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A Tutorial Introduction to Inverse Problems in Magnetic Resonance.

Richard G Spencer1, Chuan Bi1

  • 1National Institute on Aging, National Institutes of Health, Baltimore, Maryland, U.S.A.

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|August 18, 2020
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Summary
This summary is machine-generated.

This study introduces the inverse problem framework for magnetic resonance parameter estimation. It covers linear and nonlinear methods, regularization, and compressed sensing for practical applications.

Keywords:
Compressed sensingLinear inverse problemsNonlinear inverse problemsParameter estimation

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Area of Science:

  • Magnetic Resonance Imaging
  • Computational Physics
  • Applied Mathematics

Background:

  • The inverse problem framework is increasingly applied to parameter estimation in magnetic resonance (MR).
  • A comprehensive review of all basic and modern developments is infeasible in a single article.
  • This work serves as a practical introduction and literature guide.

Purpose of the Study:

  • To provide a foundational understanding of the inverse problem framework in MR.
  • To introduce key concepts and solution methods for linear and nonlinear inverse problems.
  • To offer an introduction to compressed sensing within the context of inverse problems.

Main Methods:

  • Formulation of linear and nonlinear inverse problems, focusing on MR signal equations.
  • Description of the Fredholm equation of the first kind as a model problem.
  • Detailed discussion of solution methods, including regularization and stability analysis for linear and nonlinear cases.
  • Introduction to compressed sensing and its formulation as an inverse problem.

Main Results:

  • Outlines of theoretical concepts and practical numerical examples are presented.
  • Emphasis is placed on understanding the core principles rather than exhaustive mathematical detail.
  • The material provides a pathway to further literature on the subject.

Conclusions:

  • The inverse problem framework offers a robust approach to parameter estimation in magnetic resonance.
  • Understanding regularization and stability is crucial for solving these problems.
  • Compressed sensing represents a modern advancement, leveraging inverse problem principles for signal reconstruction.