1,703 research outputs found
Enhancing Compressed Sensing 4D Photoacoustic Tomography by Simultaneous Motion Estimation
A crucial limitation of current high-resolution 3D photoacoustic tomography
(PAT) devices that employ sequential scanning is their long acquisition time.
In previous work, we demonstrated how to use compressed sensing techniques to
improve upon this: images with good spatial resolution and contrast can be
obtained from suitably sub-sampled PAT data acquired by novel acoustic scanning
systems if sparsity-constrained image reconstruction techniques such as total
variation regularization are used. Now, we show how a further increase of image
quality can be achieved for imaging dynamic processes in living tissue (4D
PAT). The key idea is to exploit the additional temporal redundancy of the data
by coupling the previously used spatial image reconstruction models with
sparsity-constrained motion estimation models. While simulated data from a
two-dimensional numerical phantom will be used to illustrate the main
properties of this recently developed
joint-image-reconstruction-and-motion-estimation framework, measured data from
a dynamic experimental phantom will also be used to demonstrate their potential
for challenging, large-scale, real-world, three-dimensional scenarios. The
latter only becomes feasible if a carefully designed combination of tailored
optimization schemes is employed, which we describe and examine in more detail
An L1 Penalty Method for General Obstacle Problems
We construct an efficient numerical scheme for solving obstacle problems in
divergence form. The numerical method is based on a reformulation of the
obstacle in terms of an L1-like penalty on the variational problem. The
reformulation is an exact regularizer in the sense that for large (but finite)
penalty parameter, we recover the exact solution. Our formulation is applied to
classical elliptic obstacle problems as well as some related free boundary
problems, for example the two-phase membrane problem and the Hele-Shaw model.
One advantage of the proposed method is that the free boundary inherent in the
obstacle problem arises naturally in our energy minimization without any need
for problem specific or complicated discretization. In addition, our scheme
also works for nonlinear variational inequalities arising from convex
minimization problems.Comment: 20 pages, 18 figure
Vector Approximate Message Passing for the Generalized Linear Model
The generalized linear model (GLM), where a random vector is
observed through a noisy, possibly nonlinear, function of a linear transform
output , arises in a range of applications such
as robust regression, binary classification, quantized compressed sensing,
phase retrieval, photon-limited imaging, and inference from neural spike
trains. When is large and i.i.d. Gaussian, the generalized
approximate message passing (GAMP) algorithm is an efficient means of MAP or
marginal inference, and its performance can be rigorously characterized by a
scalar state evolution. For general , though, GAMP can
misbehave. Damping and sequential-updating help to robustify GAMP, but their
effects are limited. Recently, a "vector AMP" (VAMP) algorithm was proposed for
additive white Gaussian noise channels. VAMP extends AMP's guarantees from
i.i.d. Gaussian to the larger class of rotationally invariant
. In this paper, we show how VAMP can be extended to the GLM.
Numerical experiments show that the proposed GLM-VAMP is much more robust to
ill-conditioning in than damped GAMP
Inference for Generalized Linear Models via Alternating Directions and Bethe Free Energy Minimization
Generalized Linear Models (GLMs), where a random vector is
observed through a noisy, possibly nonlinear, function of a linear transform
arise in a range of applications in nonlinear
filtering and regression. Approximate Message Passing (AMP) methods, based on
loopy belief propagation, are a promising class of approaches for approximate
inference in these models. AMP methods are computationally simple, general, and
admit precise analyses with testable conditions for optimality for large i.i.d.
transforms . However, the algorithms can easily diverge for general
. This paper presents a convergent approach to the generalized AMP
(GAMP) algorithm based on direct minimization of a large-system limit
approximation of the Bethe Free Energy (LSL-BFE). The proposed method uses a
double-loop procedure, where the outer loop successively linearizes the LSL-BFE
and the inner loop minimizes the linearized LSL-BFE using the Alternating
Direction Method of Multipliers (ADMM). The proposed method, called ADMM-GAMP,
is similar in structure to the original GAMP method, but with an additional
least-squares minimization. It is shown that for strictly convex, smooth
penalties, ADMM-GAMP is guaranteed to converge to a local minima of the
LSL-BFE, thus providing a convergent alternative to GAMP that is stable under
arbitrary transforms. Simulations are also presented that demonstrate the
robustness of the method for non-convex penalties as well
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