Tomographic reconstruction from a small number of views.

Summary This thesis is devoted to the tomographic reconstruction of a 3d object from a small number of views, under symmetry assumptions. As classical inversions are very unstable, I approached this problem by modeling the object: the reconstruction consists in determining the parameters of the model by comparing the projections of the latter to the data. The illumination being parallel, I first studied the reconstruction of a plane section. The 1d model is then described by zones and has n parameters. The search for these parameters is expressed as the minimization of a quadratic criterion under constraints. As the latter is not differentiable, I have proposed three techniques to regularize it. I then studied the sensitivity of the parameterized vector to the noise present on the data after having determined the covariance matrix of this vector. I finally proposed a reconstruction of all the slices of the object by iteration of this 1d approach by adding a smoothing term between the slices. I then turned to the construction of a 3d model of the objects, characterized by six quantities: three separating surfaces (generated by the rotation of three plane curves) and the density fields on these interfaces. The first difficulty was to fill a 3d field from the density fields on the interfaces. I then proposed an adaptive elliptic operator to perform this operation. With fixed interfaces, I implemented the search for densities on each of them. I then formalized the deformation of these surfaces, i.e. of the plane generators. It is characterized by the search for an optimal base of deformations obtained by PCA on a set of examples. This set is constructed in a random way: the realizations are obtained by integration of the solution of a linear stochastic differential equation.