Abstract : The first part of the thesis is dedicated to the convexity in the discrete plane Z2 or more generally Zn. In fact there exist many notions of discrete convexity: the simple convexity along some prescribed directions, the total convexity (the usual convexity in the continuous), etc. The Q-convexity is a new class of convexity which generalizes both the totally convex sets and the HV-convex polyominos. We study the links between all these notions, and the properties of special points of these sets such the median points and the salient points.
In all the second part we are interested in the main problem of discrete tomography : reconstructing a subset of Z2 from the number of its points in each line parallel to some prescribed directions. The polynomial algorithm already known for the HV-convex polyominoes and the horizontal and vertical directions can be generalized to work with Q-convex sets and any directions. On another hand the uniqueness result which shows that 7 directions are sufficient to determine completely a totally convex set from its projections can also be generalized to Q-convex sets. We deduce that when there are enough directions to have uniqueness the reconstruction of totally convex sets can be made in polynomial time. We also have a polynomial algorithm to find Q-convex sets from their approximative projections.