Hello community,
here is the log from the commit of package octave-forge-optics for
openSUSE:Factory checked in at 2019-11-27 16:05:00
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Comparing /work/SRC/openSUSE:Factory/octave-forge-optics (Old)
and /work/SRC/openSUSE:Factory/.octave-forge-optics.new.26869 (New)
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Package is "octave-forge-optics"
Wed Nov 27 16:05:00 2019 rev:2 rq:751304 version:0.1.4
Changes:
--------
--- /work/SRC/openSUSE:Factory/octave-forge-optics/octave-forge-optics.changes
2015-06-16 15:13:18.000000000 +0200
+++
/work/SRC/openSUSE:Factory/.octave-forge-optics.new.26869/octave-forge-optics.changes
2019-11-27 16:05:01.941116114 +0100
@@ -1,0 +2,8 @@
+Wed Nov 27 11:47:48 UTC 2019 - Stefan Brüns <[email protected]>
+
+- Update to version 0.1.4:
+ * Two new zernike functions
+ + zernike_osa_ansi_to_mn
+ + zernikes_and_derivatives_cartesian_OSA
+
+-------------------------------------------------------------------
Old:
----
optics-0.1.3.tar.gz
New:
----
optics-0.1.4.tar.gz
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Other differences:
------------------
++++++ octave-forge-optics.spec ++++++
--- /var/tmp/diff_new_pack.Y6u7kL/_old 2019-11-27 16:05:02.473115933 +0100
+++ /var/tmp/diff_new_pack.Y6u7kL/_new 2019-11-27 16:05:02.473115933 +0100
@@ -1,7 +1,7 @@
#
# spec file for package octave-forge-optics
#
-# Copyright (c) 2015 SUSE LINUX GmbH, Nuernberg, Germany.
+# Copyright (c) 2019 SUSE LLC
#
# All modifications and additions to the file contributed by third parties
# remain the property of their copyright owners, unless otherwise agreed
@@ -12,19 +12,19 @@
# license that conforms to the Open Source Definition (Version 1.9)
# published by the Open Source Initiative.
-# Please submit bugfixes or comments via http://bugs.opensuse.org/
+# Please submit bugfixes or comments via https://bugs.opensuse.org/
#
%define octpkg optics
Name: octave-forge-%{octpkg}
-Version: 0.1.3
+Version: 0.1.4
Release: 0
Summary: Functions covering various aspects of optics for Octave
-License: GPL-3.0+
+License: GPL-3.0-or-later
Group: Productivity/Scientific/Math
-Url: http://octave.sourceforge.net
-Source0:
http://downloads.sourceforge.net/octave/%{octpkg}-%{version}.tar.gz
+URL: https://octave.sourceforge.io
+Source0:
https://downloads.sourceforge.net/octave/%{octpkg}-%{version}.tar.gz
BuildArch: noarch
BuildRequires: octave-devel
Requires: octave-cli >= 3.2.0
++++++ optics-0.1.3.tar.gz -> optics-0.1.4.tar.gz ++++++
diff -urN '--exclude=CVS' '--exclude=.cvsignore' '--exclude=.svn'
'--exclude=.svnignore' old/optics-0.1.3/DESCRIPTION new/optics-0.1.4/DESCRIPTION
--- old/optics-0.1.3/DESCRIPTION 2015-05-06 19:16:52.799576264 +0200
+++ new/optics-0.1.4/DESCRIPTION 2019-09-20 15:58:52.198839838 +0200
@@ -1,8 +1,8 @@
Name: optics
-Version: 0.1.3
-Date: 2015-05-06
-Author: Martin Vogel <[email protected]>, Andreas Weber
<[email protected]>
-Maintainer: Andreas Weber <[email protected]>, Martin Vogel
