Mathematical symbols and computing methods in high dimensional biomimetic informatics and their applications

High dimensional biomimetic informatics (HDBI) is a novel theory of informatics developed in recent years. Its primary object of research is points in high dimensional Euclidean space, and its exploratory and resolving procedures are based on simple geometric computations. However, the mathematical descriptions and computing of geometric objects are inconvenient because of the characters of geometry. With the increase of the dimension and the multiformity of geometric objects, these descriptions are more complicated and prolix especially in high dimensional space. In this paper, we give some definitions and mathematical symbols, and discuss some symbolic computing methods in high dimensional space systematically from the viewpoint of HDBI. With these methods, some multi-variables problems in high dimensional space can be solved easily. Three detailed algorithms are presented as examples to show the efficiency of our symbolic computing methods: the algorithm for judging the center of a circle given three points on this circle, the algorithm for judging whether two points are on the same side of a hyperplane, and the algorithm for judging whether a point is in a simplex constructed by points in high dimensional space. Two experiments in blurred image restoration and uneven lighting image correction are presented for all these algorithms to show their good behaviors.