Estimation of Error in Curvature Computation on Multi-Scale Free-Form Surfaces

Authors: Mokhtarian, F.1; Khalili, N.2; Yuen, P.3

Source: International Journal of Computer Vision, Volume 48, Number 2, July 2002 , pp. 131-149(19)

Publisher: Springer

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Abstract:

A novel technique for multi-scale curvature computation on a free-form 3-D surface is presented. This is achieved by convolving local parametrisations of the surface with 2-D Gaussian filters iteratively. In our technique, semigeodesic coordinates are constructed at each vertex of the mesh. Smoothing results are shown for 3-D surfaces with different shapes indicating that surface noise is eliminated and surface details are removed gradually. A number of evolution properties of 3-D surfaces are described. Next, the surface Gaussian and mean curvature values are estimated accurately at multiple scales which are then mapped to colours and displayed directly on the surface. The performance of the technique when selecting different directions as an arbitrary direction for the geodesic at each vertex are also presented. The results indicate that the error observed for the estimation of Gaussian and mean curvatures is quite low after only one iteration. Furthermore, as the surface is smoothed iteratively, the error is further reduced. The results also show that the estimation error of Gaussian curvature is less than that of mean curvature. Our experiments demonstrate that estimation of smoothed surface curvatures are very accurate and not affected by the arbitrary direction of the first geodesic line when constructing semigeodesic coordinates. Our technique is independent of the underlying triangulation and is also more efficient than volumetric diffusion techniques since 2-D rather than 3-D convolutions are employed. Finally, the method presented here is a generalisation of the Curvature Scale Space method for 2-D contours. The CSS method has outperformed comparable techniques within the MPEG-7 evaluation framework. As a result, it has been selected for inclusion in the MPEG-7 package of standards.

Keywords: Gaussian and mean curvatures; estimation error; free-form surfaces; local parametrisation; multi-scale description; semigeodesic coordinates

Document Type: Regular Paper

Affiliations: 1: Centre for Vision, Speech, and Signal Processing, School of Electronic Engineering, Information Technology and Mathematics, University of Surrey, Guildford, GU2 7XH, UK. F.Mokhtarian@ee.surrey.ac.uk 2: Centre for Vision, Speech, and Signal Processing, School of Electronic Engineering, Information Technology and Mathematics, University of Surrey, Guildford, GU2 7XH, UK. N.Khalili@ee.surrey.ac.uk 3: Centre for Vision, Speech, and Signal Processing, School of Electronic Engineering, Information Technology and Mathematics, University of Surrey, Guildford, GU2 7XH, UK. P.Yuen@ee.surrey.ac.uk

Publication date: July 1, 2002

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