Geometric Iteration Based Approximation Algorithm And Its Applications

Data fitting is one of the basic tools for solving real-world scientific and engineering problems. Progressive-iterative approximation (PIA) is a new data fitting technique developed recently for blending curves and surfaces. Taking the given data points as the initial control points, PIA constructs a series of fitting curves or surfaces by adjusting the control points along the difference vector between data points and corresponding points on curves or surfaces iteratively, while the limit curve or surface interpolates the data points. It has been shown that the blending curves and tensor product blending patches with normalized totally positive have the PIA property. Moreover, PIA has local property that adjusting a subset of control points can interpolate the subset of data points, corresponding to the adjusting points.Although PIA method has so many advantages, there are still limitations of the PIA method. We develop an extended PIA (EPIA) format, which allows that the number of the control points is less than the given data points. And it is proved that the limit curves or patches of EPIA are the same as the solution of the least square linear system with suitable data grouping and weighting. We design an incremental data fitting method by the EPIA format and the results show that our algorithm has a good efficiency and fitting result.Vector graphics has been increasingly adopted to represent images because of its compactness, scalability and editability. In this paper, we try to use an incremental T-spline image fitting algorithm for global image vectorizations. Comparing with B-spline, T-spline has the adaptive property and reduce the superfluous control points, which is used to keep the structure of the B-spline mesh. Moreover, we prove the PIA property of T-spline and present the algorithm of global image vectorization. Finally, we illustrate the result of image vectorization with some discussions.