Gaussian Process for Tomography

Tomographic imaging refers to the reconstruction of a 3D object from its 2D projections by sectioning the object, through the use of any kind of penetrating wave, from many different directions. It has had a revolutionary impact in a number of fields ranging from biology, physics, and chemistry to astronomy [1, 2]. The technique requires an accurate image reconstruction, however, and the resulting reconstruction problem is an ill-posed optimization problem because of insufficient measurements [3]. A direct consequence of ill-posedness is that the reconstruction does not have a unique solution. Therefore, quantifying the solution quality is challenging, given the absence of ground truth and the presence of measurement noise. Moreover, ill-posedness creates a requirement for regularization that imports new parameters to the problem. Regularization parameter choice can lead to substantial variations in reconstruction, and ascertaining optimal values of such parameters is difficult without availing oneself of ground truth [4]. The transition from an optimization perspective on tomographic inversion to a Bayesian statistical perspective can provide a useful reframing of these issues. In particular, the ill-posedness of the optimization view can be replaced by quantified uncertainty in the statistical view, whereas regularization now appears in the guise of parameter estimation.