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  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 , 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.
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.
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.
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.
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.
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.
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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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.
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.