircase effect and preserve accurately edges and structural details of the image while removing the noise.
keywords: fractional-order derivative first-order primal dual algorithm. Image denoising
A nonstandard smoothing in reconstruction of apparent diffusion coefficient profiles from diffusion weighted images
Yunmei Chen Weihong Guo Qingguo Zeng Yijun Liu
Inverse Problems & Imaging 2008, 2(2): 205-224 doi: 10.3934/ipi.2008.2.205
We present a new variational framework for simultaneous smoothing and estimation of apparent diffusion coefficient (ADC) profiles from High Angular Resolution Diffusion-weighted MRI. The model approximates the ADC profiles at each voxel by a 4th order spherical harmonic series (SHS). The coefficients in SHS are obtained by solving a constrained minimization problem. The smoothing with feature preserved is achieved by minimizing a variable exponent, linear growth functional, and the data constraint is determined by the original Stejskal-Tanner equation. The antipodal symmetry and positiveness of the ADC are accommodated in the model. We use these coefficients and variance of the ADC profiles from its mean to classify the diffusion in each voxel as isotropic, anisotropic with single fiber orientation, or two fiber orientations. The proposed model has been applied to both simulated data and HARD MRI human brain data . The experiments demonstrated the effectiveness of our method in estimation and smoothing of ADC profiles and in enhancement of diffusion anisotropy. Further characterization of non-Gaussian diffusion based on the proposed model showed a consistency between our results and known neuroanatomy.
keywords: Reconstruction smoothing spherical harmonic series. diffusion weighted images
A novel method and fast algorithm for MR image reconstruction with significantly under-sampled data
Yunmei Chen Xiaojing Ye Feng Huang
Inverse Problems & Imaging 2010, 4(2): 223-240 doi: 10.3934/ipi.2010.4.223
The aim of this work is to improve the accuracy, robustness and efficiency of the compressed sensing reconstruction technique in magnetic resonance imaging. We propose a novel variational model that enforces the sparsity of the underlying image in terms of its spatial finite differences and representation with respect to a dictionary. The dictionary is trained using prior information to improve accuracy in reconstruction. In the meantime the proposed model enforces the consistency of the underlying image with acquired data by using the maximum likelihood estimator of the reconstruction error in partial $k$-space to improve the robustness to parameter selection. Moreover, a simple and fast numerical scheme is provided to solve this model. The experimental results on both synthetic and in vivo data indicate the improvement of the proposed model in preservation of fine structures, flexibility of parameter decision, and reduction of computational cost.
keywords: dictionary compressed sensing convex optimization. sparse representation
Tony F. Chan Yunmei Chen Nikos Paragios
Inverse Problems & Imaging 2010, 4(2): i-iii doi: 10.3934/ipi.2010.4.2i
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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Total variation and wavelet regularization of orientation distribution functions in diffusion MRI
Yuyuan Ouyang Yunmei Chen Ying Wu
Inverse Problems & Imaging 2013, 7(2): 565-583 doi: 10.3934/ipi.2013.7.565
We introduce a variational model and a numerical method for simultaneous ODF smoothing and reconstruction. The model uses the sparsity of MR images in finite difference domain and wavelet domain as the spatial regularization means in ODF's reconstruction. The model also incorporates angular regularization using Laplace-Beltrami operator on the unit sphere. A primal-dual scheme is applied to solve the model efficiently. The experimental results indicate that with spatial and angular regularization in the process of reconstruction, we can get better directional structures of reconstructed ODFs.
keywords: orientation distribution function primal dual hybrid gradient method. Diffusion magnetic resonance image
A fast algorithm for global minimization of maximum likelihood based on ultrasound image segmentation
Jie Huang Xiaoping Yang Yunmei Chen
Inverse Problems & Imaging 2011, 5(3): 645-657 doi: 10.3934/ipi.2011.5.645
This paper presents a novel variational model for ultrasound image segmentation that uses a maximum likelihood estimator based on Fisher-Tippett distribution of the intensities of ultrasound images. A convex relaxation method is applied to get a convex model of the subproblem with fixed distribution parameters. The relaxed subproblem, which is convex, can be fast solved by using a primal-dual hybrid gradient algorithm. The experimental results on simulated and real ultrasound images indicate the effectiveness of the method presented.
keywords: primal dual method. Ultrasound image segmentation global minimization

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