Studies on geometric shape reconstruction

Abstract

This thesis, on geometric shape modeling problems, contains two major chapters. In the first chapter, we propose a fast method for fitting planar Bspline curves to unorganized data points. In traditional methods, optimization of control points and foot points are performed in two alternating timeconsuming steps in every iteration: 1) control points are updated by setting up and solving a linear system of equations; and 2) foot points are computed by projecting each data point onto a B-spline curve. Our method uses the LBFGS optimization method to optimize control points and foot points simultaneously and therefore it does not need to solve a linear system of equations or performing foot point projection in every iteration. As a result, the proposed method is much faster than existing methods. In the second chapter, we propose a new shape description method using a radial basis function built on the medial axis of the shape. By formulating our approach as a constrained L^1-minimization problem, our method produces sparse reconstruction result which uses much fewer basis functions than previous approaches. Besides the sparse representation capacity, our method also has advantages in two aspects: 1) Our method does not rely on normal information of input points. 2) Our method has stronger capacity in representing multi-scale shapes compared with existing methods. All these characteristics will be illustrated in the corresponding chapters and sections.