Remeshing aims to produce a more regular mesh from a given input mesh, while representing the original geometry as accurately as possible. Many existing remeshing methods focus on where to place new mesh vertices; these samples are placed exactly on the input mesh. However, considering the output mesh as a piecewise linear approximation of some geometry, this simple scheme leads to significant systematic error in non-planar regions. Here, we use parameterised meshes and the recent mathematical development of orthogonal approximation using Sobolev-type inner products to develop a novel sampling scheme which allows vertices to lie in space near the input surface, rather than exactly on it. The algorithm requires little extra computational effort and can be readily incorporated into many remeshing approaches. Experimental results show that on average, approximation error can be reduced by 40% with the same number of vertices.

Non-rigid registration of 3D shapes is an essential task of increasing importance as commodity depth sensors become more widely available for scanning dynamic scenes. Non-rigid registration is much more challenging than rigid registration as it estimates a set of local transformations instead of a single global transformation, and hence is prone to the overfitting issue due to underdetermination. The common wisdom in previous methods is to impose an l2-norm regularization on the local transformation differences. However, the l2-norm regularization tends to bias the solution towards outliers and noise with heavy-tailed distribution, which is verified by the poor goodness- of-fit of the Gaussian distribution over transformation differences. On the contrary, Laplacian distribution fits well with the transformation differences, suggesting the use of a sparsity prior. We propose a sparse non-rigid registration (SNR) method with an l1-norm regularized model for transformation estimation, which is effectively solved by an alternate direction method (ADM) under the augmented Lagrangian framework. We also devise a multi-resolution scheme for robust and progressive registration. Results on both public datasets and our scanned datasets show the superiority of our method, particularly in handling large-scale deformations as well as outliers and noise.

Shape registration is fundamental to 3D object acquisition; it is used to fuse scans from multiple views. Existing algorithms mainly utilize geometric information to determine alignment, but this typically results in noticeable misalignment of textures (i.e. surface colors) when using RGB-depth cameras. We address this problem using a novel approach to color-aware registration, which takes both color and geometry into consideration simultaneously. Color information is exploited throughout the pipeline to provide more effective sampling, correspondence and alignment, in particular for surfaces with detailed textures. Our method can furthermore tackle both rigid and non-rigid registration problems (arising, for example, due to small changes in the object during scanning, or camera distortions). We demonstrate that our approach produces significantly better results than previous methods.

This paper deals with the merging problem, i.e. to approximate two adjacent Bézier curves by a single Bézier curve. A novel approach for approximate merging is introduced in the paper by using the constrained optimization method. The basic idea of this method is to find conditions for the precise merging of Bézier curves first, and then compute the constrained optimization solution by moving the control points. “Discrete” coefficient norm in L2 sense and “squared difference integral” norm are used in our method. Continuity at the endpoints of curves are considered in the merging process, and approximate merging with points constraints are also discussed. Further, it is shown that the degree elevation of original Bézier curves will reduce the merging error.

In this paper, we propose the generalized B divided difference, with which the $(k−1)$th derivative of B-spline curves of order $k$ can be obtained directly without the need to compute the first $(k−2)$ derivatives as before. Based on the generalized B divided difference, the necessary and sufficient condition for degree-reducible B-spline curves is presented. Algorithms for degree reduction of B-spline curves are proposed using the constrained optimization methods.