Adaptive Thinning for Terrain Modelling and Image Compression
Adaptive thinning algorithms are greedy point removal schemes for bivariate scattered data sets with corresponding function values, where the points are recursively removed according to some data-dependent criterion. Each subset of points, together with its function values, defines a linear spline over its Delaunay triangulation. The basic criterion for the removal of the next point is to minimise the error between the resulting linear spline at the bivariate data points and the original function values. This leads to a hierarchy of linear splines of coarser and coarser resolutions.
This paper surveys the various removal strategies developed in our earlier papers, and the application of adaptive thinning to terrain modelling and to image compression. In our image test examples, we found that our thinning scheme, adapted to diminish the least squares error, combined with a post-processing least squares optimisation and a customised coding scheme, often gives better or comparable results to the wavelet-based scheme SPIHT.
KeywordsImage Compression Delaunay Triangulation Scattered Data Compression Scheme Pixel Position
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- 1.Å. Björck. Numerical Methods for Least Squares Problems, SIAM, Philadelphia, 1996.Google Scholar
- 2.T. H. Cormen, C. E. Leiserson, R. L. Rivest, and C. Stein. Introduction to Algorithms, 2nd edition. MIT Press, Cambridge, Massachusetts, 2001.Google Scholar
- 3.G. M. Davis and A. Nosratinia. Wavelet-based image coding: an overview, Appl. Comp. Control, Signal & Circuits, B. N. Datta (ed), Birkhauser, 205–269, 1999.Google Scholar
- 4.L. De Floriani, P. Magillo, and E. Puppo. Building and traversing a surface at variable resolution. Proceedings of IEEE Visualization 97:103–110, 1997.Google Scholar
- 5.L. Demaret and A. Iske. Scattered data coding in digital image compression. Curve and Surface Fitting: Saint-Malo 2002, A. Cohen, J.-L. Merrien, and L. L. Schumaker (eds.), Nashboro Press, Brentwood, 107–117, 2003.Google Scholar
- 6.L. Demaret and A. Iske. Advances in digital image compression by adaptive thinning. To appear in the MCFA Annals, Volume III, 2004.Google Scholar
- 7.N. Dyn, M. S. Floater, and A. Iske. Univariate adaptive thinning. Mathematical Methods for Curves and Surfaces: Oslo 2000, T. Lyche and L. L. Schumaker (eds.), Vanderbilt University Press, Nashville, 123–134, 2001.Google Scholar
- 11.R. J. Fowler and J. J. Little. Automatic extraction of irregular network digital terrain models. Computer Graphics 13:199–207, 1979.Google Scholar
- 12.C. Gotsman, S. Gumhold, and L. Kobbelt. Simplification and Compression of 3D Meshes. Tutorials on Multiresolution in Geometric Modelling, A. Iske, E. Quak, and M. S. Floater (eds.), Springer-Verlag, Heidelberg, 319–361, 2002.Google Scholar
- 13.P. S. Heckbert and M. Garland. Survey of surface simplification algorithms. Technical Report, Computer Science Dept., Carnegie Mellon University, 1997.Google Scholar
- 14.D. S. Hochbaum (ed.). Approximation algorithms for NP-hard problems. PWS Publishing Company, Boston, 1997.Google Scholar
- 16.J. Lee. Comparison of existing methods for building triangular irregular network models of terrain from grid digital elevation models. Int. J. of Geographical Information Systems 5(3):267–285, 1991.Google Scholar
- 17.F. P. Preparata and M. I. Shamos. Computational Geometry. Springer, New York, 1988.Google Scholar
- 19.L. L. Schumaker. Triangulation methods. Topics in Multivariate Approximation, C. K. Chui, L. L. Schumaker, and F. Utreras (eds.), Academic Press, New York, 219–232, 1987.Google Scholar
- 21.D. Taubman. High performance scalable image compression with EBCOT, IEEE Trans. on Image Processing, July 2000, 1158–1170, 2000.Google Scholar
- 22.D. Taubman and M. W. Marcellin. JPEG2000: Image Compression Fundamentals, Standards and Practice, Kluwer, Boston, 2002.Google Scholar