Acceleration and filtering in the generalized landweber iteration using a variable shaping matrix

We use the generalized Landweber iteration with a variable shaping matrix to solve the large linear system of equations arising in the image reconstruction problem of emission tomography. Our method is based on the property that once a spatial frequency image component is almost recovered within ∊ in the generalized Landweber iteration, this component will still stay within ∊ during subsequent iterations with a different shaping matrix, as long as this shaping matrix satisfies the convergence criterion for the component. Two different shaping matrices are used: The first recovers low-frequency image components; and the second may be used either to accelerate the reconstruction of high-frequency image components, or to attenuate these components to filter the image. The variable shaping matrix gives results similar to truncated inverse filtering, but requires much less computation and memory, since it does not rely on the singular value decomposition.