Quasi-Newton Approach to Nonnegative Image Restorations
M. Hanke, J. Nagy and C. Vogel
Image restoration, or deblurring, is the process of attempting
to correct for degradation in a recorded image. Typically the
blurring system is assumed to be linear and spatially invariant,
and fast Fourier transform based schemes result in efficient
computational image restoration methods. However, real images
have properties that cannot always be handled by linear methods.
In particular, an image consists of positive light intensities,
and thus a nonnegativity constraint should be enforced.
This constraint and other ways of incorporating a priori
information have been suggested in various applications, and
can lead to substantial improvements in the reconstructions.
Nevertheless, such constraints are rarely implemented because
they lead to nonlinear problems which require demanding computations.
We suggest efficient implementations for
three nonnegatively
constrained restorations schemes: constrained least squares,
maximum likelihood and maximum entropy. We show that
with a certain parameterization, and using a Quasi-Newton
scheme, these methods are very similar. In addition,
our formulation reveals a connection between our approach
for maximum likelihood
and the expectation-maximization
method used extensively by astronomers.
Numerical experiments illustrate that our approach
is superior to expectation-maximization both in terms of
accuracy and efficiency.