Yutaka Ohtake

A multi-scale approach to 3D scattered data interpolation with compactly supported basis functions

We propose a hierarchical approach to 3D scattered data interpolation with compactly supported basis functions. Our numerical experiments suggest that the approach integrates the best aspects of scattered data fitting with locally and globally supported basis functions. Employing locally supported functions leads to an efficient computational procedure, while a coarse-to-fine hierarchy makes our method insensitive to the density of scattered data and allows us to restore large parts of missed data. Given a point cloud distributed along a surface, we first use spatial down sampling to construct a coarse-to-fine hierarchy of point sets. Then we interpolate the sets starting from the coarsest level. We interpolate a point set of the hierarchy, as an offsetting of the interpolating function computed at the previous level. An original point set and its coarse-to-fine hierarchy of interpolated sets is presented. According to our numerical experiments, the method is essentially faster than the state-of-the-art scattered data approximation with globally supported RBFs (Carr et al., 2001) and much simpler to implement.

Mesh smoothing via mean and median filtering applied to face normals

This paper presents frameworks to extend the mean and median filtering schemes in image processing to smoothing noisy 3D shapes given by triangle meshes. The frameworks consist of the application of the mean and median filters to face normals on triangle meshes and the editing of mesh vertex positions to make them fit the modified normals. We also give a quantitative evaluation of the proposed mesh filtering schemes and compare them with conventional mesh smoothing methods such as Laplacian smoothing flow and mean curvature flow. The quantitative evaluation is performed in error metrics on mesh vertices and normals. Experimental results demonstrate that our mesh mean and median filtering methods are more stable than conventional Laplacian and mean curvature flows. We propose thee new mesh smoothing methods as one possible solution of the oversmoothing problem.

Dual/Primal mesh optimization for polygonized implicit surfaces

A new method for improving polygonizations of implicit surfaces with sharp features is proposed. The method is based on the observation that, given an implicit surface with sharp features, a triangle mesh whose triangles are tangent to the implicit surface at certain inner triangle points gives a better approximation of the implicit surface than the standard marching cubes mesh Lorensen(in our experiments we use VTK marching cubes VTK). First, given an initial triangle mesh, its dual mesh composed of the triangle centroids is considered. Then the dual mesh is modified such that its vertices are placed on the implicit surface and the mesh dual to the modified dual mesh is considered. Finally the vertex positions of that double dual mesh are optimized by minimizing a quadratic energy measuring a deviation of the mesh normals from the implicit surface normals computed at the vertices of the modified dual mesh. In order to achieve an accurate approximation of fine surface features, these basic steps are combined with adaptive mesh subdivision and curvature-weighted vertex resampling. The proposed method outperforms approaches based on the mesh evolution paradigm in speed and accuracy.

Mesh denoising via iterative alpha-trimming and nonlinear diffusion of normals with automatic thresholding

We present two simple yet effective mesh denoising methods. The methods combine a diffusion (repeated local averaging) of mesh normals with histogram-based filtering procedures. According to our experimental results, the methods outperform conventional smoothing techniques in denoising meshes with sharp features.

Dynamic mesh optimization for polygonized implicit surfaces with sharp features

The paper presents a novel approach for accurate polygonization of implicit surfaces with sharp features. The approach is based on mesh evolution towards a given implicit surface with simultaneous control of the mesh vertex positions and mesh normals. Given an initial polygonization of an implicit surface, a mesh evolution process initialized by the polygonization is used. The evolving mesh converges to a limit mesh which delivers a high quality approximation of the implicit surface. For analyzing how close the evolving mesh approachesthe implicit surface we use two error metrics. The metrics measure deviations of the mesh vertices from the implicit surface and deviations of mesh normals from the normals of the implicit surface.

Multi-scale and Adaptive CS-RBFs for Shape Reconstruction from Clouds of Points

We describe a multi-scale approach for interpolation and approximation of a point set surface by compactly supported radial basis functions. Given a set of points scattered over a surface, we first use down-sampling to construct a point set hierarchy. Then starting from the coarsest level, for each level of the hierarchy, we use compactly supported RBFs to approximate the set of points at the level as an offset of the RBF approximation computed at the previous level. A simple RBF centre reduction scheme combined with the multi-scale approach accelerates the latter and allows us to achieve high quality approximations using relatively small number of RBF centres.

Interpolatory subdivision curves via diffusion of normals

We propose a new interpolatory subdivision scheme for generating nice-looking curvature-continuous curves of round shapes. The scheme is based on a diffusion of normals. Given a subdivided polyline, the new polyline vertices inserted at the splitting step are updated in order to fit diffused (averaged with appropriate weights) normals. Although the resulting interpolatory subdivision scheme is nonstationary, nonlinear, and nonuniform from the traditional point of view, the scheme is easy to implement because the same simple geometric procedure for generating new vertices is used at each subdivision step. According to our experiments, the scheme is robust and demonstrates very good convergence properties.