Efficient EMD and Hilbert spectra computation for 3D geometry processing and analysis via space-filling curve
Empirical Mode Decomposition (EMD) has proved to be an effective and powerful analytical tool for non-stationary time series and starts to exhibit its modeling potential for 3D geometry analysis. Yet, existing EMD-based geometry processing algorithms only concentrate on multi-scale data decomposition by way of computing intrinsic mode functions. More in-depth analytical properties, such as Hilbert spectra, are hard to study for 3D surface signals due to the lack of theoretical and algorithmic tools. This has hindered much more broader penetration of EMD-centric algorithms into various new applications on 3D surface. To tackle this challenge, in this paper we propose a novel and efficient EMD and Hilbert spectra computational scheme for 3D geometry processing and analysis. At the core of our scheme is the strategy of dimensionality reduction via space-filling curve. This strategy transforms the problem of 3D geometry analysis to 1D time series processing, leading to two major advantages. First, the envelope computation is carried out for 1D signal by cubic spline interpolation, which is much faster than existing envelope computation directly over 3D surface. Second, it enables us to calculate Hilbert spectra directly on 3D surface. We could take advantages of Hilbert spectra that contain a wealth of unexploited properties and utilize them as a viable indicator to guide our EMD-based 3D surface processing. Furthermore, to preserve sharp features, we develop a divide-and-conquer scheme of EMD by explicitly separating the feature signals from non-feature signals. Extensive experiments have been carried out to demonstrate that our new EMD and Hilbert spectra based method is both fast and powerful for 3D surface processing and analysis.
KeywordsEmpirical mode decomposition Hilbert spectra Space-filling curve Surface processing
We are thankful to Dr. Topraj Gurung and Dr. Mark Luffel for discussing and providing the code of Hamilton cycle generation. This work is supported in part by National Science Foundation of USA (IIS-0949467, IIS-1047715, and IIS-1049448), National Natural Science Foundation of China (No. 61190120, 61190121, 61190125, 61202261), China Postdoctoral Science Foundation (No. 2014M560875), Scientific and Technological Development Plan of Jilin Province (No. 20130522113JH). Models are courtesy of AIM@SHAPE Repository.
- 4.Belyaev, A., Ohtake, Y.: A comparison of mesh smoothing methods. In: Israel-Korea bi-national conference on geometric modeling and computer graphics, pp. 83–87 (2003)Google Scholar
- 10.Guskov, I., Sweldens, W., Schröder, P.: Multiresolution signal processing for meshes. In: SIGGRAPH’99, pp. 325–334 (1999)Google Scholar
- 13.Huang, N.E., Shen, Z., Long, S.R., Wu, M.C., Shih, H.H., Zheng, Q., Yen, N.C., Tung, C.C., Liu, H.H.: The empirical mode decomposition and the hilbert spectrum for nonlinear and non-stationary time series analysis. Proc. R. Soc. Lond. A 454(1971), 903–995 (1998)Google Scholar
- 18.Qin, X., Hong, C.X., Zhang, S.Q., Wang, W.H.: Emd based fairing algorithm for mesh surface. In: CAD/Graphics ’09., pp. 606–609 (2009)Google Scholar
- 21.Schröder, P., Sweldens, W.: Spherical wavelets: efficiently representing functions on the sphere. In: SIGGRAPH ’95, pp. 161–172 (1995)Google Scholar
- 22.Solomon, J., Crane, K., Butscher, A., Wojtan, C.: A general framework for bilateral and mean shift filtering. CoRR abs/1405.4734 (2014)Google Scholar
- 24.Taubin, G.: A signal processing approach to fair surface design. In: SIGGRAPH, pp. 351–358 (1995)Google Scholar
- 25.Taubin, G.: Constructing hamiltonian triangle strips on quadrilateral meshes. In: Visualization and Mathematics III, Mathematics and Visualization, pp. 69–91 (2003)Google Scholar
- 32.Yoshizawa, S., Belyaev, A., peter Seidel, H.: Smoothing by example: mesh denoising by averaging with similarity based weights. In: Proceedings of the IEEE international conference on shape modeling and applications (2006), pp. 38–44. IEEE (2006)Google Scholar