Abstract
In this article we address the numeric inversion of optoacoustic signals to initial stress profiles. Therefore we study a Volterra integral equation of the second kind that describes the shape transformation of propagating stress waves in the paraxial approximation of the underlying wave-equation. Expanding the optoacoustic convolution kernel in terms of a Fourier-series, a best fit to a pair of observed near-field and far-field signals allows to obtain a sequence of expansion coefficients that describe a given “apparative” setup. The resulting effective kernel is used to solve the optoacoustic source reconstruction problem using a Picard-Lindelöf correction scheme. We verify the validity of the proposed inversion protocol for synthetic input signals and explore the feasibility of our approach to also account for the shape transformation of signals beyond the paraxial approximation including the inversion of experimental data stemming from measurements on melanin doped PVA hydrogel tissue phantoms.
Highlights
The inverse optoacoustic (OA) problem is concerned with the reconstruction of “internal” medium properties from “external” measurements of acoustic pressure signals
Note that while problem I.1 is well established in the field of optoacoustics, we here make a first attempt at solving problem I.2, i.e. the kernel reconstruction problem, and demonstrate how it can be utilized for the reconstruction of initial stress profiles from observed OA signals
In the presented article we have introduced and discussed the kernel reconstruction problem in the paraxial approximation of the optoacoustic wave equation for both, synthetic input data and experimental data resulting from controlled measurements on melanin doped PVA hydrogel tissue phantoms
Summary
The inverse optoacoustic (OA) problem is concerned with the reconstruction of “internal” medium properties from “external” measurements of acoustic pressure signals. In contrast to the direct OA problem, referring to the calculation of a diffraction-transformed pressure signal at a desired field point for a given initial stress profile [1,2,3,4,5,6,7], one can distinguish two inverse OA problems: (I.1) the source reconstruction problem, where the aim is to invert measured OA signals to initial stress profiles upon knowledge of the mathematical model that mediates the underlying diffraction transformation [8,6,9,10,11], and, (I.2) a kernel reconstruction problem, where the task is to find a convolution kernel that accounts for the apparent diffraction transformation shown by the OA signal The latter arises quite naturally in a paraxial approximation wherein both signals can be related via a Volterra integral equation of the second kind [12].
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