Kronecker Product and SVD Approximations in Image Restoration
J. Kamm and J. Nagy
Image restoration applications often result in
ill-posed least squares problems
involving large, structured matrices.
One approach used extensively is to restore the image in the
frequency domain, thus providing fast algorithms using FFTs.
This is equivalent to using a circulant approximation to a given matrix.
Iterative methods may also be used effectively by exploiting the
structure of the matrix. While iterative schemes are more expensive
than FFT-based methods,
it has been demonstrated that they are capable of providing better
restorations. As an alternative, we propose an approximate singular
value decomposition, which can be used in a variety of
applications.
Used as a direct method, the computed restorations are comparable
to iterative methods but are computationally
less expensive. In addition, the
approximate SVD may be used with the generalized
cross validation method to choose regularization parameters.
It is also demonstrated that the approximate SVD can be an effective
preconditioner for iterative methods.