SIAM J. SCI. COMPUT. c  2013 Society for Industrial and Applied Mathematics Vol. 35, No. 6, pp. A3052–A3068 AN EFFICIENT METHOD FOR FITTING LARGE DATA SETS USING T-SPLINES ∗ HONGWEI LIN † AND ZHIYU ZHANG † Abstract. 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 large- image fitting to validate its efficiency and effectiveness. Key words. data fitting, iterative method, T-splines, image fitting AMS subject classifications. 65D07, 65D10, 65D17, 65D18 DOI. 10.1137/120888569 1. Introduction. Data fitting is a fundamental tool in scientific research and engineering applications. With the development of the data capture devices, real objects or models can be easily measured or scanned, outputting highly precise and regular measurement data in larger and larger quantities [2]. This naturally raises the problems of how to fit these larger and larger data sets efficiently, and how to make the fitting representation as compact as possible. Actually, there are two ingredients in solving the data fitting problem. One is the fitting method; the other is the fitting representation. When the data sets are very large, the linear systems that result from data fitting are too large to fit in the main memory-storage systems of usual devices. Therefore, direct-solution techniques such as Cholesky decomposition become impractical, and iteration methods are usually employed in fitting large data sets. On the other hand, spline functions, especially B-splines, are commonly taken as the representation in fitting large data sets, because spline functions can avoid the Runge phenomenon appearing in data fitting with a high degree of polynomials. Nevertheless, the control net of the B-spline surface is a regular grid, so it requires many superfluous control points simply to satisfy the topological structure constraints of the control net, especially in fitting large data sets. To overcome the topological ∗ Submitted to the journal’s Methods and Algorithms for Scientific Computing section August 20, 2012; accepted for publication (in revised form) October 9, 2013; published electronically December 19, 2013. This research was supported by the Natural Science Foundation of China (61379072, 60933008, 60970150). http://www.siam.org/journals/sisc/35-6/88856.html † Department of Mathematics, State Key Lab of CAD&CG, Zhejiang University, Hangzhou 310027, China (hwlin@zju.edu.cn, zhangzhiyu@zjucadcg.cn). A3052