306 research outputs found
Fast Image Recovery Using Variable Splitting and Constrained Optimization
We propose a new fast algorithm for solving one of the standard formulations
of image restoration and reconstruction which consists of an unconstrained
optimization problem where the objective includes an data-fidelity
term and a non-smooth regularizer. This formulation allows both wavelet-based
(with orthogonal or frame-based representations) regularization or
total-variation regularization. Our approach is based on a variable splitting
to obtain an equivalent constrained optimization formulation, which is then
addressed with an augmented Lagrangian method. The proposed algorithm is an
instance of the so-called "alternating direction method of multipliers", for
which convergence has been proved. Experiments on a set of image restoration
and reconstruction benchmark problems show that the proposed algorithm is
faster than the current state of the art methods.Comment: Submitted; 11 pages, 7 figures, 6 table
An Augmented Lagrangian Approach to the Constrained Optimization Formulation of Imaging Inverse Problems
We propose a new fast algorithm for solving one of the standard approaches to
ill-posed linear inverse problems (IPLIP), where a (possibly non-smooth)
regularizer is minimized under the constraint that the solution explains the
observations sufficiently well. Although the regularizer and constraint are
usually convex, several particular features of these problems (huge
dimensionality, non-smoothness) preclude the use of off-the-shelf optimization
tools and have stimulated a considerable amount of research. In this paper, we
propose a new efficient algorithm to handle one class of constrained problems
(often known as basis pursuit denoising) tailored to image recovery
applications. The proposed algorithm, which belongs to the family of augmented
Lagrangian methods, can be used to deal with a variety of imaging IPLIP,
including deconvolution and reconstruction from compressive observations (such
as MRI), using either total-variation or wavelet-based (or, more generally,
frame-based) regularization. The proposed algorithm is an instance of the
so-called "alternating direction method of multipliers", for which convergence
sufficient conditions are known; we show that these conditions are satisfied by
the proposed algorithm. Experiments on a set of image restoration and
reconstruction benchmark problems show that the proposed algorithm is a strong
contender for the state-of-the-art.Comment: 13 pages, 8 figure, 8 tables. Submitted to the IEEE Transactions on
Image Processin
Acceleration Methods for MRI
Acceleration methods are a critical area of research for MRI. Two of the most important acceleration techniques involve parallel imaging and compressed sensing. These advanced signal processing techniques have the potential to drastically reduce scan times and provide radiologists with new information for diagnosing disease. However, many of these new techniques require solving difficult optimization problems, which motivates the development of more advanced algorithms to solve them. In addition, acceleration methods have not reached maturity in some applications, which motivates the development of new models tailored to these applications. This dissertation makes advances in three different areas of accelerations. The first is the development of a new algorithm (called B1-Based, Adaptive Restart, Iterative Soft Thresholding Algorithm or BARISTA), that solves a parallel MRI optimization problem with compressed sensing assumptions. BARISTA is shown to be 2-3 times faster and more robust to parameter selection than current state-of-the-art variable splitting methods. The second contribution is the extension of BARISTA ideas to non-Cartesian trajectories that also leads to a 2-3 times acceleration over previous methods. The third contribution is the development of a new model for functional MRI that enables a 3-4 factor of acceleration of effective temporal resolution in functional MRI scans. Several variations of the new model are proposed, with an ROC curve analysis showing that a combination low-rank/sparsity model giving the best performance in identifying the resting-state motor network.PhDBiomedical EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/120841/1/mmuckley_1.pd
Graph Spectral Image Processing
Recent advent of graph signal processing (GSP) has spurred intensive studies
of signals that live naturally on irregular data kernels described by graphs
(e.g., social networks, wireless sensor networks). Though a digital image
contains pixels that reside on a regularly sampled 2D grid, if one can design
an appropriate underlying graph connecting pixels with weights that reflect the
image structure, then one can interpret the image (or image patch) as a signal
on a graph, and apply GSP tools for processing and analysis of the signal in
graph spectral domain. In this article, we overview recent graph spectral
techniques in GSP specifically for image / video processing. The topics covered
include image compression, image restoration, image filtering and image
segmentation
Beyond Low Rank + Sparse: Multi-scale Low Rank Matrix Decomposition
We present a natural generalization of the recent low rank + sparse matrix
decomposition and consider the decomposition of matrices into components of
multiple scales. Such decomposition is well motivated in practice as data
matrices often exhibit local correlations in multiple scales. Concretely, we
propose a multi-scale low rank modeling that represents a data matrix as a sum
of block-wise low rank matrices with increasing scales of block sizes. We then
consider the inverse problem of decomposing the data matrix into its
multi-scale low rank components and approach the problem via a convex
formulation. Theoretically, we show that under various incoherence conditions,
the convex program recovers the multi-scale low rank components \revised{either
exactly or approximately}. Practically, we provide guidance on selecting the
regularization parameters and incorporate cycle spinning to reduce blocking
artifacts. Experimentally, we show that the multi-scale low rank decomposition
provides a more intuitive decomposition than conventional low rank methods and
demonstrate its effectiveness in four applications, including illumination
normalization for face images, motion separation for surveillance videos,
multi-scale modeling of the dynamic contrast enhanced magnetic resonance
imaging and collaborative filtering exploiting age information
Space-Varying Coefficient Models for Diffusion Tensor Imaging using 3d Wavelets
In this paper, the space-varying coefficients model on the basis of B-splines (Heim et al., (2006)) is adapted to wavelet basis functions and re-examined using artificial and real data. For an introduction to diffusion tensor imaging refer to Heim et al. (2005, Chap. 2). First, wavelet theory is introduced and explained by means of 1d and 2d examples (Sections 1.1 { 1.3). Section 1.4 is dedicated to the most common thresholding techniques that serve as regularization concepts for wavelet based models. Prior to application of the 3d wavelet decomposition to the space-varying coe cient elds, the SVCM needs to be rewritten. The necessary steps are outlined in Section 2 together with the incorporation of the positive de niteness constraint using log-Cholesky parametrization. Section 3 provides a simulation study as well as a comparison with the results obtained through B-splines and standard kernel application. Finally, a real data example is presented and discussed. The theoretical parts are based on books of Gen cay et al. (2002, Chap. 1, 4-6), Härdle et al. (1998), Ogden (1997) and Jansen (2001) if not stated otherwise
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