<[email protected]>
+Version: 0.1.4
+Date: 2019-09-12
+Author: Martin Vogel <[email protected]>, Ramom Flores
<[email protected]>, Andreas Weber <[email protected]>
+Maintainer: Andreas Weber <[email protected]>
Title: Optics
Description: Functions covering various aspects of optics
Categories: physics
diff -urN '--exclude=CVS' '--exclude=.cvsignore' '--exclude=.svn'
'--exclude=.svnignore' old/optics-0.1.3/INDEX new/optics-0.1.4/INDEX
--- old/optics-0.1.3/INDEX 2015-05-06 19:16:52.799576264 +0200
+++ new/optics-0.1.4/INDEX 2019-09-20 15:58:52.198839838 +0200
@@ -54,3 +54,5 @@
zernike_noll_to_mn
zernike_polar
zernike_R_poly
+ zernike_osa_ansi_to_mn
+ zernikes_and_derivatives_cartesian_OSA
diff -urN '--exclude=CVS' '--exclude=.cvsignore' '--exclude=.svn'
'--exclude=.svnignore' old/optics-0.1.3/NEWS new/optics-0.1.4/NEWS
--- old/optics-0.1.3/NEWS 2015-05-06 19:16:52.799576264 +0200
+++ new/optics-0.1.4/NEWS 2019-09-20 15:58:52.198839838 +0200
@@ -1,3 +1,10 @@
+ Summary of important user-visible changes for optics 0.1.4 (2019/09/12):
+-------------------------------------------------------------------------
+
+ ** Two new zernike functions
+ zernike_osa_ansi_to_mn
+ zernikes_and_derivatives_cartesian_OSA
+
Summary of important user-visible changes for optics 0.1.3 (2015/05/06):
-------------------------------------------------------------------------
diff -urN '--exclude=CVS' '--exclude=.cvsignore' '--exclude=.svn'
'--exclude=.svnignore' old/optics-0.1.3/devel/TODO new/optics-0.1.4/devel/TODO
--- old/optics-0.1.3/devel/TODO 1970-01-01 01:00:00.000000000 +0100
+++ new/optics-0.1.4/devel/TODO 2019-09-20 15:58:52.198839838 +0200
@@ -0,0 +1,2 @@
+* A common API (at least on the image package), is to either plot stuff
+ or return depending on nargout (instead of using a plot_flag)
diff -urN '--exclude=CVS' '--exclude=.cvsignore' '--exclude=.svn'
'--exclude=.svnignore'
old/optics-0.1.3/doc/zernikes_and_derivatives_cartesian_OSA.tex
new/optics-0.1.4/doc/zernikes_and_derivatives_cartesian_OSA.tex
--- old/optics-0.1.3/doc/zernikes_and_derivatives_cartesian_OSA.tex
1970-01-01 01:00:00.000000000 +0100
+++ new/optics-0.1.4/doc/zernikes_and_derivatives_cartesian_OSA.tex
2019-09-20 15:58:52.198839838 +0200
@@ -0,0 +1,356 @@
+%% LyX 2.3.3 created this file. For more info, see http://www.lyx.org/.
+%% Do not edit unless you really know what you are doing.
+\documentclass[english]{article}
+\usepackage[T1]{fontenc}
+\usepackage[latin9]{inputenc}
+\usepackage[a4paper]{geometry}
+\geometry{verbose,tmargin=2.5cm,bmargin=2.5cm,lmargin=2.5cm,rmargin=2.5cm}
+\usepackage{color}
+\usepackage{babel}
+\usepackage{array}
+\usepackage{amsmath}
+\usepackage[unicode=true,pdfusetitle,
+ bookmarks=true,bookmarksnumbered=false,bookmarksopen=false,
+ breaklinks=false,pdfborder={0 0
0},pdfborderstyle={},backref=false,colorlinks=true]
+ {hyperref}
+
+\makeatletter
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% LyX specific LaTeX commands.
