An Efficient Scattered Data Approximation Using Multilevel B-splines Based on Quasi-Interpolants Byung-Gook Lee, Joon Jae Lee Division of Computer and Information Engineering, Dongseo University, Busan 617-716, Korea lbg,jjlee@dongseo.ac.kr Jaechil Yoo Department of Mathematics, Dongeui University, Busan 614-714, Korea yoo@deu.ac.kr Abstract In this paper, we propose an efficient approxima- tion algorithm using multilevel B-splines based on quasi- interpolants. Multilevel technique uses a coarse to fine hi- erarchy to generate a sequence of bicubic B-spline func- tions whose sum approaches the desired interpolation func- tion. To compute a set of control points, quasi-interpolants gives a procedure for deriving local spline approximation methods where a B-spline coefficient only depends on data points taken from the neighborhood of the support corre- sponding the B-spline. Experimental results show that the smooth surface reconstruction with high accuracy can be obtained from a selected set of scattered or dense irregular samples. 1 Introduction Recently, there are many interesting approaches for re- constructing smooth 3D surface from discrete uniform data points or scattered data points. The problem of reconstruct- ing smooth surfaces arises in many fields of science and engineering, and the data sources include measured values such as laser range scanning. The problem of recovering a surface from a set of data is simple in concept but tricky when we get into the detail. Since the real world is made up of continuous surfaces, not discrete points, we want to create a continuous surface from the unorganized data points. The ultimate goal of this pa- per is to find a surface reconstruction method as getting a smooth and high fidelity of 3D surface from a large num- bers of scattered data points. In particular, the description should be sufficiently completed to reconstruct the 3D sur- face within a certain tolerance error, given their relative lo- cations and expected noise. There exist many techniques for surface approximation to improve the approximate continuity and smoothness in handling a large number of data. Tensor product of B- splines surfaces is widely used to approximate rather than to work with other types of approximation because of the advantages inherent in working with tensor products. Ten- sor product guarantees internal continuity if the knot vectors are set properly. Multilevel idea has been adopted to reduce the approxi- mation error. Therefore this paper is based on the multilevel B-splines approximation techniques presented in 1997, the publication of Lee, Wolberg and Shin[7]. They named the schemes multilevel B-splines. In the previous work, Forsey and Bartels[4] developed a surface fitting method which is adaptive on hierarchical spline functions. However, this method cannot deal with scattered data. Lee presented a multilevel B-spline algorithm to fit a uniform bicubic B- spline surface to scatterd data where multilevel or hierar- chy is used to reduce the approximation errors. The method does not guarantee a reasonable global approximation at ini- tial level even though it has an advantage of local process- ing. The splines approximation technique used in this pa- per is quasi-interpolants, first developed by de Boor and Fix[2]. The quasi-interpolants operators were later gener- alized by Lyche and Schumaker[5], and it was their version that used in the alternative surface approximation technique. A quasi-interpolants operator approximates a curve by cal- culating coefficients that are used to weight samplings of the curve to be approximated. Lyche and Schumaker quasi- interpolants operator uses coefficients that are inexpensive to calculate and samplings that are relatively expensive to calculate. It turns out to produce splines approximation with the required accuracy. According to the trend in recent algo- rithms, the hierarchical, multiresolution technique has been used for scattered data and irregular samples. Proceedings of the 5th Int’l Conf. on 3-D Digital Imaging and Modeling (3DIM 2005) 1550-6185/05 $20.00 © 2005 IEEE