Singular Values of the Attenuated Photoacoustic Imaging Operator

TL;DR

This paper investigates how attenuation affects the mathematical properties of the photoacoustic imaging operator, revealing exponential decay of singular values in strong attenuation and similar decay rates to the non-attenuated case in weak attenuation.

Contribution

It introduces an attenuated photoacoustic operator and characterizes the asymptotic behavior of its singular values based on attenuation strength.

Findings

Strong attenuation causes exponential decay of singular values.

Weak attenuation results in decay rates similar to non-attenuated case.

Provides a mathematical framework for understanding ill-posedness in attenuated photoacoustic imaging.

Abstract

We analyse the ill-posedness of the photoacoustic imaging problem in the case of an attenuating medium. To this end, we introduce an attenuated photoacoustic operator and determine the asymptotic behaviour of its singular values. Dividing the known attenuation models into strong and weak attenuation classes, we show that for strong attenuation, the singular values of the attenuated photoacoustic operator decay exponentially, and in the weak attenuation case the singular values of the attenuated photoacoustic operator decay with the same rate as the singular values of the non-attenuated photoacoustic operator.