The purpose of this thesis has been to study several problems typical for shape analysis, computer vision and image processing in general. First, a problem of sampling planar shape and continuous curves is considered. It has been shown that samples of a discrete curve or a surface should contain the information about the object itself as well as the coordinate system used to produce a functional parameterization. The sampling algorithm has been
designed as a two-cost optimization problem. Secondly, we have proposed a new algorithmic approach to implementing a simulated annealing type of optimization, which is based on the notion of scale. The scale is defined as a unit time interval used when converting a continuous-time evolution into a discrete one. It has been shown that by varying the scale as opposed to keeping it constant, one can achieve a better performance of an optimization
algorithm, all in the absence of a complicated acceptance/rejection Markov chain mechanism. Finally, we have studied the problem of identification of symmetric planar shapes from a single view. It has been shown that the proposed notion of a skeleton of an arbitrary view of the shape is instrumental in constructing invariants enabling us to perform identification. It has further been demonstrated that for the case of noisy data one can describe the distribution of the skeleton points and thus construct an optimal skeleton estimation technique.
