| A class of efficient algorithms is presented for reconstructing an image from noisy, blurred data. This methodology is based on Tikhonov regularization with a regularization functional of total variation type. Use of total variation in image processing was pioneered by Rudin and Osher. Minimization yields a nonlinear integro-differential equation which, when discretized using cell-centered finite differences, yields a full matrix equation. A fixed point iteration is applied, and the intermediate linear equations are solved via a preconditioned conjugate gradient method. A multigrid preconditioner, due to Ewing and Shen, is applied to the differential operator, and a spectral preconditioner is applied to the integral operator. A multi-level quadrature technique, due to Brandt and Lubrecht is employed to find the action of the integral operator on a function. Application to laser confocal microscopy is discussed, and a numerical reconstruction of two-dimensional data from a laser confocal scanning microscope is presented. In addition, reconstructions of synthetic data and a numerical study of convergence rates are given. |
