Journal of Computer Science and Technology ›› 2022, Vol. 37 ›› Issue (2): 487-504.doi: 10.1007/s11390-020-0183-1

Special Issue: Artificial Intelligence and Pattern Recognition; Computer Graphics and Multimedia

• Regular Paper • Previous Articles    

Conjugate-Gradient Progressive-Iterative Approximation for Loop and Catmull-Clark Subdivision Surface Interpolation

Yusuf Fatihu Hamza1 and Hong-Wei Lin1,2,* (蔺宏伟), Member, CCF        

  1. 1School of Mathematical Science, Zhejiang University, Hangzhou 310027, China
    2State Key Laboratory of CAD & CG, Zhejiang University, Hangzhou 310058, China
  • Received:2019-11-18 Revised:2020-08-14 Accepted:2020-08-20 Online:2022-03-31 Published:2022-03-31
  • Contact: Hong-Wei Lin
  • About author:Hong-Wei Lin received his B.Sc. degree in applied mathematics from the Department of Applied Mathematics at Zhejiang University, Hangzhou, in 1996, and his Ph.D. degree in applied mathematics from Department of Mathematics at Zhejiang University, Hangzhou, in 2004. He is a professor in the School of Mathematical Science and State Key Laboratory of CAD & CG, Zhejiang University, Hangzhou. His current research interests include geometric design, computer graphics, and topology data analysis.
  • Supported by:
    This work is supported by the National Natural Science Foundation of China under Grant Nos. 61872316 and 61932018.

Loop and Catmull-Clark are the most famous approximation subdivision schemes, but their limit surfaces do not interpolate the vertices of the given mesh. Progressive-iterative approximation (PIA) is an efficient method for data interpolation and has a wide range of applications in many fields such as subdivision surface fitting, parametric curve and surface fitting among others. However, the convergence rate of classical PIA is slow. In this paper, we present a new and fast PIA format for constructing interpolation subdivision surface that interpolates the vertices of a mesh with arbitrary topology. The proposed method, named Conjugate-Gradient Progressive-Iterative Approximation (CG-PIA), is based on the Conjugate-Gradient Iterative algorithm and the Progressive Iterative Approximation (PIA) algorithm. The method is presented using Loop and Catmull-Clark subdivision surfaces. CG-PIA preserves the features of the classical PIA method, such as the advantages of both the local and global scheme and resemblance with the given mesh. Moreover, CG-PIA has the following features. 1) It has a faster convergence rate compared with the classical PIA and W-PIA. 2) CG-PIA avoids the selection of weights compared with W-PIA. 3) CG-PIA does not need to modify the subdivision schemes compared with other methods with fairness measure. Numerous examples for Loop and Catmull-Clark subdivision surfaces are provided in this paper to demonstrate the efficiency and effectiveness of CG-PIA.

Key words: progressive-iterative approximation; Loop subdivision; Catmull-Clark subdivision; conjugate-gradient method ;

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