Spectral processing of point-sampled geometry

ABSTRACT

We present a new framework for processing point-sampled objects using spectral methods. By establishing a concept of local frequencies on geometry, we introduce a versatile spectral representation that provides a rich repository of signal processing algorithms. Based on an adaptive tesselation of the model surface into regularly resampled displacement fields, our method computes a set of windowed Fourier transforms creating a spectral decomposition of the model. Direct analysis and manipulation of the spectral coefficients supports effective filtering, resampling, power spectrum analysis and local error control. Our algorithms operate directly on points and normals, requiring no vertex connectivity information. They are computationally efficient, robust and amenable to hardware acceleration. We demonstrate the performance of our framework on a selection of example applications including noise removal, enhancement, restoration and subsampling.

References

1. 1.Amenta, N., Bern, M., Kamvysselis, M. A New Voronoi-Based Surface Reconstruction Algorithm. SIGGRAPH 98 Conference Proceedings, 1998.
2. 2.Bracewell, R.N. The Fourier Transform and Its Applications. McGraw-Hill, New York, 2nd rev. ed., 1986.
3. 3.Gartner, B. Fast and Robust Smallest Enclosing Balls. Proc. 7th Annual European Symposium on Algorithms (ESA), Lecture Notes in Computer Science 1643, Springer-Verlag, 1999.
4. 4.Desbrun, M., Meyer, M. Schroder, P., Barr, A.H. Implicit Fairing of Irregular Meshes Using Diffusion and Curvature Flow. SIGGRAPH 99 Conference Proceedings, 1999.
5. 5.Dudgeon, D.E., Mersereau, R.M. Multidimensional Digital Signal Processing, Prentice-Hall, 1984.
6. 6.Freeman, H., Shapira, R. Determining the minimal-area encasing rectangle for an arbitrary closed curve. Communications of ACM, 18, 409413, 1975.
7. 7.Gonzalez, R.C., Woods, R.E. Digital Image Processing. Addision- Wesley, 1993.
8. 8.Gortler, S.J., Grzeszczuk, R., Szeliski, R., Cohen, M.F. The Lumigraph. SIGGRAPH 96 Conference Proceedings, 1996.
9. 9.Guskov, I., Sweldens, W., Schroder, P. Multiresolution Signal Processing for Meshes. SIGGRAPH 99 Conference Proceedings, 1999.
10. 10.Guskov, I., Vidimce, K., Sweldens, W., Schroder, P. Normal Meshes. SIGGRAPH 00 Conference Proceedings, 2000.
11. 11.Jain, A.K. Fundamentals of Digital Image Processing, Prentice Hall, 1989.
12. 12.Karni, Z. Gotsman, C. Spectral Compression of Mesh Geometry. SIGGRAPH 00 Conference Proceedings, 2000.
13. 13.Kobbelt, L. Discrete Fairing. Proc. of the 7th IMA Conference on the Mathematics of Surfaces '97, 1997.
14. 14.Lee, A., Moreton, H., Hoppe, H. Displaced Subdivision Surfaces. SIGGRAPH 00 Conference Proceedings, 2000.
15. 15.Levoy, M., Pulli, K., Curless, B., Rusinkiewicz, S., Koller, D., Pereira, L., Ginzton, M., Anderson, S., Davis, J., Ginsberg, J., Shade, J., Fulk, D. The Digital Michelangelo Project: 3D Scanning of Large Statues. SIGGRAPH 00 Conference Proceedings, 2000.
16. 16.Mitchell, D.P. Generating antialiased images at low sampling densities. SIGGRAPH 87 Conference Proceedings, 1987.
17. 17.Papoulis, A. Signal Analysis, McGraw Hill, 1977.
18. 18.Pfister,H., Zwicker, M., van Baar, J., Gross, M. Surfels: Surface Elements as Rendering Primitives. SIGGRAPH 00 Conference Proceedings, 2000.
19. 19.Rusinkiewicz, S., Levoy, M. QSplat: A Multiresolution Point Rendering System for Large Meshes. SIGGRAPH 00 Conference Proceedings, 2000.
20. 20.Taubin, G. A Signal Processing Approach to Fair Surface Design. SIGGRAPH 95 Conference Proceedings, 1995.

Index Terms

• Computing methodologies

  • Artificial intelligence

    • Computer vision

      • Image and video acquisition

        • 3D imaging

  • Computer graphics

    • Animation

    • Shape modeling

• Theory of computation

  • Randomness, geometry and discrete structures

    • Computational geometry