Properties Of Generalized Barycentric Coordinates And Application Study

Barycentric coordinates were first defined on polygons by A.F.M?bius in 1827,which have become a powerful tool for image processing because of its geometry-free construction.After the efforts of the relevant scholars for more than a century,the definition,property and application of GBCs(generalized barycentric coordinates)have been greatly expanded.In the first chapter,the definitions of GBCs are summarized.Barycentric coordinates were generalized from plane to high dimension and from discrete domain to continuous domain.In addition,several GBCs are listed,which urge to find a general approach for constructing GBCs.In the second chapter,a new idea of constructing GBCs is given from the view of the solution space of a system of linear equations,and properties of GBCs are discussed in two situations.First,the recursion on GBCs and its limited condition are given by the study of recursive property of GBCs.Unlike the classical way to construct GBCs,which based on geometric meaning of coordinate functions,we provide a new method to construct GBCs if a suitable coordinate function is chosen.As a result,we restore an existing coordinates and construct a new set of coordinates.Second,during the study of continuity of GBCs,we prove that Wachspress coordinates become polynomial function in parallelogram which is rational in general.While piecewise Wachspress coordinates reach onlyC~0continuity in joint parallelograms.Finally,a set of birational GBCs on is given on quadrilateral which can avoid the distortion in image warping and restoration.And the singularities of the birational GBCs are pointed out.In the third chapter,applications of GBCs are given in barycentric interpolation and image warping.The advantage of birational GBCs is mentioned in image warping and restoration.In the last chapter,the sduties are summarized.Meanwhile,the research directions are given about the constructing approach,the continuity and birational property.