Add pre-print for arxiv (#145)

.gitignore

@@ -8,16 +8,21 @@ _version.py
 .pytest_cache
 .tox
 .vscode
+*.aux
 *.c
 *.egg-info
+*.fdb_latexmk
+*.fls
 *.html
 *.ipynb*
+*.log
 *.mat
 *.npy
 *.pdf
 *.png
 *.pyc
 *.so
+*.synctex.gz
 *.tmp
 build
 Coeff_Height_and_Depth_to2190_DTM2006.0

paper/arxiv.tex

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+% Options for packages loaded elsewhere
+\PassOptionsToPackage{unicode}{hyperref}
+\PassOptionsToPackage{hyphens}{url}
+\PassOptionsToPackage{dvipsnames,svgnames,x11names}{xcolor}
+%
+\documentclass[
+]{article}
+\usepackage{amsmath,amssymb}
+\usepackage{lmodern}
+\usepackage{iftex}
+\ifPDFTeX
+  \usepackage[T1]{fontenc}
+  \usepackage[utf8]{inputenc}
+  \usepackage{textcomp} % provide euro and other symbols
+\else % if luatex or xetex
+  \usepackage{unicode-math}
+  \defaultfontfeatures{Scale=MatchLowercase}
+  \defaultfontfeatures[\rmfamily]{Ligatures=TeX,Scale=1}
+\fi
+% Use upquote if available, for straight quotes in verbatim environments
+\IfFileExists{upquote.sty}{\usepackage{upquote}}{}
+\IfFileExists{microtype.sty}{% use microtype if available
+  \usepackage[]{microtype}
+  \UseMicrotypeSet[protrusion]{basicmath} % disable protrusion for tt fonts
+}{}
+\makeatletter
+\@ifundefined{KOMAClassName}{% if non-KOMA class
+  \IfFileExists{parskip.sty}{%
+    \usepackage{parskip}
+  }{% else
+    \setlength{\parindent}{0pt}
+    \setlength{\parskip}{6pt plus 2pt minus 1pt}}
+}{% if KOMA class
+  \KOMAoptions{parskip=half}}
+\makeatother
+\usepackage{xcolor}
+\setlength{\emergencystretch}{3em} % prevent overfull lines
+\providecommand{\tightlist}{%
+  \setlength{\itemsep}{0pt}\setlength{\parskip}{0pt}}
+\setcounter{secnumdepth}{-\maxdimen} % remove section numbering
+\newlength{\cslhangindent}
+\setlength{\cslhangindent}{1.5em}
+\newlength{\csllabelwidth}
+\setlength{\csllabelwidth}{3em}
+\newlength{\cslentryspacingunit} % times entry-spacing
+\setlength{\cslentryspacingunit}{\parskip}
+\newenvironment{CSLReferences}[2] % #1 hanging-ident, #2 entry spacing
+ {% don't indent paragraphs
+  \setlength{\parindent}{0pt}
+  % turn on hanging indent if param 1 is 1
+  \ifodd #1
+  \let\oldpar\par
+  \def\par{\hangindent=\cslhangindent\oldpar}
+  \fi
+  % set entry spacing
+  \setlength{\parskip}{#2\cslentryspacingunit}
+ }%
+ {}
+\usepackage{calc}
+\newcommand{\CSLBlock}[1]{#1\hfill\break}
+\newcommand{\CSLLeftMargin}[1]{\parbox[t]{\csllabelwidth}{#1}}
+\newcommand{\CSLRightInline}[1]{\parbox[t]{\linewidth - \csllabelwidth}{#1}\break}
+\newcommand{\CSLIndent}[1]{\hspace{\cslhangindent}#1}
+\ifLuaTeX
+\usepackage[bidi=basic]{babel}
+\else
+\usepackage[bidi=default]{babel}
+\fi
+\babelprovide[main,import]{american}
+% get rid of language-specific shorthands (see #6817):
+\let\LanguageShortHands\languageshorthands
+\def\languageshorthands#1{}
+\ifLuaTeX
+  \usepackage{selnolig}  % disable illegal ligatures
+\fi
+\IfFileExists{bookmark.sty}{\usepackage{bookmark}}{\usepackage{hyperref}}
+\IfFileExists{xurl.sty}{\usepackage{xurl}}{} % add URL line breaks if available
+\urlstyle{same} % disable monospaced font for URLs
+\hypersetup{
+  pdftitle={SLEPLET: Slepian Scale-Discretised Wavelets in Python},
+  pdfauthor={Patrick J. Roddy},
