Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Inverse photoacoustic tomography problem in media with fractional attenuation

Sebastian Acosta, Benjamin Palacios

TL;DR

The paper extends inverse photoacoustic tomography to media with fractional attenuation modeled by a Caputo derivative of order $α∈(0,1)$, showing uniqueness under a geometric foliation via fractional Carleman estimates, and establishing stability through a continuity framework for fractional time-derivatives. A Neumann-series reconstruction is derived under full data and small damping, leveraging a time-reversal approach and energy estimates. The work combines generalized Caputo calculus, microlocal analysis, and PDE techniques to address memory effects in attenuated wave propagation, with potential broad applicability to PDEs with memory. These results broaden the mathematical foundation of PAT in lossy, memory-bearing tissues and offer tools for analyzing memory-augmented wave models in imaging and beyond.

Abstract

We investigate the inverse problem of recovering an initial source for the wave equation with fractional attenuation, motivated by photoacoustic tomography (PAT). The attenuation is modeled by a Caputo fractional derivative of order $α\in(0,1)$. We establish uniqueness under a geometric foliation condition via an adaptation of two types of Carleman estimates to the fractional setting, prove stability through continuity inequalities for fractional time-derivatives of wave solutions, and derive a reconstruction scheme based on a Neumann series. While our results apply directly to PAT, we expect that the analytic approach and tools employed might be of broader relevance to the analysis of PDEs with memory effects and to inverse problems for attenuated wave models.

Inverse photoacoustic tomography problem in media with fractional attenuation

TL;DR

The paper extends inverse photoacoustic tomography to media with fractional attenuation modeled by a Caputo derivative of order , showing uniqueness under a geometric foliation via fractional Carleman estimates, and establishing stability through a continuity framework for fractional time-derivatives. A Neumann-series reconstruction is derived under full data and small damping, leveraging a time-reversal approach and energy estimates. The work combines generalized Caputo calculus, microlocal analysis, and PDE techniques to address memory effects in attenuated wave propagation, with potential broad applicability to PDEs with memory. These results broaden the mathematical foundation of PAT in lossy, memory-bearing tissues and offer tools for analyzing memory-augmented wave models in imaging and beyond.

Abstract

We investigate the inverse problem of recovering an initial source for the wave equation with fractional attenuation, motivated by photoacoustic tomography (PAT). The attenuation is modeled by a Caputo fractional derivative of order . We establish uniqueness under a geometric foliation condition via an adaptation of two types of Carleman estimates to the fractional setting, prove stability through continuity inequalities for fractional time-derivatives of wave solutions, and derive a reconstruction scheme based on a Neumann series. While our results apply directly to PAT, we expect that the analytic approach and tools employed might be of broader relevance to the analysis of PDEs with memory effects and to inverse problems for attenuated wave models.

Paper Structure

This paper contains 15 sections, 20 theorems, 312 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Previous work
  3. Organization of the paper
  4. Preliminaries
  5. Notation
  6. Caputo's fractional derivative for locally integrable functions
  7. The initial value problem in PAT with fractional attenuation
  8. Some properties of the forward problem
  9. Uniqueness
  10. Stability
  11. Reconstruction
  12. Interpolation and Sobolev Spaces
  13. Integration on Banach spaces
  14. Strict convexity of Riemannian hypersurfaces
  15. Symbol classes, FIO's and $\Psi$DO's

Key Result

Lemma 1

Figures (1)

  • Figure 1: An example of an admissible foliation and its domain of influence. Two integral curve segments for the unit normal vector field $\nu$ are depicted, as well as the limiting leaves $\Sigma_{s_0}$ and $\Sigma_{s_T}$.

Theorems & Definitions (44)

  • Definition 1: Definition 2.1 in LiLiu
  • Remark 1
  • Lemma 1: Lemma 2.2 in LiLiu
  • Definition 2: Definition 3.4 in LiLiu
  • Remark 2
  • Remark 3
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • ...and 34 more