The reconstruction of a curve or surface from an unorganized set of points is a well known problem. It is important in image analysis, but also in computeraided design and the reverse engineering of surfaces. One of the methods for solving the surface reconstruction problem uses radial basis functions to construct an implicit surface interpolating each point of the input set. There are, however, no theoretical guarantees concerning the topological correctness of such an approximation. This master's thesis deals with the reconstruction of convex curves in the plane using radial basis functions. In this case, an implicit curve is constructed as the collection of points where a function evaluates to zero. The function is constructed as a linear combination of radial basis functions centered at the input points. A linear system results from the requirement that each input point belongs to the curve. Using the solution to the linear system the function is found and the curve can be reconstructed from the set of points where the function evaluates to zero. For samplings of a circle and the identity as the radial basis function, it is proven that the reconstruction is a topologically correct approximation of a circle. The same result is achieved for general convex curves under a general condition on the coefficients of the linear system.
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