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.