Reconstruction of surfaces from planar contours

Abstract

This dissertation is concerned with the problem of reconstructing the surfaces of three-dimensional objects from a collection of planar contours representing cross sections through the objects. This problem has important applications in diverse fields, including bio-medical research and instruction, solid modeling, geology, archaeology, oceanography and industrial inspection. The problem can be broken into four subproblems, the correspondence problem, the tiling problem, the branching problem, and the surface fitting problem. This dissertation is concerned with improving the quality of automated solutions to the tiling and branching problems. A new method for solving the tiling problem is developed that makes use of multiresolution analysis to improve performance of an optimizing tiling algorithm both in terms of execution time and space requirement. The method computes an optimized tiling at low resolution, then adds detail to the low resolution tiling by alternating between adding wavelet coefficients and local optimization until the original resolution has been reached. By halting the reconstruction before reaching the original resolution, the method also provides an efficient and effective way to reduce the number of vertices used to represent a contour, with savings in the time required to display a reconstruction and in the space required to store it, all with minimum loss of detail. A new method for solving the branching problem is also developed. It makes use of the medial axis transformation to classify branches into one of three types. The strategy used for constructing a solution to the branching problem is different for each of the classes. Depending on branch class, post-branch contours are merged to form composite contours or pre-branch contours are split into sub-contours, establishing a one-to-one correspondence of contours between sections, so that a standard tiling algorithm can be used. The new method considers the shapes of the pre-branch and post-branch contours when modification are made. Previous methods generally have not considered contour shape when forming composites, and the results have often been of poor quality. The new method is a significant improvement over previously available methods.