Photoacoustic image reconstruction in an attenuating medium using singular-value decomposition

Attenuation effects can be significant in photoacoustic tomography (PAT) since the measured pressure signals are broadband and ignoring them may lead to image artifacts and blurring. Previous work by our group had derived a method for modeling the attenuation effect and correcting for it in the image reconstruction. This was done by relating the ideal, unattenuated pressure signals to the attenuated pressure signals via an integral operator. In this work, we explore singular-value decomposition (SVD) of a previously derived 3D integral equation that relates the Fourier transform of the measured pressure with respect to time and two spatial components to the 2D spatial Fourier transform of the optical absorption function. We find that the smallest singular values correspond to wavelet-like eigenvectors in which most of the energy is concentrated at times corresponding to greater depths in tissue. This allows us characterize the ill posedness of recovering absorption information from depth in an attenuating medium. This integral equation can be inverted using standard SVD methods and the optical absorption function can be recovered. We will conduct simulations and derive algorithm for image reconstruction using SVD of this integral operator.