30,268 research outputs found
Fourier Based Fast Multipole Method for the Helmholtz Equation
The fast multipole method (FMM) has had great success in reducing the
computational complexity of solving the boundary integral form of the Helmholtz
equation. We present a formulation of the Helmholtz FMM that uses Fourier basis
functions rather than spherical harmonics. By modifying the transfer function
in the precomputation stage of the FMM, time-critical stages of the algorithm
are accelerated by causing the interpolation operators to become
straightforward applications of fast Fourier transforms, retaining the
diagonality of the transfer function, and providing a simplified error
analysis. Using Fourier analysis, constructive algorithms are derived to a
priori determine an integration quadrature for a given error tolerance. Sharp
error bounds are derived and verified numerically. Various optimizations are
considered to reduce the number of quadrature points and reduce the cost of
computing the transfer function.Comment: 24 pages, 13 figure
From 3D Point Clouds to Pose-Normalised Depth Maps
We consider the problem of generating either pairwise-aligned or pose-normalised depth maps from noisy 3D point clouds in a relatively unrestricted poses. Our system is deployed in a 3D face alignment application and consists of the following four stages: (i) data filtering, (ii) nose tip identification and sub-vertex localisation, (iii) computation of the (relative) face orientation, (iv) generation of either a pose aligned or a pose normalised depth map. We generate an implicit radial basis function (RBF) model of the facial surface and this is employed within all four stages of the process. For example, in stage (ii), construction of novel invariant features is based on sampling this RBF over a set of concentric spheres to give a spherically-sampled RBF (SSR) shape histogram. In stage (iii), a second novel descriptor, called an isoradius contour curvature signal, is defined, which allows rotational alignment to be determined using a simple process of 1D correlation. We test our system on both the University of York (UoY) 3D face dataset and the Face Recognition Grand Challenge (FRGC) 3D data. For the more challenging UoY data, our SSR descriptors significantly outperform three variants of spin images, successfully identifying nose vertices at a rate of 99.6%. Nose localisation performance on the higher quality FRGC data, which has only small pose variations, is 99.9%. Our best system successfully normalises the pose of 3D faces at rates of 99.1% (UoY data) and 99.6% (FRGC data)
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