+  pdflang={en-US},
+  colorlinks=true,
+  linkcolor={Maroon},
+  filecolor={Maroon},
+  citecolor={Blue},
+  urlcolor={Blue},
+  pdfcreator={LaTeX via pandoc}}
+
+\title{SLEPLET: Slepian Scale-Discretised Wavelets in Python}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% Authors and Affiliations
+
+\usepackage[affil-it]{authblk}
+\usepackage{orcidlink}
+%% \renewcommand\Authsep{, }
+\setlength{\affilsep}{1em}
+\author[1%
+  %
+  ]{Patrick J. Roddy%
+    \,\orcidlink{0000-0002-6271-1700}\,%
+    }
+
+\affil[1]{Advanced Research Computing, University College London, UK}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\date{20\textsuperscript{th} April 2023}
+
+\begin{document}
+\maketitle
+
+\hypertarget{summary}{%
+\section{Summary}\label{summary}}
+
+Wavelets are widely used in various disciplines to analyse signals both
+in space and scale. Whilst many fields measure data on manifolds (i.e.,
+the sphere), often data are only observed on a partial region of the
+manifold. Wavelets are a typical approach to data of this form, but the
+wavelet coefficients that overlap with the boundary become contaminated
+and must be removed for accurate analysis. Another approach is to
+estimate the region of missing data and to use existing whole-manifold
+methods for analysis. However, both approaches introduce uncertainty
+into any analysis. Slepian wavelets enable one to work directly with
+only the data present, thus avoiding the problems discussed above.
+Applications of Slepian wavelets to areas of research measuring data on
+the partial sphere include gravitational/magnetic fields in geodesy,
+ground-based measurements in astronomy, measurements of whole-planet
+properties in planetary science, geomagnetism of the Earth, and cosmic
+microwave background analyses.
+
+\hypertarget{statement-of-need}{%
+\section{Statement of Need}\label{statement-of-need}}
+
+Many fields in science and engineering measure data that inherently live
+on non-Euclidean geometries, such as the sphere. Techniques developed in
+the Euclidean setting must be extended to other geometries. Due to
+recent interest in geometric deep learning, analogues of Euclidean
+techniques must also handle general manifolds or graphs. Often, data are
+only observed over partial regions of manifolds, and thus standard
+whole-manifold techniques may not yield accurate predictions. Slepian
+wavelets are designed for datasets like these. Slepian wavelets are
+built upon the eigenfunctions of the Slepian concentration problem of
+the manifold (\protect\hyperlink{ref-Landau1961}{Landau \& Pollak,
+1961}, \protect\hyperlink{ref-Landau1962}{1962};
+\protect\hyperlink{ref-Slepian1961}{Slepian \& Pollak, 1961}): a set of
+bandlimited functions that are maximally concentrated within a given
+region. Wavelets are constructed through a tiling of the Slepian
+harmonic line by leveraging the existing scale-discretised framework
+(\protect\hyperlink{ref-Leistedt2013}{Leistedt, B. et al., 2013};
+\protect\hyperlink{ref-Wiaux2008}{Wiaux et al., 2008}). Whilst these
+wavelets were inspired by spherical datasets, like in cosmology, the
+wavelet construction may be utilised for manifold or graph data.
