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Regularization of linear inverse problems with irregular noise using embedding operators

Xinyan Li, Simon Hubmer, Shuai Lu, Ronny Ramlau

TL;DR

By introducing the consequent preprocessed problem, the case that the noise can be preprocessed by certain adjoint embedding operators is considered, and convergence analysis for general regularization schemes under standard assumptions is provided.

Abstract

In this paper, we investigate regularization of linear inverse problems with irregular noise. In particular, we consider the case that the noise can be preprocessed by certain adjoint embedding operators. By introducing the consequent preprocessed problem, we provide convergence analysis for general regularization schemes under standard assumptions. Furthermore, for a special case of Tikhonov regularization in Computerized Tomography, we show that our approach leads to a novel (Fourier-based) filtered backprojection algorithm. Numerical examples with different parameter choice rules verify the efficiency of our proposed algorithm.

Regularization of linear inverse problems with irregular noise using embedding operators

TL;DR

By introducing the consequent preprocessed problem, the case that the noise can be preprocessed by certain adjoint embedding operators is considered, and convergence analysis for general regularization schemes under standard assumptions is provided.

Abstract

In this paper, we investigate regularization of linear inverse problems with irregular noise. In particular, we consider the case that the noise can be preprocessed by certain adjoint embedding operators. By introducing the consequent preprocessed problem, we provide convergence analysis for general regularization schemes under standard assumptions. Furthermore, for a special case of Tikhonov regularization in Computerized Tomography, we show that our approach leads to a novel (Fourier-based) filtered backprojection algorithm. Numerical examples with different parameter choice rules verify the efficiency of our proposed algorithm.
Paper Structure (11 sections, 4 theorems, 71 equations, 2 figures, 6 tables, 1 algorithm)

This paper contains 11 sections, 4 theorems, 71 equations, 2 figures, 6 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background on Sobolev embedding operators
  3. Real order Sobolev spaces
  4. Sobolev embedding operators
  5. Error bound analysis for general regularization schemes
  6. Tikhonov regularization for computerized tomography via Sobolev embedding operators
  7. Brief overview of the Radon transform
  8. Fourier-based inversion algorithm for the Radon transform
  9. Numerical experiments
  10. Numerical reconstruction algorithm
  11. Numerical examples

Key Result

Lemma 3.1

Let Assumption ass_linkcond hold and denote $\Theta(\lambda) := \sqrt{\lambda}\psi(\lambda)$. Then there holds

Figures (2)

  • Figure 5.1: Original image of carved cheese (upper left) and images reconstructed with the Fourier-based Tikhonov regularization method \ref{['re']} with $\alpha=10^{-5},10^{-8},10^{-10}$.
  • Figure 5.2: Original image of walnut (upper left) and images reconstructed with the Fourier-based Tikhonov regularization method \ref{['re']} with $\alpha=10^{-5},10^{-8},10^{-10}$.

Theorems & Definitions (17)

  • Definition 1.1
  • Example 1.1
  • Example 3.1
  • Definition 3.1
  • Example 3.2
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • Remark 3.1
  • ...and 7 more