Delaunay meshes (DM) are a special type of triangle mesh where the local Delaunay condition holds everywhere. We present an efficient algorithm to convert an arbitrary manifold triangle mesh M into a Delaunay mesh. We show that the constructed DM has O(Kn) vertices, where n is the number of vertices in M and K is a model-dependent constant. We also develop a novel algorithm to simplify Delaunay meshes, allowing a smooth choice of detail levels. Our methods are conceptually simple, theoretically sound and easy to implement. The DM construction algorithm also scales well due to its O(nK logK) time complexity. Delaunay meshes have many favorable geometric and numerical properties. For example, a DM has exactly the same geometry as the input mesh, and it can be encoded by any mesh data structure. Moreover, the empty geodesic circumcircle property implies that the commonly used cotangent Laplace-Beltrami operator has non-negative weights. Therefore, the existing digital geometry processing algorithms can benefit the numerical stability of DM without changing any codes. We observe that DMs can improve the accuracy of the heat method for computing geodesic distances. Also, popular parameterization techniques, such as discrete harmonic mapping, produce more stable results on the DMs than on the input meshes.

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Given a set of mixtures, blind source separation attempts to retrieve the source signals without or with very little information of the mixing process. We present a geometric approach for blind separation of nonnegative linear mixtures termed <i>facet component analysis</i>. The approach is based on facet identification of the underlying cone structure of the data. Earlier works focus on recovering the cone by locating its vertices (vertex component analysis) based on a mutual sparsity condition which requires each source signal to possess a stand-alone peak in its spectrum. We formulate alternative conditions so that enough data points fall on the facets of a cone instead of accumulating around the vertices. To find a regime of unique solvability, we make use of both geometric and density properties of the data points and develop an efficient facet identification method by combining data classification and linear

Automatically discovering high-level facade structures in unorganized 3D point clouds of urban scenes is crucial for applications like digitalization of real cities. However, this problem is challenging due to poor-quality input data, contaminated with severe missing areas, noise and outliers. This work introduces the concept of adaptive partitioning to automatically derive a flexible and hierarchical representation of 3D urban facades. Our key observation is that urban facades are largely governed by concatenated and/or interlaced grids. Hence, unlike previous automatic facade analysis works which are typically restricted to globally rectilinear grids, we propose to automatically partition the facade in an adaptive manner, in which the splitting direction, the number and location of splitting planes are all adaptively determined. Such an adaptive partition operation is performed recursively to generate a hierarchical representation of the facade. We show that the concept of adaptive partitioning is also applicable to flexible and robust analysis of image facades. We evaluate our method on a dozen of LiDAR scans of various complexity and styles, and the image facades from the eTRIMS database and the Ecole Centrale Paris database. A series of applications that benefit from our approach are also demonstrated.

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.

The medial axis is an important shape representation that finds a wide range of applications in shape analysis. For large-scale shapes of high resolution, a progressive medial axis representation that starts with the lowest resolution and gradually adds more details is desired. In this paper, we propose a fast and robust geometric algorithm that computes progressive medial axes of a large-scale planar shape. The key ingredient of our method is a novel structural analysis of merging medial axes of two planar shapes along a shared boundary. Our method is robust by separating the analysis of topological structure from numerical computation. Our method is also fast and we show that the time complexity of merging two medial axes is O(n log nv), where n is the number of total boundary generators, nv is strictly smaller than n and behaves as a small constant in all our experiments. Experiments on large-scale polygonal data and comparison with state-of-the-art methods show the efficiency and effectiveness of the proposed method.

In this paper, two explicit conversion formulae between triangular and rectangular Bézier patches are derived. Using the formulae, one triangular Bézier patch of degree n can be converted into one rectangular Bézier patch of degree $n \times n$. And one rectangular Bézier patch of degree m × n can be converted into two triangular Bézier patches of degree $m + n$ . Besides, two stable recursive algorithms corresponding to the two conversion formulae are given. Using the algorithms, when converting triangular Bézier patches to rectangular Bézier patches, we can computer the relations between the control points of the two types of patches for any $n\ge2$ based on the relationships for $n=1$. When converting rectangular Bézier patches to triangular Bézier patches, we can computer the relations between the control points of the two types of patches for any $m\ge 2, n \subset N^+ $ and $n \ge 2, m \subset N^+$ based on the relationships for $m=n=1$.

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The huge number of points scanned from pipeline plants make the plant reconstruction very difficult. Traditional cylinder detection methods cannot be applied directly due to the high computational complexity. In this paper, we explore the structural characteristics of point cloud in pipeline plants and define a structure feature. Based on the structure feature, we propose a hierarchical structure detection and decomposition method that reduces the difficult pipeline-plant reconstruction problem in IR3 into a set of simple circle detection problems in IR2. Experiments with industrial applications are presented, which demonstrate the efficiency of the proposed structure detection method.

The ability to directly manipulate an embedded object in the free-form deformation (FFD) method improves controllability. However, the existing solution to this problem involves a pseudo-inverse matrix that requires complicated calculations. This paper solves the problem using a constrained optimization method. We derive the explicit solutions for deforming an object which is to pass through a given target point. For constraints with multiple target points, the proposed solution also involves simple calculations, only requiring solving a system of linear equations.We show that the direct manipulations exhibit the commutative group property, namely commutative, associative, and invertible properties, which further enhance the controllability of FFD. In addition, we show that multiple point constraints can be decomposed into separate manipulations of single point constraints, thus providing the user the freedom of specifying the constraints in any appropriate order.