+%% Because html converters don't know tabularnewline
+\providecommand{\tabularnewline}{\\}
+
+\makeatother
+
+\begin{document}
+\title{Calculation of the Zernike polynomials and their partial derivatives
+in Cartesian coordinates using the standard indexes OSA/ANSI.}
+\author{Jos� Ramom Flores das Seixas}
+\date{August 4, 2019}
+
+\maketitle
+\tableofcontents{}
+
+\section{Introduction}
+
+The \texttt{zernikes\_and\_derivatives\_cartesian\_OSA} function calculates
+the Zernike polynomials and their partial derivatives in Cartesian
+coordinates. For this purpose, it uses the algorithm described in
+reference \cite{Andersen_2018}. However, it should be noted that
+in the cited article the author use unit-normalized Zernike polynomials
+arranged according to the azimuthal scheme set forth by Rimmer and
+Wyant \cite{Rimmer=000026Wyant_1975}, while to implement the function
+has been used the OSA/ANSI standard notation described in references
+\cite{Thibos_et_al_2002,ANSI-2017}. This means the index scheme is
+different and, moreover, the polynomials are not unit-normalized but
+normalized to $\pi$.
+
+\section{Notation}
+
+Zernike's polynomials are usually ordered by a double-index, $Z_{n}^{m}$,
+being $n$ the radial order and $m$ the angular frequency, both integers.
+Although in programming it is usually utilized a single index, $Z_{j}$.
+
+The scheme used in reference \cite{Andersen_2018}, consider a double-index
+with $n\geq0$ and $0\leq m\leq n$. However, the standard OSA/ANSI\footnote{In
fact, this double index schema is the most usual, and is employed
+by other notations than the standard OSA/ANSI.} uses a double-index with
$n\geq0$ and $-n\leq m\leq n$, as shown
+in the following table:
+\noindent \begin{center}
+\begin{tabular}{|c|c|c|c|>{\centering}p{3cm}|}
+\multicolumn{2}{|c|}{Rimmer\&Wyant schema} & & \multicolumn{2}{c|}{~~~~OSA
standard~~~~}\tabularnewline
+\cline{1-2} \cline{2-2} \cline{4-5} \cline{5-5}
+$\tilde{n}$ & $\tilde{m}$ & & $n$ & $m$\tabularnewline
+\cline{1-2} \cline{2-2} \cline{4-5} \cline{5-5}
+0 & 0 & & 0 & 0\tabularnewline
+\cline{1-2} \cline{2-2} \cline{4-5} \cline{5-5}
+1 & 0, 1 & & 1 & -1, 1\tabularnewline
+\cline{1-2} \cline{2-2} \cline{4-5} \cline{5-5}
+2 & 0, 1, 2 & & 2 & -2, 0, 2\tabularnewline
+\cline{1-2} \cline{2-2} \cline{4-5} \cline{5-5}
+3 & 0, 1, 2, 3 & & 3 & -3, -1, 1, 3\tabularnewline
+\cline{1-2} \cline{2-2} \cline{4-5} \cline{5-5}
+4 & 0, 1, 2, 3, 4 & & 4 & -4, -2, 0, 2, 4\tabularnewline
+\cline{1-2} \cline{2-2} \cline{4-5} \cline{5-5}
+$\vdots$ & $\vdots$ & & $\vdots$ & $\vdots$\tabularnewline
+\cline{1-2} \cline{2-2} \cline{4-5} \cline{5-5}
+\end{tabular}
+\par\end{center}
+
+Therefore, by the way of example, the following polynomials --using
+the two schemes seen-- are equivalent:
+\noindent \begin{center}
+\begin{tabular}{ccc}
+Rimmer\&Wyant schema & & ~~~~OSA standard~~~~\tabularnewline
+\cline{1-1} \cline{3-3}
+$Z_{1}^{0}\left(x,y\right)$ & ~~~~ &
$Z_{1}^{-1}\left(x,y\right)$\tabularnewline
+$Z_{3}^{0}\left(x,y\right)$ & & $Z_{3}^{-3}\left(x,y\right)$\tabularnewline
+$Z_{3}^{1}\left(x,y\right)$ & & $Z_{3}^{-1}\left(x,y\right)$\tabularnewline
+$Z_{4}^{4}\left(x,y\right)$ & & $Z_{4}^{4}\left(x,y\right)$\tabularnewline
+$Z_{8}^{6}\left(x,y\right)$ & & $Z_{8}^{4}\left(x,y\right)$\tabularnewline
+\end{tabular}
+\par\end{center}
+
+Using $\left(\tilde{n},\tilde{m}\right)$ to denote the R\&W indices,
+and $\left(n,m\right)$ those of the OSA scheme, both pairs of indices
+are related as follow:
+\begin{equation}
+n=\tilde{n}\qquad m=2\cdot\tilde{m}-\tilde{n}\label{eq:R=000026W_to_OSA}
+\end{equation}
+
+
+\subsection{Single index}
+
+As said before, it is more convenient to use a simple index to program.
