32,541 research outputs found
Task adapted reconstruction for inverse problems
The paper considers the problem of performing a task defined on a model
parameter that is only observed indirectly through noisy data in an ill-posed
inverse problem. A key aspect is to formalize the steps of reconstruction and
task as appropriate estimators (non-randomized decision rules) in statistical
estimation problems. The implementation makes use of (deep) neural networks to
provide a differentiable parametrization of the family of estimators for both
steps. These networks are combined and jointly trained against suitable
supervised training data in order to minimize a joint differentiable loss
function, resulting in an end-to-end task adapted reconstruction method. The
suggested framework is generic, yet adaptable, with a plug-and-play structure
for adjusting both the inverse problem and the task at hand. More precisely,
the data model (forward operator and statistical model of the noise) associated
with the inverse problem is exchangeable, e.g., by using neural network
architecture given by a learned iterative method. Furthermore, any task that is
encodable as a trainable neural network can be used. The approach is
demonstrated on joint tomographic image reconstruction, classification and
joint tomographic image reconstruction segmentation
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
Comparing D-Bar and Common Regularization-Based Methods for Electrical Impedance Tomography
Objective: To compare D-bar difference reconstruction with regularized linear reconstruction in electrical impedance tomography. Approach: A standard regularized linear approach using a Laplacian penalty and the GREIT method for comparison to the D-bar difference images. Simulated data was generated using a circular phantom with small objects, as well as a \u27Pac-Man\u27 shaped conductivity target. An L-curve method was used for parameter selection in both D-bar and the regularized methods. Main results: We found that the D-bar method had a more position independent point spread function, was less sensitive to errors in electrode position and behaved differently with respect to additive noise than the regularized methods. Significance: The results allow a novel pathway between traditional and D-bar algorithm comparison
Inversion of band-limited discrete Fourier transforms of binary images: Uniqueness and algorithms
Inversion of the two-dimensional discrete Fourier transform (DFT) typically
requires all DFT coefficients to be known. When only band-limited DFT
coefficients of a matrix are known, the original matrix can not be uniquely
recovered. Using a priori information that the matrix is binary (all elements
are either 0 or 1) can overcome the missing high-frequency DFT coefficients and
restore uniqueness. We theoretically investigate the smallest pass band that
can be applied while still guaranteeing unique recovery of an arbitrary binary
matrix. The results depend on the dimensions of the matrix. Uniqueness results
are proven for the dimensions , , and , where are primes numbers and an integer. An
inversion algorithm is proposed for practically recovering the unique binary
matrix. This algorithm is based on integer linear programming methods and
significantly outperforms naive implementations. The algorithm efficiently
reconstructs binary matrices using 81 out of the total 289 DFT
coefficients.Comment: 12 page
Geometric deep learning: going beyond Euclidean data
Many scientific fields study data with an underlying structure that is a
non-Euclidean space. Some examples include social networks in computational
social sciences, sensor networks in communications, functional networks in
brain imaging, regulatory networks in genetics, and meshed surfaces in computer
graphics. In many applications, such geometric data are large and complex (in
the case of social networks, on the scale of billions), and are natural targets
for machine learning techniques. In particular, we would like to use deep
neural networks, which have recently proven to be powerful tools for a broad
range of problems from computer vision, natural language processing, and audio
analysis. However, these tools have been most successful on data with an
underlying Euclidean or grid-like structure, and in cases where the invariances
of these structures are built into networks used to model them. Geometric deep
learning is an umbrella term for emerging techniques attempting to generalize
(structured) deep neural models to non-Euclidean domains such as graphs and
manifolds. The purpose of this paper is to overview different examples of
geometric deep learning problems and present available solutions, key
difficulties, applications, and future research directions in this nascent
field
- …