Study On The Rbf Implicit Surface And Its Application

In computer graphics field, implicit surface is an important.geometric representation, because it is a single analytic function, and it is suitable for collision detection, deformation, blending, warping, Boolean operations and many other aspects. Among them, the radial basis function (short for RBF) implicit surface can accurately and stably solve the 3D reconstruction issue of scattered points, and it is the most important implicit surface algorithms in recent years. This paper innovatively introduces the RBF implicit surface to the image-based modeling, minimal surface problem, and study on the geometric transformation of RBF implicit surface. Relevant research results can be applied to virtual reality, scene modeling, computer vision, graphics, visualization and other fields, research on minimal surfaces can be applied to plastic surgery and dental surgery, packaging design, molecular engineering and materials science, art, modern membranes engineering and many other fields. Therefore, this study has important significance on science and worthiness in practical application.Focused on RBF implicit surface, the image-based modeling, the geometric transformation of RBF implicit surface and minimal surface problem are studied and a number of novel algorithms are proposed in this dissertation. The main contributions include:1) In image-base modeling field, we present a grid-based feature detection algorithm, which can match more points among a sequence of images, and then we introduce the RBF implicit surface reconstruction to obtain the 3D model of the object. We make relatively dense grids and look for the best features near the grid points. Then, the sub-pixel coordinates of the corners are found by iteration. Then we calculate optical flow for a sparse feature set using iterative Lucas-Kanade method in pyramids. By this technique, we can reconstruct comparatively even and dense 3D point-cloud with self-calibration algorithm. Implicit surface reconstruction was introduced to generate the surface model of the object. By experimenting with a number of image sequences, the reconstruction results show that the algorithm can obtain satisfied surfaces for a rich texture scene.2) We derived the relationship between the initial and the transformed RBF coefficients. When we use a Radial Basis Function (RBF) implicit surface, we may need to transform it. The conventional algorithm to transform an RBF implicit surface is to apply inverse transformation to the given point and to evaluate the original function at the inversely transformed point. The algorithm keeps the initial RBF centers. Sometimes, we need the transformed RBF centers. In these cases, if we still use the conventional algorithm, we need to keep both the initial and the transformed RBF centers. Obviously, this is a problem that wastes the memory. We have derived the relationship between the initial and the transformed RBF coefficients, which can solve the previous problem. Our method only needs to keep the transformed RBF centers, and save much memory. By this method, we can get the new RBF surface quickly and it works for both globally and compactly supported RBF. We also compare our algorithm with the conventional algorithm about the time efficiency in details. The theoretical analysis and experiment results show that our algorithm is faster than the conventional algorithm in many cases. We also applied our method on RBF-based collision detection and Boolean operations。We also propose a method to speed up the Boolean operations of implicit surface. Moreover, we present a solution to relieve the bumps in CSRBF Boolean operations.3) The minimal surface problem is a classic problem in mathematics, especially its multiple-solutions problem has not yet resolved. This paper proposed a new algorithm, which may iteratively obtain a number of different minimal surfaces from an initial surface (i.e. get a number of different solutions), including unstable minimal surface. Giving a space curve as the boundary, finding out the surfaces which have minimal area is called "Plateau's problem" or "minimal suface problem". For a given boundary, our algorithm will take an initial arbitrary surface and then derive a minimal surface. For the same initial surface, when we select different parameters, the algorithm may derive different minimal surfaces, which other existing methods cannot achieve. Our algorithm simulates a non-linear spring model in the impact of air drag to derive the results. For the problem of how to generate the initial surface, there are a few literatures that report the related algorithms. We propose an algorithm using RBF implicit surface, which can generate specified genus initial surface semi-automatically, only a small amount of user interaction is needed. Our method is valid for any number of boundary curves. Moreover, for a single regular boundary curve, we propose an algorithm to generate the initial surface automatically.