Smooth Surface Constructions via a Higher-Order Level-Set Method Chandrajit L. Bajaj ∗ CVC, Department of Computer Science, Institute for Computational Engineering and Sciences University of Texas, Austin, TX 78712 Guoliang Xu † , Qin Zhang LSEC, Institute of Computational Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing, 100080 Abstract We present a general framework for a higher-order spline level-set (HLS) method and apply this to smooth surface constructions. Starting from a first order energy functional, we derive the general level set formulation, and provide an efficient solution of a second order geometric partial differential equation using a C 2 spline basis. We also present a fast cubic spline interpolation algorithm based on convolution and the Z-transform, which exploits the local relationship of interpolatory cubic spline coefficients with respect to given function data values. We provide two demonstrative smooth surface construction examples of our HLS method. The first is the construction of a smooth surface model (an implicit solvation interface) of bio-molecules in solvent, given their individual atomic coordinates and solvated radii. The second is the smooth surface reconstruction from a cloud of points generated from a 3D surface scanner. Key words: Higher-order spline level-set; Geometric partial differential equation; Smooth Molecular surfaces; Surface reconstruction from Point Clouds. 1 Introduction Given a non-negative function g(x) over a domain Ω ∈ R 3 . Find a surface Γ in Ω, such that the energy functional E(Γ) =  Γ g(x)dx + ǫ  Γ h(x, n)dx (1.1) * Supported in part by NSF grants IIS-0325550, CNS-0540033 and NIH contracts P20-RR020647, R01- GM074258, R01-GM07308. † Support in part by NSFC grant 10371130 and National Key Basic Research Project of China (2004CB318000). 1