+In the OSA standard, the conversion of indices is
\href{https://en.wikipedia.org/wiki/Zernike_polynomials\#OSA/ANSI_standard_indices}{as
follows}:
+\[
+j=\frac{n\left(n+2\right)+m}{2}\qquad\begin{array}{l}
+n=roundup\left[\dfrac{-3+\sqrt{9+8\,j}}{2}\right]\\
+\\
+m=2\,j-n\left(n+2\right)
+\end{array}
+\]
+
+
+\subsection{U-polynomials}
+
+Employing the notation used in reference \cite{Andersen_2018}, the
+unit-normalized Zernike's polynomials are called $U_{nm}$, and are
+related to the $\pi$-normalized ones as follows:
+\[
+Z_{n}^{m}\left(x,y\right)=N_{nm}\cdot U_{nm}\left(x,y\right)
+\]
+ with
+\[
+N_{nm}=\sqrt{\frac{2\left(n+1\right)}{1+\delta_{m0}}}\qquad\textrm{being}\quad\delta_{m0}=\left\{
\begin{array}{ll}
+1 & m=0\\
+0 & m\neq0
+\end{array}\right.
+\]
+
+
+\section{Algorithm}
+
+It is well known that for high-order Zernike's polynomials, computing
+the values using explicit expressions is not the best strategy, since
+it is inefficient and suffers form large cancellation errors. So,
+several schemes using recurrence relations have been devised.
+
+Reference \cite{Andersen_2018} presents one recurrence relation with
+coefficients that do not depend on radial or azimuthal orders and
+which contains no singularities. In addition, it also presents a recurrence
+relation to compute the partial derivatives in Cartesian coordinates.
+
+\subsection{Recurrence relations in Cartesian coordinates for Zernike
U-polynomials}
+
+The general recurrence relation is
+\[
+U_{\tilde{n},\tilde{m}}=xU_{\tilde{n}-1,\tilde{m}}+yU_{\tilde{n}-1,\tilde{n}-1-\tilde{m}}+xU_{\tilde{n}-1,\tilde{m}-1}-yU_{\tilde{n}-1,\tilde{n}-\tilde{m}}-U_{\tilde{n}-2,\tilde{m}-1}
+\]
+But there are several exceptions:
+\begin{itemize}
+\item for $\tilde{m}=0$$\qquad
U_{\tilde{n},0}=xU_{\tilde{n}-1,0}+yU_{\tilde{n}-1,\tilde{n}-1}$
+\item for $\tilde{m}=\tilde{n}$$\qquad
U_{\tilde{n},\tilde{n}}=xU_{\tilde{n}-1,\tilde{n}-1}-yU_{\tilde{n}-1,0}$
+\item for $\tilde{n}$ odd and $\tilde{m}=\dfrac{\tilde{n}-1}{2}$
+\[
+U_{\tilde{n},\tilde{m}}=yU_{\tilde{n}-1,\tilde{n}-1-\tilde{m}}+xU_{\tilde{n}-1,\tilde{m}-1}-yU_{\tilde{n}-1,\tilde{n}-\tilde{m}}-U_{\tilde{n}-2,\tilde{m}-1}
+\]
+\item for $\tilde{n}$ odd and $\tilde{m}=\dfrac{\tilde{n}-1}{2}+1$
+\[
+U_{\tilde{n},\tilde{m}}=xU_{\tilde{n}-1,\tilde{m}}+yU_{\tilde{n}-1,\tilde{n}-1-\tilde{m}}+yU_{\tilde{n}-1,\tilde{m}-1}-U_{\tilde{n}-2,\tilde{m}-1}
+\]
+\item for $\tilde{n}$ even and $\tilde{m}=\dfrac{\tilde{n}}{2}$
+\[
+U_{\tilde{n},\tilde{m}}=2xU_{\tilde{n}-1,\tilde{m}}+2yU_{\tilde{n}-1,\tilde{m}-1}-U_{\tilde{n}-2,\tilde{m}-1}
+\]
+\end{itemize}
+The starting polynomials being
+\[
+U_{0,0}=1,\quad U_{1,0}=y,\quad U_{1,1}=x\quad.
