Abstract (in English)

This thesis deals with the modelling and the estimation of parametric functions in Computer Vision. It focuses on three main topics: range surface fitting, image registration, and 3D reconstruction of smooth surfaces. In addition to these three main topics, we consider other transversal elements. In particular, we focus on the general problem caused by the hyperparameters in parametric model estimation, which includes regularization problems. All these topics are related by a single objective which is nowadays one of the most important goals in Computer Vision: the reconstruction of an arbitrary surface from images taken in an arbitrarily deforming environment. This thesis can be divided into four main parts.

The first part deals with the basics. It includes background on optimization and on parameter estimation. The problems related to the hyperparameters are also explained and illustrated. The rest of the thesis is centred on our original contributions.

The second part of this thesis deals with the problem of fitting a surface to range data. This problem consists in finding a parametric smooth surface that approximates accurately a sparse set of 3D points. We consider two main problems. First, we propose methods to automatically tune the hyperparameters such as a regularization weight. Second, we show how heteroskedastic noise may be handled. Heteroskedastic noise is an important problem since it is typical of range sensors, for instance Time-of-Flight cameras.

The third part of this thesis is dedicated to the problem of image registration. We propose three contributions in this topic. First, we present a new warp (image deformation function) able to correctly model the effect of perspective projection. Second, we show how to solve an important problem that typically happens in direct image registration: the problem of the region of interest. Third, we propose a new framework to estimate in a reliable way the hyperparameters needed in feature-based image registration (threshold of an M-estimator, regularization weight, number of control points, etc).

The last part of this thesis deals with the problem of reconstructing an inextensible surface from a monocular sequence of images. We also use the hypothesis that a reference shape is known. Using only the motion cue, the problem is ill-posed but, nonetheless, satisfying and plausible results can be obtained. We propose two new formulations to reconstruct the surface: the first one reconstruct a sparse set of points using a second order cone program, and the second one reconstruct a smooth parametric surface using a least-squares minimization problem.