An Efficient Method for Fitting Large Data Sets Using T-Splines

Data fitting is a fundamental tool in scientific research and engineering applications. Generally, there are two ingredients in solving data fitting problems. One is the fitting representation, and the other is the fitting method. Nowadays, the fitting of larger and larger quantities of data sets requires more compact fitting representation and faster fitting methods. The T-spline is a recently invented spline representation, whose control mesh (T-mesh) allows a row of control points to terminate, thus reducing the number of superfluous control points in the B-spline representation significantly. This property makes T-splines more compact than B-splines in fitting large data sets. However, the adaptivity of the T-spline causes the coefficient matrix of a least-squares fitting linear system to lose its block structure. Thus, when fitting large data sets with T-splines by iterative methods, only point iterative methods can be used, and the iteration speeds of typically employed point iterative methods are rapidly slowed with the increasing number of unknowns. In this paper, we present a progressive T-spline data fitting algorithm for fitting large data sets with a T-spline representation. As an iterative method, the iteration speed of our method is steady and insensitive to the growing number of unknown T-mesh vertices; thus, it is able to fit large data sets efficiently. Additionally, our method can handle data sets with or without holes in a unified framework, without any special processing. Finally, we apply the progressive T-spline data fitting algorithm in largeimage fitting to validate its efficiency and effectiveness.