Computationally attractive reconstruction of bandlimited images from irregular samples

An efficient method for the reconstruction of bandlimited images and the approximation of arbitrary images from nonuniform sampling values is developed. The novel method is based on the observation that the reconstruction problem can be formulated as linear system of equations using two-dimensional (2-D) trigonometric polynomials, where the matrix is of block- Toeplitz type with Toeplitz blocks. This system is solved iteratively by the conjugate gradient (CG) method. We show that the use of so-called adaptive weights in the establishment of the block Toeplitz matrix can be seen as efficient preconditioning. The superiority of the new method over conventional approaches is demonstrated by numerical experiments.