+\]
+
+
+\subsection{Recurrence relations for the OSA/ANSI scheme}
+
+If we now ``translate'' the previous equations into the OSA scheme,
+we obtain that the general recurrence relation is
+\[
+U_{n.m}=xU_{n-1,m+1}+yU_{n-1,-\left(m+1\right)}+xU_{n-1,m-1}-yU_{n-1,1-m}-U_{n-2,m}
+\]
+the exceptions are given by
+\begin{itemize}
+\item for $m=-n$$\qquad U_{n,-n}=xU_{n-1,1-n}+yU_{n-1,n-1}$
+\item for $m=n$$\qquad U_{n,n}=xU_{n-1,n-1}-yU_{n-1,1-n}$
+\item for $m=-1$$\qquad U_{n,-1}=yU_{n-1,0}+xU_{n-1,-2}-yU_{n-1,2}-U_{n-2,-1}$
+\item for $m=1$$\qquad U_{n,1}=xU_{n-1,2}+yU_{n-1,-2}+xU_{n-1,0}-U_{n-2,1}$
+\item for $m=0$$\qquad U_{n,0}=2xU_{n-1,1}+2yU_{n-1,-1}-U_{n-2,0}$
+\end{itemize}
+and the starting polynomials are
+\[
+U_{0,0}=1,\quad U_{1,-1}=y\quad U_{1,1}=x\quad.
+\]
+
+Note that using the OSA scheme the conditions of the exceptions have
+been simplified, which makes it easier to compute.
+
+\subsubsection{Recurrence relations with a single index}
+
+As mentioned earlier, when programming it is more convenient to use
+a simple index. In this case, recurrence relations are given as follows.
+Let us start now with the exceptions:
+\begin{itemize}
+\item for $m=-n\quad\Rightarrow\quad j=\dfrac{\left(n+1\right)n}{n}$
+\[
+U_{j}=xU_{j-n}+yU_{j-1}
+\]
+\item for $m=n\quad\Rightarrow\quad j=\dfrac{n\left(n+3\right)}{2}$
+\[
+U_{j}=xU_{j-\left(n+1\right)}-yU_{j-2n}
+\]
+\item for $m=-1\quad\Rightarrow\quad j=\dfrac{n\left(n+2\right)-1}{2}$
+\[
+U_{j}=xU_{j-\left(n+1\right)}+yU_{j-n}-yU_{j-\left(n-1\right)}-U_{j-2n}
+\]
+\item for $m=1\quad\Rightarrow\quad j=\dfrac{n\left(n+2\right)+1}{2}$
+\[
+U_{j}=xU_{j-n}+xU_{j-\left(n+1\right)}+yU_{j-\left(n+2\right)}-U_{j-2n}
+\]
+\item for $m=0\quad\Rightarrow\quad j=\dfrac{n\left(n+2\right)}{2}$
+\[
+U_{j}=2xU_{j-n}+2yU_{j-\left(n+1\right)}-U_{j-2n}\,.
