Multi-Scale Free-Form Surface Description
and Curvature Estimation
Estimation of Gaussian & Mean Curvatures and Curvature Zero-Crossing Contours
Examples of original surfaces and their smoothed shapes
At the Centre for Vision, Speech, and Signal Processing in the 5* rated Department of Electronic and Electrical Engineering at the University of Surrey, we have developed a novel technique for Multi-Scale Shape Description of free-form 3-D surfaces. The work has been carried out by Dr Nasser Khalili and Peter Yuen and has been supervised by Dr Farzin Mokhtarian. The figures above show several test objects and the final results of their multi-scale description.
Complete triangulated models of 3-D objects were constructed through automatic fusion of range images and used in our experiments. Diffusion of the surface is acheived through convolutions of local parametrizations of the surface with a 2-D Gaussian filter. Our method for local parametrization makes use of semigeodesic or goedesic polar coordinates as a natural and efficient way of sampling the local surface shape. The smoothing eliminates surface noise and small surface detail gradually. This process is repeated at each point of the surface and the new point positions after filtering define the smoothed surface. To achieve multi-scale description of a 3-D surface, the surface is then smoothed iteratively. This process is equivalent to heat diffusion of the surface. The theory described above is the generalization of the curvature scale space method to 3-D surfaces, and has been applied to a number of 3-D triangulated meshes. The curvature scale space shape descriptor has been selected for MPEG-7 standardization.
Related WorkRelated work on multi-scale surface smoothing includes:
(1) Volumetric smoothing (Koenderink, Solid Shape, MIT press, 1990)
(2) Level-set methods (Sethian, Level-set methods, Cambridge University press, 1996)
(3) Mesh-based smoothing (Taubin, ICCV'95 and ECCV'96)
The main problem with (1) and (2) is that they lack local support, and can not be used for matching-with-occlusion problems since the entire object must be present, and it should not be close to any other objects. They also add an extra dimension to the initial inherently 2-D problem which renders it less efficient. The major problem with (3) is that the smoothing process is very much influenced by the structure of the underlying mesh.
Gaussian and Mean Curvatures
The Gaussian and mean curvature values of surface points can also be estimated at multiple scales. This is achieved through convolutions of the first and second partial derivatives of the Gaussian with local parametrizations of the surface. As a result, our technique integrates surface smoothing and curvature estimation into the same process, and renders curvature estimation more accurate. Curvature values are then mapped to colours or greyscales, using the Visualization ToolKit (VTK), and displayed directly on the surface. When colour is used, red designates maximum or near-maximum values, and blue designates minimum or near-minimum values. Once curvature values have been estimated, curvature zero-crossing contours can be detected and also displayed on the surface using VTK.
Animation of multi-scale description as well as curvature estimation and curvature zero-crossing contour can be observed for each of the following objects:
I: Complete 3D Surfaces
II: Incomplete 3D Surface
- Details of this technique can be found in:1. F. Mokhtarian, N. Khalili, and P. Yuen, ``Multi-Scale Free-Form 3-D Object Recognition using 3-D Models,'' Image and Vision Computing, Volume 19, Number 5, pp. 271-281, 2001.as well as: Proc. BMVC98, BMVC99, SCIA99, CAIP99 and other publications.
2. P. Yuen, F. Mokhtarian, N. Khalili, and J. Illingworth, ``Curvature and Torsion Feature Extraction from Free-Form 3-D Meshes at Multiple Scales,'' IEE Proceedings on Vision, Image, and Signal Processing, Volume 147, Number 5, pp. 454-462, 2000.
3. F. Mokhtarian, N. Khalili, and P. Yuen, ``Curvature Computation on Free-Form 3-D Meshes at Multiple Scales,'' Computer Vision and Image Understanding, Volume 83, No. 2, pp. 118-139, 2001.
4. F. Mokhtarian, N. Khalili, and P. Yuen, ``Estimation of Error in Curvature Computation on Multi-Scale free-Form Surfaces,'' International Journal of Computer Vision, Volume 48, No. 2, pp. 131-149, 2002.
- This work was supported by an EPSRC grant (no. GR/K79727).
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F.Mokhtarian@ee.surrey.ac.uk Last Update: Aug 2003