Department of Electrical Engineering
Syed Babar Ali School of Science and Engineering
Lahore University of Management Sciences.

Novel Spherical Harmonic Transform (NSHT) - MATLAB Toolbox

Introduction

NSHT Toolbox is a MATLAB based toolbox for spherical harmonic analysis on the sphere. It implements the novel spherical harmonic transform (NSHT) of a band-limited signal using the optimal number of samples.

Overview

For the accurate representation of a signal on the sphere band-limited at degree $L$ with $L^2$ degrees of freedom in the spectral domain, we developed a new sampling scheme in [1] that requires $L^2$ on the sphere. Thus, the sampling scheme matches the spectral dimensionality of the signal.

For the sampling scheme, we also developed spherical harmonic transform, referred as novel spherical harmonic transform, which is implemented in this toolbox. The spherical harmonic transforms (forward and inverse) allow the computation of transforms with sufficient accuracy and manageable computational complexity and do not require any pre-computation associated with spherical harmonic transforms. The pre-computation is only required to determine the optimal placement of samples on the sphere. Once optimal sample positions are determined, the implementation of SHT does not require any pre-computation.

If you use NSHT toolbox for harmonic analysis which results in publication, please reference the site (http:www.zubairkhalid.org/nsht) and our related academic paper:

Authors and Affiliations

1. Zubair Khalid
   Research School of Engineering, The Australian National University

2. Rodney A. Kennedy
   Research School of Engineering, The Australian National University

3. Jason D. McEwen
   Mullard Space Science Laboratory, University College London
   
   

Download

The source code of the NSHT toolbox can be downloaded from here (version: 0.9b).

We provide this code in the hope that it will be useful, but WITHOUT ANY WARRANTY. If you notice an error or bug or need support, please write to zubair.khalid@anu.edu.au.

Version History

Current version: 0.9b (15 Aug. 2014), Public release of first beta.

License

NSHT toolbox to compute spherical harmonic transforms and carry out harmonic analysis Copyright (C) 2014 Zubair Khalid

This program is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version.

This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.

You should have received a copy of the GNU General Public License along with this program. If not, see http:www.gnu.org/licenses.

Installation Instructions

Once you download the toolbox as nsht.zip, unzip the code to the directory ‘nsht’.

The MATLAB working directory or path should be set for directory ‘nsht’ which contains the ‘src’ directory, file ‘nsht_install.m’ and nsht demos.

For working directory: nsht, execute the file ‘nsht_install.m’. The installation creates a mex function for the computation of associated Legendre polynomials, which significantly improves the computation time. The installation also checks that the pre-computed optimal sample positions are available in the src/Data directory.

Notes: The band-limit supported in the current version (0.9b) is $L=722$. The NSHT is shown to compute sufficiently accurate spherical harmonic transform up to band-limit $L=2048$. However, the optimal sample positions have been pre-computed up to $L=722$.

Installation Requirements: MATLAB with the support of codegen and mex compiler.

Notes on Installation Requirements: For efficient implementation of spherical harmonic transforms, the associated Legendre functions are computed in C. This computation is implemented using MATLAB codegen that generates a C code and mex function from MATLAB code. The mex function is then used for computation. If MATLAB is not supported by mex compiler, the toolbox can still be used. However, it takes more computation time to compute spherical harmonic transform.

Documentation

There is documentation for the MATLAB functions within the source files, which can be accessed through the help or doc command in MATLAB.

Functions

List of MATLAB functions with brief description is provided here.

The documentation for each function in MATLAB can be obtained using the command: doc <function name>.

Demos (Examples)

Accuracy Analysis

We randomly generate the complex valued band-limited test signal $f_{rm t}$ with real and imaginary parts uniformly distributed in the interval $-1,\,1$ over the $L^2$ samples of our sampling scheme. The forward SHT, followed by inverse SHT is applied on the test signal to obtain the reconstructed signal $f_{rm r}$ band-limited at $L$.

We repeat the experiment 10 times and then obtain the average values for the maximum reconstruction error $E_textrm{max}$ and the mean reconstruction error $E_textrm{mean}$ between the test signal $f_{rm t}$ and reconstructed signal $f_{rm r}$. See our paper [1] for details.

Both $E_textrm{max}$ and $E_textrm{max}$ are plotted below for the band-limit in the range $16leq Lleq 1024$.

Computation Times

We measure the computation times, denoted by $tau_I$ and $tau_F$ to carry out the proposed inverse and forward transforms, respectively.

The computation times are recorded using NSHT MATLAB Toolbox running on a machine equipped with 2.6 GHz Intel Core i7 processor and 64 GB of RAM.

The computation times $tau_I$ and $tau_F$ in seconds, averaged over 10 different signals, are plotted below.