Statistics Seminar, "Spline smoothing in 3D with positivity constraints: An application to flame tomography", Johan Lindström, Mathematical Sciences, Lund University
Tomography can be seen as a linear reconstruction problem where each observation is represented by the sum of intensities for all voxels along a line. The aim is to use these observations to reconstruct the observed 3D-shape, represented by intensities in a number of voxels. The problem is often under-determined with fewer observations than voxels and a smoothness prior is imposed to make the solution identifiable. By expressing the smoothness prior as a squared penalty on derivatives the problem can be transformed to a latent Gaussian field with Gaussian observations. The large size of the data results in memory and computational limitations which can be solved using iterative methods and, for gridded data, FFT.
Since we are reconstructing flame intensity the voxels have to be positive, and the observations give informations regarding certain voxels that are known to be exactly equal to zero. Reconstruction and simulation of Gaussian fields under these restrictions can be formulated as a convex optimization problem with inequality constraints. For the case with only equality constraints the parameters can be estimated using a Monte-Carlo EM algorithm that utilizes the iterative solvers for fast simulation in the E-step. Finally we note that correct discretization of the smoothness penalty allows an expression of the Gaussian field that correctly accounts for the voxel size, allowing parameters estimated at a coarse spatial scale to be used for reconstructions at a much higher resolution substantially speeding up parameter estimation.
Tid: 2022-12-16 13:00 till 14:00
dragi [at] maths [dot] lth [dot] se