+\]
+\end{itemize}
+The general case is
+\[
+U_{j}=xU_{j-n}+yU_{j-\left(n+m+1\right)}+xU_{j-\left(n+1\right)}-yU_{j-\left(m+m\right)}-U_{j-2n}
+\]
+and the starting polynomials are
+\[
+U_{0}=1,\quad U_{1}=y,\quad U_{2}=x\quad.
+\]
+
+
+\subsection{Recurrence relations for Cartesian derivatives for Zernike
U-polynomials}
+
+The general recursive relations for partial derivatives are:
+\[
+\frac{\partial U_{\tilde{n},\tilde{m}}}{\partial
x}=\tilde{n}U_{\tilde{n}-1,\tilde{m}}+\tilde{n}U_{\tilde{n}-1,\tilde{m}-1}+\frac{\partial
U_{\tilde{n}-2,\tilde{m}-1}}{\partial x}
+\]
+and
+\[
+\frac{\partial U_{\tilde{n},\tilde{m}}}{\partial
y}=\tilde{n}U_{\tilde{n}-1}-\tilde{n}U_{\tilde{n}-1,\tilde{n}-\tilde{m}}+\frac{\partial
U_{\tilde{n}-2,\tilde{m}-1}}{\partial y}
+\]
+But, as before, there are several exceptions:
+\begin{itemize}
+\item for $\tilde{m}=0$
+\[
+\frac{\partial U_{\tilde{n},0}}{\partial
x}=\tilde{n}U_{\tilde{n}-1,0}\qquad\frac{\partial U_{\tilde{n},0}}{\partial
y}=\tilde{n}U_{\tilde{n}-1,\tilde{n}-1}
+\]
+\item for $\tilde{m}=\tilde{n}$
+\[
+\frac{\partial U_{\tilde{n},\tilde{n}}}{\partial
x}=-\tilde{n}U_{\tilde{n}-1,\tilde{n}-1}\qquad\frac{\partial
U_{\tilde{n},\tilde{n}}}{\partial y}=-\tilde{n}U_{\tilde{n}-1,0}
+\]
+\item for $\tilde{n}$ odd and $\tilde{m}=\dfrac{\tilde{n}-1}{2}$
+\[
+\frac{\partial U_{\tilde{n},\tilde{m}}}{\partial
x}=\tilde{n}U_{\tilde{n}-1,\tilde{m}-1}+\frac{\partial
U_{\tilde{n}-2,\tilde{m}-1}}{\partial x}\qquad\frac{\partial
U_{\tilde{n},\tilde{m}}}{\partial
y}=\tilde{n}U_{\tilde{n}-1,\tilde{n}-\tilde{m}-1}-\tilde{n}U_{\tilde{n}-1,\tilde{n}-\tilde{m}}+\frac{\partial
U_{\tilde{n}-2,\tilde{m}-1}}{\partial y}
+\]
+\item for $\tilde{n}$ odd and $\tilde{m}=\dfrac{\tilde{n}-1}{2}+1$
+\[
+\frac{\partial U_{\tilde{n},\tilde{m}}}{\partial
x}=\tilde{n}U_{\tilde{n}-1,\tilde{m}}+\tilde{n}U_{\tilde{n}-1,\tilde{m}-1}+\frac{\partial
U_{\tilde{n}-2,\tilde{m}-1}}{\partial x}\qquad\frac{\partial
U_{\tilde{n},\tilde{m}}}{\partial
y}=\tilde{n}U_{\tilde{n}-1,\tilde{n}-\tilde{m}-1}+\frac{\partial
U_{\tilde{n}-2,\tilde{m}-1}}{\partial y}
+\]
+\item for $\tilde{n}$ even and $\tilde{m}=\dfrac{\tilde{n}}{2}$
+\[
+\frac{\partial U_{\tilde{n},\tilde{m}}}{\partial
x}=2\tilde{n}U_{\tilde{n}-1,\tilde{m}}+\frac{\partial
U_{\tilde{n}-2,\tilde{m}-1}}{\partial x}\qquad\frac{\partial
U_{\tilde{n},\tilde{m}}}{\partial
y}=2\tilde{n}U_{\tilde{n}-1,\tilde{n}-\tilde{m}-1}+\frac{\partial
U_{\tilde{n}-2,\tilde{m}-1}}{\partial y}
+\]
+\end{itemize}
