2014/07/02 by Peter Kuchment, Kuchment, Peter, Dustin Steinhauer +1
Engineering · Mathematics · #35R30 #Advanced X-ray and CT Imaging #Analysis of PDEs (math.AP) #Electrical and Bioimpedance Tomography #FOS: Mathematics #Numerical methods in inverse problems #Photoacoustic and Ultrasonic Imaging
paper · pdf · doi:10.48550/arxiv.1407.0763
openalex publication_date 2014/07/02 · openalex created_date 2022/10/06 · openalex updated_date 2026/07/28
In the previous paper "Stabilizing Inverse Problems by Internal Data", the\nauthors introduced a simple procedure that allows one to detect whether and\nexplain why internal information arising in several novel coupled physics\n(hybrid) imaging modalities could turn extremely unstable techniques, such as\noptical tomography or electrical impedance tomography, into stable,\ngood-resolution procedures. It was shown that in all cases of interest, the\nFrechet derivative of the forward mapping is a pseudo-differential operator\nwith an explicitly computable principal symbol. If one can set up the imaging\nprocedure in such a way that the symbol is elliptic, this would indicate that\nthe problem was stabilized. In the cases when the symbol is not elliptic, the\ntechnique suggests how to change the procedure (e.g., by adding extra\nmeasurements) to achieve ellipticity.\n In this article, we consider the situation arising in acousto-optical\ntomography (also called ultrasound modulated optical tomography), where the\ninternal data available involves the Green's function, and thus depends\nglobally on the unknown parameter(s) of the equation and its solution. It is\nshown that the technique of "Stabilizing Inverse Problems by Internal Data" can\nbe successfully adopted to this situation as well. We also obtain results on\ngeneric uniqueness for the linearized problem in a variety of situations,\nincluding those arising in acousto-electric and quantitative photoacoustic\ntomography.\n