+
+To the author's knowledge, there is no public software that allows one
+to compute Slepian wavelets (or a similar approach) on the sphere or
+general manifolds/meshes. \texttt{SHTools}
+(\protect\hyperlink{ref-Wieczorek2018}{Wieczorek \& Meschede, 2018}) is
+a \texttt{Python} code used for spherical harmonic transforms, which
+allows one to compute the Slepian functions of the spherical polar cap
+(\protect\hyperlink{ref-Simons2006}{Frederik J. Simons et al., 2006}). A
+series of \texttt{MATLAB} scripts exist in \texttt{slepian\_alpha}
+(\protect\hyperlink{ref-Simons2020}{Frederik J. Simons et al., 2020}),
+which permits the calculation of the Slepian functions on the sphere.
+However, these scripts are very specialised and hard to generalise.
+
+\texttt{SLEPLET} (\protect\hyperlink{ref-Roddy2023a}{Roddy, 2023}) is a
+Python package for the construction of Slepian wavelets in the spherical
+and manifold (via meshes) settings. In contrast to the aforementioned
+codes, \texttt{SLEPLET} handles any spherical region as well as the
+general manifold setting. The API is documented and easily extendible,
+designed in an object-orientated manner. Upon installation,
+\texttt{SLEPLET} comes with two command line interfaces -
+\texttt{sphere} and \texttt{mesh} - that allow one to easily generate
+plots on the sphere and a set of meshes using \texttt{plotly}. Whilst
+these scripts are the primary intended use, \texttt{SLEPLET} may be used
+directly to generate the Slepian coefficients in the spherical/manifold
+setting and use methods to convert these into real space for
+visualisation or other intended purposes. The construction of the
+sifting convolution (\protect\hyperlink{ref-Roddy2021}{Roddy \& McEwen,
+2021}) was required to create Slepian wavelets. As a result, there are
+also many examples of functions on the sphere in harmonic space (rather
+than Slepian) that were used to demonstrate its effectiveness.
+\texttt{SLEPLET} has been used in the development of
+(\protect\hyperlink{ref-Roddy2022a}{Roddy, 2022};
+\protect\hyperlink{ref-Roddy2021}{Roddy \& McEwen, 2021},
+\protect\hyperlink{ref-Roddy2022}{2022},
+\protect\hyperlink{ref-Roddy2023}{2023}).
+
+Whilst Slepian wavelets may be trivially computed from a set of Slepian
+functions, the computation of the spherical Slepian functions themselves
+are computationally complex, where the matrix scales as
+\(\mathcal{O}(L^{4})\). Although symmetries of this matrix and the
+spherical harmonics have been exploited, filling in this matrix is
+inherently slow due to the many integrals performed. The matrix is
+filled in parallel in \texttt{Python} using \texttt{concurrent.futures},
+and the spherical harmonic transforms are computed in \texttt{C} using
+\texttt{SSHT}. This may be sped up further by utilising the new
+\texttt{ducc0} backend for \texttt{SSHT}, which may allow for a
+multithreaded solution. Ultimately, the eigenproblem must be solved to
+compute the Slepian functions, requiring sophisticated algorithms to
+balance speed and accuracy. Therefore, to work with high-resolution data
+such as these, one requires high-performance computing methods on
+supercomputers with massive memory and storage. To this end, Slepian
+wavelets may be exploited at present at low resolutions, but further
+work is required for them to be fully scalable.
+
+\hypertarget{acknowledgements}{%
+\section{Acknowledgements}\label{acknowledgements}}
+
+The author would like to thank Jason D. McEwen for his advice and
+guidance on the mathematics behind \texttt{SLEPLET}. Further, the author
+would like to thank Zubair Khalid for providing his \texttt{MATLAB}
+implementation to compute the Slepian functions of a polar cap region,
+as well as the formulation for a limited colatitude-longitude region
+(\protect\hyperlink{ref-Bates2017}{Bates et al., 2017}).