+The starting expressions for the Cartesian derivatives being:
+\[
+\frac{\partial U_{0,0}}{\partial x}=\frac{\partial U_{0,0}}{\partial
y}=0,\quad\frac{\partial U_{1,0}}{\partial x}=\frac{\partial U_{1,1}}{\partial
y}=0,\quad\frac{\partial U_{1,1}}{\partial x}=\frac{\partial U_{1,0}}{\partial
y}=1
+\]
+
+
+\subsubsection{Recurrence relations for the OSA/ANSI scheme}
+
+If we now ``translate'' the previous equations into the OSA scheme,
+we obtain that the general recurrence relations for partial derivativess
+are
+\[
+\frac{\partial U_{n,m}}{\partial x}=nU_{n-1,m+1}+nU_{n-1,m-1}+\frac{\partial
U_{n-2,m}}{\partial x}
+\]
+and
+\[
+\frac{\partial U_{n,m}}{\partial
y}=nU_{n-1,-\left(m+1\right)}-nU_{n-1,1-m}+\frac{\partial U_{n-2,m}}{\partial y}
+\]
+the exceptions are give by
+\begin{itemize}
+\item for $m=-n$$\qquad\dfrac{\partial U_{n,-n}}{\partial
x}=nU_{n-1,1-n}\qquad\dfrac{\partial U_{n,-n}}{\partial y}=nU_{n-1,n-1}$
+\item for $m=n$$\qquad\dfrac{\partial U_{n,-n}}{\partial
x}=nU_{n-1,n-1}\qquad\dfrac{\partial U_{n,-n}}{\partial y}=-nU_{n-1,1-n}$
+\item for $m=-1$
+\[
+\dfrac{\partial U_{n,-1}}{\partial x}=nU_{n-1,-2}+\dfrac{\partial
U_{n-2,-1}}{\partial x}\qquad\dfrac{\partial U_{n,-1}}{\partial
y}=nU_{n-1,0}-nU_{n-1,2}+\dfrac{\partial U_{n-2,-1}}{\partial y}
+\]
+\item for $m=1$
+\[
+\dfrac{\partial U_{n,1}}{\partial x}=nU_{n-1,2}+nU_{n-1,0}+\dfrac{\partial
U_{n-2,1}}{\partial x}\qquad\dfrac{\partial U_{n,1}}{\partial
y}=nU_{n-1,-2}+\dfrac{\partial U_{n-2,1}}{\partial y}
+\]
+\item for $m=0$
+\[
+\dfrac{\partial U_{n,0}}{\partial x}=2nU_{n-1,1}+\dfrac{\partial
U_{n-2,0}}{\partial x}\qquad\dfrac{\partial U_{n,0}}{\partial
y}=2nU_{n-1,-1}+\dfrac{\partial U_{n-2,0}}{\partial y}
+\]
+\end{itemize}
+and the starting derivatives are
+\[
+\frac{\partial U_{0,0}}{\partial x}=\frac{\partial U_{0,0}}{\partial
y}=0,\quad\frac{\partial U_{1,-1}}{\partial x}=\frac{\partial U_{1,1}}{\partial
y}=0,\quad\frac{\partial U_{1,1}}{\partial x}=\frac{\partial U_{1,-1}}{\partial
y}=1
+\]
+
+
+\subsubsection{Recurrence relations with a single index}
+
+Let us start now with the exceptions:
+\begin{itemize}
+\item for $m=-n\quad\Rightarrow\quad j=\dfrac{\left(n+1\right)n}{n}$
+\[
+\frac{\partial U_{j}}{\partial x}=nU_{j-n}\qquad\frac{\partial U_{j}}{\partial
y}=nU_{j-1}
+\]
+\item for $m=n\quad\Rightarrow\quad j=\dfrac{n\left(n+3\right)}{2}$
+\[
+\frac{\partial U_{j}}{\partial x}=nU_{j-\left(n+1\right)}\qquad\frac{\partial
U_{j}}{\partial y}=-nU_{j-2n}
+\]
+\item for $m=-1\quad\Rightarrow\quad j=\dfrac{n\left(n+2\right)-1}{2}$
