Generalized Newton methods for energy formulations in image procesing

Many problems in image processing are solved via the minimization of a cost functional. The most widely used optimization technique is the gradient descent, often used due to its simplicity and applicability where other optimization techniques, e.g., those coming from discrete optimization, can not be used. Yet, gradient descent suffers from a slow convergence, and often to just local minima which highly depends on the condition number of the functional Hessian. Newton- type methods, on the other hand, are known to have a rapid (quadratic) convergence. In its classical form, the Newton method relies on the L²-type norm to define the descent direction. In this paper, we generalize and reformulate this very important optimization method by introducing a novel Newton method based on general norms. This generalization opens up new possibilities in the extraction of the Newton step, including benefits such as mathematical stability and smoothness constraints. We first present the derivation of the modified Newton step in the calculus of variation framework. Then we demonstrate the method with two common objective functionals: variational image deblurring and geodesic active contours. We show that in addition to the fast convergence, different selections norm yield different and superior results.