Variational Analysis On The Signed Bregman Distance Function

The signed distance function(or oriented distance function)of a set in a metric space determines the distance of a given point from the set.It is an improvement of the usual metric function.If the point is outside the set,the metric function is positive and the point is negative inside the set.Symbolic metric function retains the good properties of the metric function,and also produces classical geometric properties related to the set and its boundary.The smoothness of the signed distance function near the given set boundary is equivalent to the local smoothness of the set boundary.The signed distance function has been widely studied and applied in the fields of level set method,computer vision,image processing and so on.The main results,obtained in this dissertation,may be summarized as follows:1.In chapter 1,we give brief introduction to the research background and content of signed distance function,Bregman distance and proximal mapping.And we also summarize the developments of above these problems.Finally,we outline the contents studied in this thesis.2.In Chapter 2,for the proximal mappings in n-dimensional Euclidean spaces,the norm is generalized to a more general Bregman distance.The basic properties of the proximal mappings in the sense of Bregman distance are studied.Under what conditions are the proximal mappings a singleton.A concrete example is given for the conclusion of the theorem and generalization of the second proximal mapping theorem under Bregman distance.By using a special indicator function,the equivalent relation between the orthogonal projection of a set and the adjacent mapping is obtained.3.In Chapter 3,we extend the Euclidean metric commonly used in sign metric functions to a more general Bregman distance,give the definition of the signed Bregman distance function,and redefine the N-dimensional Euclidean space.The interior,closure and boundary of the set are further explored,and the analytical properties of the symbol Bregman distance are further explored.