An Novel Theory to Segment Iso-surface in Marine Gis 3d Data

Segmentation of 3D data (some time 4D data) is a very challenging problem in applications exploiting Marine GIS data. To tackle this problem, this paper proposes a topological approach based on the Digital Morse theory which is a kind of Discrete Morse theory to high dimension Grid points. The essence of the approach concerns detecting critical points in the High dimension Data, which represent parts of the topology changing. Because less or more some prior information could be got, our approach is quite robust against noise. Experimental results demonstrate the validity of our method * Corresponding author. This is useful to know for communication with the appropriate person in cases with more than one author. 1. INTRODUCTIO In the marine GIS, there are lot three dimensions Volume Data from ocean surveys of water temperature, salinity, and contami nants. The space distributing features of these data contains the key environment information of the sea from which they came. One of the most important ways to analysis the space distributing features of 3D Data is to compute iso-surface of these data, and then visualize them. There are many method to compute or segment iso-surface in 3D date, the most popular one is matching cube, but in many cases it will commit errors, this is related to a typical problem in mathematics involves attempting to understand the topology, or large-scale structure, of an object with limited information. This kind of problem also occurs in mathematical physics, dynamic systems and mechanical engineering. Morse theory is a generalization of calculus of variations, which draws the relationship between the stationary points of a smooth real-valued function on a manifold and the global topology of the manifold. Morse theory consists of two parts: one is the critical point theory and another is the application in calculus of variations. Dr. J.L.Cox and Dr. D.B.Karron from City University of New York developed a Digital Morse Theory, which expands the fundamental insight of Morse theory to the critical point and criticality graph theory in discrete set. With this powerful theory we can easily recognize iso-surface and analysis the geometry and topology of high dimensional data set. Here I gave a comprehensive introduction to Digital Morse theory in this report and showed a few simple successful applications in Marine 3D Data iso-surface segmentation. This report is intended, as far as possible, to give an exact insight into digital Morse theory to the readers, who are interested in the theory. Fore I believe it is a powerful tool to analysis high dimension temporal –space data in complicated GIS system such as Marine GIS 2. MORSE THOERY Traditional Morse Theory begins with this insight: Let f be a continuous function defined on a compact, smoothly differentiable manifold 2 C M . A Morse function has the following properties: Each critical point of f is an isolated point, and at each critical point the Hessian (matrix of second order partials) is nonsingular. In other words each criticality is a single isolated point and is a true local maximum, minimum or saddle point (there are no points of inflection). Then the topology changes of the level sets of f occur only at the critical values and are completely characterized by the number of negative Eigen values of the Hessian at each critical point, which determines the number of linearly independent down directions, and thus whether it is a maximum, a minimum, or determines the type of saddle. Morse theory can be thought of as a generalization of the classical theory of critical points (maxima, minima and saddle points) of smooth functions on Euclidean spaces. Morse theory states that for a generic function defined on a closed compact manifold (e.g. a closed surface)) the nature of its critical points determines the topology of the manifold. Morse functions are generic functions for which all the critical points are nondegenerate (the Hessian matrix of the function at the critical point is non singular). For a Morse function, the critical points determine the homology groups of the manifold, that is a sets of points for which the function is less than a given value x . Moreover these sets can fully describe the topology of the manifold. The way the manifold is embedded in the 3D space can be coded