A continuous adjoint for photo-acoustic tomography of the brain

We present an optimization framework for photo-acoustic tomography of brain based on a system of coupled equations that describe the propagation of sound waves in linear isotropic inhomogeneous and lossy elastic media with the absorption and physical dispersion following a frequency power law using fractional Laplacian operators. The adjoint of the associated continuous forward operator is derived, and a numerical framework for computing this adjoint based on a k- space pseudospectral method is presented. We analytically show that the derived continuous adjoint matches the adjoint of an associated discretised operator. We include this adjoint in a first-order positivity constrained optimization algorithm that is regularized by total variation minimization, and show that the iterates monotonically converge to a minimizer of an objective function, even in the presence of some error in estimating the physical parameters of the medium.