Matching Items (4)

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Incorporating the Sparsity of Edges into the Fourier Reconstruction of Piecewise Smooth Functions

Description

In applications such as Magnetic Resonance Imaging (MRI), data are acquired as Fourier samples. Since the underlying images are only piecewise smooth, standard recon- struction techniques will yield the Gibbs

In applications such as Magnetic Resonance Imaging (MRI), data are acquired as Fourier samples. Since the underlying images are only piecewise smooth, standard recon- struction techniques will yield the Gibbs phenomenon, which can lead to misdiagnosis. Although filtering will reduce the oscillations at jump locations, it can often have the adverse effect of blurring at these critical junctures, which can also lead to misdiagno- sis. Incorporating prior information into reconstruction methods can help reconstruct a sharper solution. For example, compressed sensing (CS) algorithms exploit the expected sparsity of some features of the image. In this thesis, we develop a method to exploit the sparsity in the edges of the underlying image. We design a convex optimization problem that exploits this sparsity to provide an approximation of the underlying image. Our method successfully reduces the Gibbs phenomenon with only minimal "blurring" at the discontinuities. In addition, we see a high rate of convergence in smooth regions.

Contributors

Agent

Created

Date Created
  • 2014-05

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Limitations of Classical Tomographic Reconstructions from Restricted Measurements and Enhancing with Physically Constrained Machine Learning

Description

This work is concerned with how best to reconstruct images from limited angle tomographic measurements. An introduction to tomography and to limited angle tomography will be provided and a

This work is concerned with how best to reconstruct images from limited angle tomographic measurements. An introduction to tomography and to limited angle tomography will be provided and a brief overview of the many fields to which this work may contribute is given.

The traditional tomographic image reconstruction approach involves Fourier domain representations. The classic Filtered Back Projection algorithm will be discussed and used for comparison throughout the work. Bayesian statistics and information entropy considerations will be described. The Maximum Entropy reconstruction method will be derived and its performance in limited angular measurement scenarios will be examined.

Many new approaches become available once the reconstruction problem is placed within an algebraic form of Ax=b in which the measurement geometry and instrument response are defined as the matrix A, the measured object as the column vector x, and the resulting measurements by b. It is straightforward to invert A. However, for the limited angle measurement scenarios of interest in this work, the inversion is highly underconstrained and has an infinite number of possible solutions x consistent with the measurements b in a high dimensional space.

The algebraic formulation leads to the need for high performing regularization approaches which add constraints based on prior information of what is being measured. These are constraints beyond the measurement matrix A added with the goal of selecting the best image from this vast uncertainty space. It is well established within this work that developing satisfactory regularization techniques is all but impossible except for the simplest pathological cases. There is a need to capture the "character" of the objects being measured.

The novel result of this effort will be in developing a reconstruction approach that will match whatever reconstruction approach has proven best for the types of objects being measured given full angular coverage. However, when confronted with limited angle tomographic situations or early in a series of measurements, the approach will rely on a prior understanding of the "character" of the objects measured. This understanding will be learned by a parallel Deep Neural Network from examples.

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Agent

Created

Date Created
  • 2020

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Recent techniques for regularization in partial differential equations and imaging

Description

Inverse problems model real world phenomena from data, where the data are often noisy and models contain errors. This leads to instabilities, multiple solution vectors and thus ill-posedness. To solve

Inverse problems model real world phenomena from data, where the data are often noisy and models contain errors. This leads to instabilities, multiple solution vectors and thus ill-posedness. To solve ill-posed inverse problems, regularization is typically used as a penalty function to induce stability and allow for the incorporation of a priori information about the desired solution. In this thesis, high order regularization techniques are developed for image and function reconstruction from noisy or misleading data. Specifically the incorporation of the Polynomial Annihilation operator allows for the accurate exploitation of the sparse representation of each function in the edge domain.

This dissertation tackles three main problems through the development of novel reconstruction techniques: (i) reconstructing one and two dimensional functions from multiple measurement vectors using variance based joint sparsity when a subset of the measurements contain false and/or misleading information, (ii) approximating discontinuous solutions to hyperbolic partial differential equations by enhancing typical solvers with l1 regularization, and (iii) reducing model assumptions in synthetic aperture radar image formation, specifically for the purpose of speckle reduction and phase error correction. While the common thread tying these problems together is the use of high order regularization, the defining characteristics of each of these problems create unique challenges.

Fast and robust numerical algorithms are also developed so that these problems can be solved efficiently without requiring fine tuning of parameters. Indeed, the numerical experiments presented in this dissertation strongly suggest that the new methodology provides more accurate and robust solutions to a variety of ill-posed inverse problems.

Contributors

Agent

Created

Date Created
  • 2018

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High-order sparsity exploiting methods with applications in imaging and PDEs

Description

High-order methods are known for their accuracy and computational performance when applied to solving partial differential equations and have widespread use

in representing images compactly. Nonetheless, high-order methods have difficulty

High-order methods are known for their accuracy and computational performance when applied to solving partial differential equations and have widespread use

in representing images compactly. Nonetheless, high-order methods have difficulty representing functions containing discontinuities or functions having slow spectral decay in the chosen basis. Certain sensing techniques such as MRI and SAR provide data in terms of Fourier coefficients, and thus prescribe a natural high-order basis. The field of compressed sensing has introduced a set of techniques based on $\ell^1$ regularization that promote sparsity and facilitate working with functions having discontinuities. In this dissertation, high-order methods and $\ell^1$ regularization are used to address three problems: reconstructing piecewise smooth functions from sparse and and noisy Fourier data, recovering edge locations in piecewise smooth functions from sparse and noisy Fourier data, and reducing time-stepping constraints when numerically solving certain time-dependent hyperbolic partial differential equations.

Contributors

Agent

Created

Date Created
  • 2016