Rational RBF-based partition of unity method for efficiently and accurately approximating 3D objects

We consider the problem of reconstructing 3D objects via meshfree interpolation methods. In this framework, we usually deal with large data sets and thus we develop an efficient local scheme via the well-known Partition of Unity (PU) method. The main contribution in this paper consists in constructing the local interpolants for the implicit interpolation by means of Rational Radial Basis Functions (RRBFs). Numerical evidence confirms that the proposed method is particularly performing when 3D objects, or more in general implicit functions defined by scattered data, need to be approximated

Three-Dimensional Time Resolved Lagrangian Flow Field Reconstruction Based on Constrained Least Squares and Stable Radial Basis Function

The three-dimensional Time-Resolved Lagrangian Particle Tracking (3D TR-LPT) technique has recently advanced flow diagnostics by providing high spatiotemporal resolution measurements under the Lagrangian framework. To fully exploit its potential, accurate and robust data processing algorithms are needed. These algorithms are responsible for reconstructing particle trajectories, velocities, and differential quantities (e.g., pressure gradients, strain- and rotation-rate tensors, and coherent structures) from raw LPT data. In this paper, we propose a three-dimensional (3D) divergence-free Lagrangian reconstruction method, where three foundation algorithms -- Constrained Least Squares (CLS), stable Radial Basis Function (RBF-QR), and Partition-of-Unity Method (PUM) -- are integrated into one comprehensive reconstruction strategy. Our method, named CLS-RBF PUM, is able to (i) directly reconstruct flow fields at scattered data points, avoiding Lagrangian-to-Eulerian data conversions; (ii) assimilate the flow diagnostics in Lagrangian and Eulerian descriptions to achieve high-accuracy flow reconstruction; (iii) process large-scale LPT data sets with more than hundreds of thousand particles in two dimensions (2D) or 3D; (iv) enable spatiotemporal super-resolution while imposing physical constraints (e.g., divergence-free for incompressible flows) at arbitrary time and location. Validation based on synthetic and experimental LPT data confirmed that our method can consistently achieve the above advantages with accuracy and robustness.Comment: 30 pages, 11 figure

Stable Computations with Flat Radial Basis Functions Using Vector-Valued Rational Approximations

One commonly finds in applications of smooth radial basis functions (RBFs) that scaling the kernels so they are \u27flat\u27 leads to smaller discretization errors. However, the direct numerical approach for computing with flat RBFs (RBF-Direct) is severely ill-conditioned. We present an algorithm for bypassing this ill-conditioning that is based on a new method for rational approximation (RA) of vector-valued analytic functions with the property that all components of the vector share the same singularities. This new algorithm (RBF-RA) is more accurate, robust, and easier to implement than the Contour-Padé method, which is similarly based on vector-valued rational approximation. In contrast to the stable RBF-QR and RBF-GA algorithms, which are based on finding a better conditioned base in the same RBF-space, the new algorithm can be used with any type of smooth radial kernel, and it is also applicable to a wider range of tasks (including calculating Hermite type implicit RBF-FD stencils). We present a series of numerical experiments demonstrating the effectiveness of this new method for computing RBF interpolants in the flat regime. We also demonstrate the flexibility of the method by using it to compute implicit RBF-FD formulas in the flat regime and then using these for solving Poisson\u27s equation in a 3-D spherical shell

Multimodal decision-level fusion for person authentication

In this paper, the use of clustering algorithms for decision-level data fusion is proposed. Person authentication results coming from several modalities (e.g., still image, speech), are combined by using fuzzy k-means (FKM), fuzzy vector quantization (FVQ) algorithms, and median radial basis function (MRBF) network. The quality measure of the modalities data is used for fuzzification. Two modifications of the FKM and FVQ algorithms, based on a novel fuzzy vector distance definition, are proposed to handle the fuzzy data and utilize the quality measure. Simulations show that fuzzy clustering algorithms have better performance compared to the classical clustering algorithms and other known fusion algorithms. MRBF has better performance especially when two modalities are combined. Moreover, the use of the quality via the proposed modified algorithms increases the performance of the fusion system

Topological shape optimization design of continuum structures via an effective level set method

© 2016 The Author(s). This open access article is distributed under a Creative Commons Attribution (CC-BY) 4.0 license. This paper proposes a new level set method for topological shape optimization of continuum structure using radial basis function (RBF) and discrete wavelet transform (DWT). The boundary of the structure is implicitly represented as the zero level set of a higher-dimensional level set function. The interpolation of the implicit surface using RBF is introduced to decouple the spatial and temporal dependence of the level set function. In doing so, the Hamilton-Jacobi partial differential equation (PDE) that defines the motion of the level set function is transformed into an explicit parametric form, without requiring the direct solution of the complicated PDE using the finite difference method. Therefore, many more efficient gradient-based optimization algorithms can be applied to solve the optimization problem, via updating the expansion coefficients of the interpolant and then evolving the level set function and the boundary. Furthermore, the DWT is employed to handle the full matrix arising from the globally supported RBF interpolation. Several high stiffness but lightweight designs with smooth and clear structural boundaries are optimized and presented. The numerical results show that the proposed method can remarkably increase the efficiency in the topology optimization design of both the 2D and 3D structures