117 research outputs found
Optimization Methods for Inverse Problems
Optimization plays an important role in solving many inverse problems.
Indeed, the task of inversion often either involves or is fully cast as a
solution of an optimization problem. In this light, the mere non-linear,
non-convex, and large-scale nature of many of these inversions gives rise to
some very challenging optimization problems. The inverse problem community has
long been developing various techniques for solving such optimization tasks.
However, other, seemingly disjoint communities, such as that of machine
learning, have developed, almost in parallel, interesting alternative methods
which might have stayed under the radar of the inverse problem community. In
this survey, we aim to change that. In doing so, we first discuss current
state-of-the-art optimization methods widely used in inverse problems. We then
survey recent related advances in addressing similar challenges in problems
faced by the machine learning community, and discuss their potential advantages
for solving inverse problems. By highlighting the similarities among the
optimization challenges faced by the inverse problem and the machine learning
communities, we hope that this survey can serve as a bridge in bringing
together these two communities and encourage cross fertilization of ideas.Comment: 13 page
Parametric Level Set Methods for Inverse Problems
In this paper, a parametric level set method for reconstruction of obstacles
in general inverse problems is considered. General evolution equations for the
reconstruction of unknown obstacles are derived in terms of the underlying
level set parameters. We show that using the appropriate form of parameterizing
the level set function results a significantly lower dimensional problem, which
bypasses many difficulties with traditional level set methods, such as
regularization, re-initialization and use of signed distance function.
Moreover, we show that from a computational point of view, low order
representation of the problem paves the path for easier use of Newton and
quasi-Newton methods. Specifically for the purposes of this paper, we
parameterize the level set function in terms of adaptive compactly supported
radial basis functions, which used in the proposed manner provides flexibility
in presenting a larger class of shapes with fewer terms. Also they provide a
"narrow-banding" advantage which can further reduce the number of active
unknowns at each step of the evolution. The performance of the proposed
approach is examined in three examples of inverse problems, i.e., electrical
resistance tomography, X-ray computed tomography and diffuse optical
tomography
High-order regularized regression in Electrical Impedance Tomography
We present a novel approach for the inverse problem in electrical impedance
tomography based on regularized quadratic regression. Our contribution
introduces a new formulation for the forward model in the form of a nonlinear
integral transform, that maps changes in the electrical properties of a domain
to their respective variations in boundary data. Using perturbation theory the
transform is approximated to yield a high-order misfit unction which is then
used to derive a regularized inverse problem. In particular, we consider the
nonlinear problem to second-order accuracy, hence our approximation method
improves upon the local linearization of the forward mapping. The inverse
problem is approached using Newton's iterative algorithm and results from
simulated experiments are presented. With a moderate increase in computational
complexity, the method yields superior results compared to those of regularized
linear regression and can be implemented to address the nonlinear inverse
problem
Solution of the inverse scattering problem by T-matrix completion. II. Simulations
This is Part II of the paper series on data-compatible T-matrix completion
(DCTMC), which is a method for solving nonlinear inverse problems. Part I of
the series contains theory and here we present simulations for inverse
scattering of scalar waves. The underlying mathematical model is the scalar
wave equation and the object function that is reconstructed is the medium
susceptibility. The simulations are relevant to ultrasound tomographic imaging
and seismic tomography. It is shown that DCTMC is a viable method for solving
strongly nonlinear inverse problems with large data sets. It provides not only
the overall shape of the object but the quantitative contrast, which can
correspond, for instance, to the variable speed of sound in the imaged medium.Comment: This is Part II of a paper series. Part I contains theory and is
available at arXiv:1401.3319 [math-ph]. Accepted in this form to Phys. Rev.
EIT Reconstruction Algorithms: Pitfalls, Challenges and Recent Developments
We review developments, issues and challenges in Electrical Impedance
Tomography (EIT), for the 4th Workshop on Biomedical Applications of EIT,
Manchester 2003. We focus on the necessity for three dimensional data
collection and reconstruction, efficient solution of the forward problem and
present and future reconstruction algorithms. We also suggest common pitfalls
or ``inverse crimes'' to avoid.Comment: A review paper for the 4th Workshop on Biomedical Applications of
EIT, Manchester, UK, 200
Parametric Level-sets Enhanced To Improve Reconstruction (PaLEnTIR)
In this paper, we consider the restoration and reconstruction of piecewise
constant objects in two and three dimensions using PaLEnTIR, a significantly
enhanced Parametric level set (PaLS) model relative to the current
state-of-the-art. The primary contribution of this paper is a new PaLS
formulation which requires only a single level set function to recover a scene
with piecewise constant objects possessing multiple unknown contrasts. Our
model offers distinct advantages over current approaches to the multi-contrast,
multi-object problem, all of which require multiple level sets and explicit
estimation of the contrast magnitudes. Given upper and lower bounds on the
contrast, our approach is able to recover objects with any distribution of
contrasts and eliminates the need to know either the number of contrasts in a
given scene or their values. We provide an iterative process for finding these
space-varying contrast limits. Relative to most PaLS methods which employ
radial basis functions (RBFs), our model makes use of non-isotropic basis
functions, thereby expanding the class of shapes that a PaLS model of a given
complexity can approximate. Finally, PaLEnTIR improves the conditioning of the
Jacobian matrix required as part of the parameter identification process and
consequently accelerates the optimization methods by controlling the magnitude
of the PaLS expansion coefficients, fixing the centers of the basis functions,
and the uniqueness of parametric to image mappings provided by the new
parameterization. We demonstrate the performance of the new approach using both
2D and 3D variants of X-ray computed tomography, diffuse optical tomography
(DOT), denoising, deconvolution problems. Application to experimental sparse CT
data and simulated data with different types of noise are performed to further
validate the proposed method.Comment: 31 pages, 56 figure
Regularization methods for diffuse optical tomography
Near-infrared light can be used as a three dimensional imaging tool, called diffuse optical tomography (DOT), in the study of human physiology. Due to differences in the extinction coefficients of oxygenated and deoxygenated haemoglobin at different wavelengths, concentrations of the haemoglobins can be resolved from measurements at a few wavelengths. Therefore, DOT is a fascinating modality for biomedical applications, such as functional brain imaging, breast cancer screening, etc. Moreover light is a safe tool, because it is non-ionizing and at intensity levels used in DOT, it does not cause burns at skin or in organs.
There are a few different models to describe light propagation in tissue-like media. One of the simplest, called the diffusion approximation (DA), was used in this thesis. The optical properties, the absorption and the scattering coefficients, are the parameters which determine the light propagation in the DA model. When optical properties are known and one is interested in estimating the photon flux at the boundary, the problem is called a forward problem. Likewise, when the photon flux at the boundary is measured and the task is to find the optical properties, the problem is called an inverse problem. The inverse problem related to DOT is ill-posed, i.e., the solution might not be unique or the solution does not depend continuously on data.
Due to ill-posedness of the inverse problem, some regularization methods should be used. In this thesis, regularization methods for a stationary and nonstationary inverse problems was considered. By the nonstationary inverse problem, it is meant that the optical properties are not static during the measurement and the whole evolution of the optical properties is reconstructed in contrast to the stationary problem, where the optical properties are assumed to be static during the measurement.
The regularization for the inverse problem could be implemented as the Tikhonov regularization or using statistical inversion theory, also known as the Bayesian framework. In this thesis, two different regularization methods for the static reconstruction problem in DOT were studied. They both allow discontinuities in the optical properties that might occur at boundaries between organs. For the nonstationary reconstruction problem, an efficient regularization model is presented
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