Explicit, approximate expressions for the resolution anda posterioricovariance of massive tomographic systems

Abstract

We present an approximate method to estimate the resolution, covariance and correlation matrix for linear tomographic systems Ax=b that are too large to be solved by singular value decomposition. An explicit expression for the approximate inverse matrix A− is found using one-step backprojections on the Penrose condition AA−≈I, from which we calculate the statistical properties of the solution. The computation of A− can easily be parallelized, each column being constructed independently. The method is validated on small systems for which the exact covariance can still be computed with singular value decomposition. Though A− is not accurate enough to actually compute the solution x, the qualitative agreement obtained for resolution and covariance is sufficient for many purposes, such as rough assessment of model precision or the reparametrization of the model by the grouping of correlating parameters. We present an example for the computation of the complete covariance matrix of a very large (69 043 × 9610) system with 5.9 × 106 non-zero elements in A. Computation time is proportional to the number of non-zero elements in A. If the correlation matrix is computed for the purpose of reparametrization by combining highly correlating unknowns xi, a further gain in efficiency can be obtained by neglecting the small elements in A, but a more accurate estimation of the correlation requires a full treatment of even the smaller Aij. We finally develop a formalism to compute a damped version of A−.