Introduction

In a nutshell, the ultimate goal of Computer Vision would be to make computers able to understand the world into which they `live'.
Here, the word `computer' must be taken in a broad sense since computing chips are now not only in classical computers but in many other devices such as smartphones, cars, *etc*.
The verb `to understand' must also be considered in a really broad sense.
Despite all the efforts spent by the scientists during the past decades, we are still far from having intelligent computers.
In a more realistic way, it would be probably more reasonable to talk about `automatic extraction of information' of images instead of `understanding the world'.
The mechanical being does not exist yet but Computer Vision is nonetheless of broad interest with useful applications in domains such as multimedia, metrology, medical imaging, robotics, *etc*.
Computer Vision has been present in the professional context for decades now, particularly in the field of medical imaging.
However, the situation is evolving quite rapidly since the beginning of this millennium.
Indeed, Computer Vision is now involved in mass products such as mobile phones, cars, and game consoles.
This has opened brand new perspectives and developments for research in Computer Vision.

**Parametric models.**- A parametric model is a family of functions that can be described by a finite set of parameters
^{1.2}. The combination of a parametric model with a set of parameters allows one to model a phenomenon (such as a surface that fits a cloud of points or the deformation between two images). Many parametric models may be used in Computer Vision, ranging from very specific models (representing, for instance, an affine transformation) to models of general use (such as the well-known B-splines that allows one to model complex transformations such as the image deformation function). The choice of a `good' parametric model,*i.e.*a model that can represent the phenomenon under study, is extremely important. In this thesis, we use many different existing parametric models and also propose new ones. **Parameter estimation.**- Parameter estimation is a central part of parametric approaches. It consists in finding an appropriate set of parameters that, combined with a fixed parametric model, `explains' correctly a data set. This is achieved by modelling the problem under study. It usually results in a `score function' (also known as criterion, cost function, loss function, or residual error). The minimization (or, sometimes, maximization) of this criterion is expected to give the right result. The modelling step is a mathematical formulation of the problem that takes into account various elements such as the nature of the data, the type of measurement noise, the presence of erroneous data, the prior knowledge one has about the solution,
*etc*. **Hyperparameter estimation.**- Hyperparameter estimation is another important part of any classical parametric approach. What we call
*hyperparameters*in this document are the additional parameters that typically arise in the optimization problems resulting of the modelling step. The number of control points of a B-spline or the strength given to a regularization prior are examples of hyperparameters. For reasons that will become clear along this manuscript, hyperparameters cannot be estimated the same way as natural parameters. Determining in an automatic way good hyperparameters is a challenging problem that is often neglected. Using appropriate hyperparameters is nonetheless crucial in order to get satisfactory results with a parametric approach. Part of our work is dedicated to this point.

**Range surface fitting**- is the problem of finding an analytic expression of a smooth parametric surface that approximates a set of 3D data point. In this document, we consider `range data'. This type of data is also known as 2.5D. It may be viewed as a set of 2D locations, each one of which being associated an altitude (or height, or depth). Range data is now of broad interest because there are some devices that allows one to get such data quite easily: Time-of-Flight cameras, laser range scanners,
*etc*. The main challenges encountered in such problems is to cope with noise, large amounts of data, and discontinuities. **Image registration**- is the problem of determining the transformation between two (or more) images of the same scene. Various types of transformations may be considered: photometric, geometric. In this document, we are mainly interested in geometric transformations. Besides, we focus on deformable environments. It means that the position (or the shape) of the objects may vary between the images to register. This implies that complex parametric models must be used to model the deformations. Consequently the parameter estimation step also becomes quite difficult in general.
**3D reconstruction of deformable surfaces.**- The last problem of Computer Vision addressed in this document is the reconstruction of a deforming 3D surface from a monocular video. Using the motion cue only makes it a fundamentally ill-posed problem since there exist an infinite number of 3D shapes that have the same reprojection in an image. We propose to overcome this problem by considering that the deformable surface is inextensible and that a reference shape is available for a template image. Although these assumptions are common, the way we enforce the underlying constraints is new: we model the reconstructed surface as a smooth surface (based on tensor-product B-splines) and impose that the surface be everywhere a local isometry.

Contributions to Parametric Image Registration and 3D Surface Reconstruction (Ph.D. dissertation, November 2010) - Florent Brunet

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