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Photoacoustic tomography (PAT) is a novel and promising technology in hybridmedical imaging that involves generating acoustic waves in the object ofinterest by stimulating electromagnetic energy. The acoustic wave is measuredoutside the object. One of the key mathematical problems in PAT is thereconstruction of the initial function that contains diagnostic informationfrom the solution of the wave equation on the surface of the acoustictransducers. Herein, we propose a wave forward operator that assigns an initialfunction to obtain the solution of the wave equation on a unit sphere. Underthe assumption of the radial variable speed of ultrasound, we obtain thesingular value decomposition of this wave forward operator by determining theorthonormal basis of a certain Hilbert space comprising eigenfunctions. Inaddition, numerical simulation results obtained using the continuous Galerkinmethod are utilized to validate the inversion resulting from the singular valuedecomposition.

Singular value decomposition of the wave forward operator with radial variable coefficients