Graph-based regularization of inverse problems in imaging

As of today, the extension of the human visual capabilities to machines remains both a cornerstone and an open challenge in the attempt to develop intelligent systems. On the one hand, the development of more and more sophisticated imaging devices, capable of sensing richer information than a plain perspective projection of the real world, is critical in order to allow machines to understand the complex environment around them. On the other hand, despite the advances in imaging, the complexity of the real world cannot be fully captured by a single imaging device, either due to intrinsic hardware limitations or to the environment complexity itself. As a consequence, the attempt to extend the human visual capabilities to machines requires inevitably to estimate some unknown quantities, which could not be measured, from the available captured data. Equivalently, imaging requires the solution of arbitrarily complex inverse problems.

In most scenarios, inverse problems are ill-posed and admit an infinite number of solutions, while only one or few of them are the desired ones. It becomes therefore crucial to reduce, equivalently \textit{to regularize}, the solution space by exploiting all the available prior information about the problem structure and, especially, about the target quantity to estimate. In this thesis we investigate the use of graph-based regularizers to encode our prior knowledge about the target quantity and to inject it directly into the inverse problem. In particular, we cast the inverse problem into an optimization task, where the target quantity is modelled as a graph whose topology captures our prior knowledge. In order to show the effectiveness and the flexibility of graph-based regularizers, we study their use in different inverse imaging problems, each one characterized by different geometrical constraints.

We start by investigating how to augment the resolution of a light field. In fact, although light field cameras permit to capture the 3D information in a scene within a single exposure, thus providing much richer information than a perspective camera, their compact design limits their spatial resolution dramatically. We present a smooth graph-based regularizer which models the geometry of a light field explicitly and we use it to augment the light field spatial resolution while relying only on the complementary information encoded in the low resolution light field views. In particular, we show that the use of a graph-based regularizer permits to enforce the light field geometric structure without the need for a precise and costly disparity estimation step.

Then we analyze the further benefits provided by adopting nonsmooth graph-based regularizers, as these better preserve edges and fine details than their smooth counterpart. In particular, we focus on a specific nonsmooth graph-based regularizer and show its effectiveness within two applications. The first application revolves again around light field super-resolution, which permits a comparison with the smooth regularizer adopted previously. The second applications is disparity estimation in omnidirectional stereo systems, where the two captured images and the target disparity map live on a spherical surface, hence a graph-based regularizer can be used to model the non trivial correlation underneath each signal.

Finally, we investigate the refinement of a depth map and the estimation of the corresponding normal map. In fact, ...