+\texttt{SLEPLET} makes use of several libraries the author would like to
+acknowledge, in particular, \texttt{libigl}
+(\protect\hyperlink{ref-Libigl2017}{Jacobson \& Panozzo, 2017}),
+\texttt{NumPy} (\protect\hyperlink{ref-Harris2020}{Harris et al.,
+2020}), \texttt{plotly} (\protect\hyperlink{ref-Plotly2015}{Inc.,
+2015}), \texttt{SSHT} (\protect\hyperlink{ref-McEwen2011}{McEwen \&
+Wiaux, 2011}), \texttt{S2LET}
+(\protect\hyperlink{ref-Leistedt2013}{Leistedt, B. et al., 2013}).
+
+\hypertarget{references}{%
+\section*{References}\label{references}}
+\addcontentsline{toc}{section}{References}
+
+\hypertarget{refs}{}
+\begin{CSLReferences}{1}{0}
+\leavevmode\vadjust pre{\hypertarget{ref-Bates2017}{}}%
+Bates, A. P., Khalid, Z., \& Kennedy, R. A. (2017). Slepian
+spatial-spectral concentration problem on the sphere: Analytical
+formulation for limited colatitude-longitude spatial region. \emph{IEEE
+Transactions on Signal Processing}, \emph{65}(6), 1527--1537.
+\url{https://doi.org/10.1109/TSP.2016.2646668}
+
+\leavevmode\vadjust pre{\hypertarget{ref-Harris2020}{}}%
+Harris, C. R., Millman, K. J., Walt, S. J. van der, Gommers, R.,
+Virtanen, P., Cournapeau, D., Wieser, E., Taylor, J., Berg, S., Smith,
+N. J., Kern, R., Picus, M., Hoyer, S., Kerkwijk, M. H. van, Brett, M.,
+Haldane, A., Río, J. F. del, Wiebe, M., Peterson, P., \ldots{} Oliphant,
+T. E. (2020). Array programming with {NumPy}. \emph{Nature},
+\emph{585}(7825), 357--362.
+\url{https://doi.org/10.1038/s41586-020-2649-2}
+
+\leavevmode\vadjust pre{\hypertarget{ref-Plotly2015}{}}%
+Inc., P. T. (2015). \emph{Collaborative data science}.
+\url{https://plot.ly}
+
+\leavevmode\vadjust pre{\hypertarget{ref-Libigl2017}{}}%
+Jacobson, A., \& Panozzo, D. (2017). Libigl: Prototyping geometry
+processing research in {C}++. \emph{SIGGRAPH Asia 2017 Courses}.
+\url{https://doi.org/10.1145/3134472.3134497}
+
+\leavevmode\vadjust pre{\hypertarget{ref-Landau1961}{}}%
+Landau, H. J., \& Pollak, H. O. (1961). Prolate spheroidal wave
+functions, {F}ourier analysis and uncertainty -- {II}. \emph{The Bell
+System Technical Journal}, \emph{40}(1), 65--84.
+\url{https://doi.org/10.1002/j.1538-7305.1961.tb03977.x}
+
+\leavevmode\vadjust pre{\hypertarget{ref-Landau1962}{}}%
+Landau, H. J., \& Pollak, H. O. (1962). Prolate spheroidal wave
+functions, {F}ourier analysis and uncertainty -- {III}: The dimension of
+the space of essentially time- and band-limited signals. \emph{The Bell
+System Technical Journal}, \emph{41}(4), 1295--1336.
+\url{https://doi.org/10.1002/j.1538-7305.1962.tb03279.x}
+
+\leavevmode\vadjust pre{\hypertarget{ref-Leistedt2013}{}}%
+Leistedt, B., McEwen, J. D., Vandergheynst, P., \& Wiaux, Y. (2013).
+S2LET: A code to perform fast wavelet analysis on the sphere.
+\emph{Astronomy \& Astrophysics}, \emph{558}, A128.
