Habilitationsvortrag | Dr. Gaël Rigaud
Regularization based on nonlinear isotropic diffusion for inverse problems with piecewise constant solutions
Regularization is a necessary step when solving ill-posed inverse problems in order to handle the stabilitiy issue. A way to achieve such a regularization is to consider constrained optimization problems as proposed in the Tikhonov-Philips regularization. When dealing with images like in image processing or imaging, the total-variation (TV) has proven itself to be a very efficient way to preserve the edges of the regularized solution. An interesting way to interpret the effect of the TV-term is using diffusion equations since the derivative of the TV functional is a special case of isotropic diffusion equations. In particular, the nature of edge preserving of TV reveals itself as a backward diffusion process in the direction of gradients with high values. Various choices of isotropic diffusion equations, such as Perona-Malik, were successfully proposed in the literature. In the case of Perona-Malik, a supplementary parameter controls backward and forward diffusion, i.e. edge preserving and smoothing.
An important aspect of images is to represent piecewise constant functions. Typically in imaging applications, each characteristic grey value describes a specific material which can usually be assumed to be known a priori. Following on from that observation and from the properties of regularization of istropic diffusion processes, this talk proposes to explore the design of a regularization term based on nonlinear isotropic diffusions and assuming to know a priori the greyscale of the solution.