Quantitative PAT with unknown ultrasound speed : uncertainty characterization and reconstruction methods
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Quantitative photoacoustic tomography (QPAT) is a hybrid medical imaging modality that combines high-resolution ultrasound tomography with high-contrast optical tomography. The objective of QPAT is to recover certain optical properties of heterogeneous media from measured ultrasound signals, generated by the photoacoustic effect, on the surfaces of the media. Mathematically, QPAT is an inverse problem where we intend to reconstruct physical parameters in a set of partial differential equations from partial knowledge of the solution of the equations. A rather complete mathematical theory for the QPAT inverse problem has been developed in the literature for the case where the speed of ultrasound inside the underlying medium is known. In practice, however, the ultrasound speed is usually not exactly known for the medium to be imaged. Using an approximated ultrasound speed in the reconstructions often yields images which contain severe artifacts. There is little study as yet to systematically investigate this issue of unknown ultrasound speed in QPAT reconstructions. The objective of this dissertation is exactly to investigate this important issue of QPAT with unknown ultrasound speed. The first part of this dissertation addresses the question of how an incorrect ultrasound speed affects the quality of the reconstructed images in QPAT. We prove stability estimates in certain settings which bound the error in the reconstructions by the uncertainty in the ultrasound speed. We also study the problem numerically by adopting a statistical framework and applying tools in uncertainty quantification to systematically characterize artifacts arising from the parameter mismatch. In the second part of this dissertation, we propose an alternative reconstruction algorithm for QPAT which does not assume knowledge of the ultrasound speed map a priori, but rather reconstructs it alongside the original optical parameters of interest using data from multiple illumination sources. We explain the advantage of this simultaneous reconstruction approach compared to the usual two-step approach to QPAT and demonstrate numerically the feasibility of our algorithm.