B-spline surface reconstruction by inverse subdivisions Khoi NGUYEN-TAN a , Romain RAFFIN b , Marc DANIEL c LSIS, UMR CNRS 6168, 13288 Marseille cedex 09 – (France) a khoi.nguyen-tan@univmed.fr, b romain.raffin@lsis.org c marc.daniel@esil.univmed.fr Cung LE CRePA, PFIEV, DaNang University DaNang, VietNam k.spkt@ud.edu.vn Abstract - This paper presents a method to reconstruct a B-spline surface from a quadrangular mesh, using an inverse Catmull- Clark subdivision. We want to minimize the surface contraction due to the approximating subdivision scheme. We introduce geometrical operations which minimize the impact of the subdivision approximation and can be used in the parametric surface reconstruction. The quality of the method is evaluated by criteria of distances, curvatures or computing time on experimental results. Keywords : Reconstruction, B-spline surface, Catmull-Clark, inverse subdivision, Data compression. I. INTRODUCTION In this paper we present a method to reconstruct a uniform bicubic B-spline surface [7] from a quadrangular mesh using an inverse Catmull-Clark subdivision. As the inverse subdivision can be stopped after each step, different surfaces can be obtained. The higher the number of steps, the better the data reduction, but the larger the reconstruction error. The two main contributions in this paper is an efficient method to reconstruct a uniform bicubic B-spline surface using an inverse Catmull-Clark subdivision scheme (UISS), and the definition of criteria to evaluate the surface reconstruction quality. II. RELATED WORKS IN SURFACE RECONSTRUCTION Reconstructing a parametric surface is a well-known problem. [6] proposes an approximation algorithm using the Catmull-Clark method for a NURBS surface. Based on this work, [8] proposes a method to convert a Catmull-Clark subdivision surface into a Bézier surface. In [9], the authors apply the method in [6] to convert a surface into a NURBS representation. Our method is based on an inverse subdivision to reconstruct a uniform bicubic B-spline surface from a quadrangular initial mesh. As we consider the Catmull-Clark subdivision scheme [1], it converges by definition towards a uniform bicubic B-spline surface (called the limit surface). The control polyhedron of this limit surface is the initial subdivision mesh [2][5]. III. CATMULL-CLARK SUBDIVISION AND INVERSION The main idea of the subdivision scheme is to define a continuous surface (the limit surface) by successive steps on a mesh. As the Catmull-Clark scheme is invertible, one can restore recursively all the previous coarser meshes by using the inverse subdivision scheme. We present here the inverse Catmull-Clark subdivision scheme introduced in [4]. It proposes the formulas to exactly reconstruct vertices of the inverse mesh M k deduced from the vertices of mesh M k+1 . At each inverse subdivision, three distinguishable classes of points are generated (see Figure 1), namely, the vertex points located in the centre of four old face points (Figure 1a), the edge points at the position corresponding of the old edge points (Figure 1b) and the corner points in place corresponding of the face points (Figure 1c). 1) Rules of inverse subdivision scheme Considering the vertex points, the face points and the edge points of quadrangular mesh M k+1 , the relationship between the vertices of quadrangular mesh M k and the corresponding vertices of quadrangular mesh M k+1 are deduced from the previous equations. The inverse vertex points of mesh M k are computed based on formulas in [4] (Figure 1a): ( ) 1 1 1 1 1 , 2 1,2 1 2 1,2 2 2 ,2 1 2 1,2 2 2,2 1 1 1 1 1 2 2,2 2 2 ,2 2 2 ,2 2 2,2 P 4P P +P P +P P P P P 4 k k k k k k ij i j i j i j i j i j k k k k i j i j i j i j + + + + + - - - - - - - - + + + + - - - - = - + + + + + (1) with i=1...m-2, j=1...n-2, where m=(h+3)%2, n=(l+3)%2. The inverse edge points on the top border of mesh M k (except the corner points): 1 1 1 , 2 ,2 1 2 ,2 2 2 ,2 1, P 4P P P P k k k k k ij i j i j i j i j + + + - - + = - - - (2) with i=0, j=1..n-2; n=(l+3)%2. Other point edges of mesh M k are computed similarly. The inverse corner points: For four corner points P 00 , P 0,n-1 , P m-1,0 , P m-1,n-1 of the mesh M k have a valence 2. They are computed based on formula in [4]. Point P 00 is computed as follows: 1 , 2 ,2 , 1 1, 1 1, P 4P P P P k k k k k ij i j ij i j i j + + + + + = - - - with i=0, j=0 (3) Other points are computed in the same way. 978-1-4244-4568-4/09/$25.00 ©2009 IEEE 1