+\url{https://doi.org/10.1051/0004-6361/201220729}
+
+\leavevmode\vadjust pre{\hypertarget{ref-McEwen2011}{}}%
+McEwen, J. D., \& Wiaux, Y. (2011). A novel sampling theorem on the
+sphere. \emph{IEEE Transactions on Signal Processing}, \emph{59}(12),
+5876--5887. \url{https://doi.org/10.1109/TSP.2011.2166394}
+
+\leavevmode\vadjust pre{\hypertarget{ref-Roddy2022a}{}}%
+Roddy, P. J. (2022). \emph{Slepian wavelets for the analysis of
+incomplete data on manifolds} {[}PhD thesis, UCL (University College
+London){]}. \url{https://paddyroddy.github.io/thesis}
+
+\leavevmode\vadjust pre{\hypertarget{ref-Roddy2023a}{}}%
+Roddy, P. J. (2023). \emph{SLEPLET: {S}lepian scale-discretised wavelets
+in {P}ython}. \url{https://doi.org/10.5281/zenodo.7268074}
+
+\leavevmode\vadjust pre{\hypertarget{ref-Roddy2021}{}}%
+Roddy, P. J., \& McEwen, J. D. (2021). Sifting convolution on the
+sphere. \emph{IEEE Signal Processing Letters}, \emph{28}, 304--308.
+\url{https://doi.org/10.1109/LSP.2021.3050961}
+
+\leavevmode\vadjust pre{\hypertarget{ref-Roddy2022}{}}%
+Roddy, P. J., \& McEwen, J. D. (2022). Slepian scale-discretised
+wavelets on the sphere. \emph{IEEE Transactions on Signal Processing},
+\emph{70}, 6142--6153. \url{https://doi.org/10.1109/TSP.2022.3233309}
+
+\leavevmode\vadjust pre{\hypertarget{ref-Roddy2023}{}}%
+Roddy, P. J., \& McEwen, J. D. (2023). \emph{Slepian scale-discretised
+wavelets on manifolds}. arXiv. \url{https://arxiv.org/abs/2302.06006}
+
+\leavevmode\vadjust pre{\hypertarget{ref-Simons2006}{}}%
+Simons, Frederik J., Dahlen, F. A., \& Wieczorek, M. A. (2006).
+Spatiospectral concentration on a sphere. \emph{SIAM Review},
+\emph{48}(3), 504--536. \url{https://doi.org/10.1137/S0036144504445765}
+
+\leavevmode\vadjust pre{\hypertarget{ref-Simons2020}{}}%
+Simons, Frederik J., Harig, C., Plattner, A., Hippel, M. von, \& Albert.
+(2020). \emph{{slepian\_alpha}}.
+\url{https://doi.org/10.5281/zenodo.4085210}
+
+\leavevmode\vadjust pre{\hypertarget{ref-Slepian1961}{}}%
+Slepian, D., \& Pollak, H. O. (1961). Prolate spheroidal wave functions,
+{F}ourier analysis and uncertainty -- {I}. \emph{The Bell System
+Technical Journal}, \emph{40}(1), 43--63.
+\url{https://doi.org/10.1002/j.1538-7305.1961.tb03976.x}
+
+\leavevmode\vadjust pre{\hypertarget{ref-Wiaux2008}{}}%
+Wiaux, Y., McEwen, J. D., Vandergheynst, P., \& Blanc, O. (2008). Exact
+reconstruction with directional wavelets on the sphere. \emph{Monthly
+Notices of the Royal Astronomical Society}, \emph{388}(2), 770--788.
+\url{https://doi.org/10.1111/j.1365-2966.2008.13448.x}
+
+\leavevmode\vadjust pre{\hypertarget{ref-Wieczorek2018}{}}%
+Wieczorek, M. A., \& Meschede, M. (2018). {SHTools}: Tools for working
+with spherical harmonics. \emph{Geochemistry, Geophysics, Geosystems},
+\emph{19}(8), 2574--2592. \url{https://doi.org/10.1029/2018GC007529}
+
+\end{CSLReferences}
+
+\end{document}