Bayesian estimation of regularization parameters for deformable surface models

In this article we build on our past attempts to reconstruct a 3D, time-varying bolus of radiotracer from first-pass data obtained by the dynamic SPECT imager, FASTSPECT, built by the University of Arizona. The object imaged is a CardioWest total artificial heart. The bolus is entirely contained in one ventricle and its associated inlet and outlet tubes. The model for the radiotracer distribution at a given time is a closed surface parameterized by 482 vertices that are connected to make 960 triangles, with nonuniform intensity variations of radiotracer allowed inside the surface on a voxel-to-voxel basis. The total curvature of the surface is minimized through the use of a weighted prior in the Bayesian framework, as is the weighted norm of the gradient of the voxellated grid. MAP estimates for the vertices, interior intensity voxels and background count level are produced. The strength of the priors, or hyperparameters, are determined by maximizing the probability of the data given the hyperparameters, called the evidence. The evidence is calculated by first assuming that the posterior is approximately normal in the values of the vertices and voxels, and then by evaluating the integral of the multidimensional normal distribution. This integral (which requires evaluating the determinant of a covariance matrix) is computed by applying a recent algorithm from Bai et. al. that calculates the needed determinant efficiently. We demonstrate that the radiotracer is highly inhomogenous in early time frames, as suspected in earlier reconstruction attempts that assumed a uniform intensity of radiotracer within the closed surface, and that the optimal choice of hyperparameters is substantially different for different time frames.