+\[
+\frac{\partial U_{j}}{\partial x}=nU_{j-\left(n+1\right)}+\frac{\partial
U_{j-2n}}{\partial x}\qquad\frac{\partial U_{j}}{\partial
y}=nU_{j-n}-nU_{j-\left(n-1\right)}+\frac{\partial U_{j-2n}}{\partial y}
+\]
+\item for $m=1\quad\Rightarrow\quad j=\dfrac{n\left(n+2\right)+1}{2}$
+\[
+\frac{\partial U_{j}}{\partial
x}=nU_{j-n}+nU_{j-\left(n+1\right)}+\frac{\partial U_{j-2n}}{\partial
x}\qquad\frac{\partial U_{j}}{\partial
y}=nU_{j-\left(n+2\right)}+\frac{\partial U_{j-2n}}{\partial y}
+\]
+\item for $m=0\quad\Rightarrow\quad j=\dfrac{n\left(n+2\right)}{2}$
+\[
+\frac{\partial U_{j}}{\partial x}=2nU_{j-n}+\frac{\partial U_{j-2n}}{\partial
x}\qquad\frac{\partial U_{j}}{\partial
y}=2nU_{j-\left(n+1\right)}+\frac{\partial U_{j-2n}}{\partial y}
+\]
+\end{itemize}
+The general case is
+\[
+\frac{\partial U_{j}}{\partial
x}=nU_{j-n}+nU_{j-\left(n+1\right)}+\frac{\partial U_{j-2n}}{\partial
x}\qquad\frac{\partial U_{j}}{\partial
y}=nU_{j-\left(n+m+1\right)}-nU_{j-\left(n+m\right)}+\frac{\partial
U_{j-2n}}{\partial y}
+\]
+and the starting derivatives are
+
+\[
+\frac{\partial U_{0}}{\partial x}=\frac{\partial U_{0}}{\partial
y}=0,\quad\frac{\partial U_{1}}{\partial x}=\frac{\partial U_{2}}{\partial
y}=0,\quad\frac{\partial U_{2}}{\partial x}=\frac{\partial U_{1}}{\partial y}=1
+\]
+
+\begin{thebibliography}{1}
+\bibitem{Andersen_2018}Andersen T.B.,
\href{https://doi.org/10.1364/OE.26.018878}{Efficient and robust recurrence
relations for the Zernike circle polynomials and their derivatives in Cartesian
coordinates}.
+\emph{Optic Express 26}(15), 18878-18896 (2018).
+
+\bibitem{Rimmer=000026Wyant_1975}Rimmer M.P. \& Wyant J.C.,
\href{https://wp.optics.arizona.edu/jcwyant/wp-content/uploads/sites/13/2016/08/LSI-Analysis.pdf}{Evaluation
of large aberrations using a lateral-shear interferometer having variable
shear}.
+\emph{Appl. Opt. 14}(1), 142--150 (1975).
+
+\bibitem{Thibos_et_al_2002}Thibos, L.N, Applegate, R.A., Schwiegerling,
+J.T. \& Webb, R.,
\href{https://pdfs.semanticscholar.org/6e71/c63afa9960dc6e452f3ae2c8b20e2185ba88.pdf}{Standards
for reporting the optical aberrations of eyes}.
+\emph{Journal of refractive surgery}, \emph{18}(5), S652-S660 (2002).
+
+\bibitem{ANSI-2017}ANSI Z80.28--2017. American National Standards
+of Ophthalmics:methods for reporting optical aberrations of eyes.
+\end{thebibliography}
+
+\end{document}
diff -urN '--exclude=CVS' '--exclude=.cvsignore' '--exclude=.svn'
'--exclude=.svnignore' old/optics-0.1.3/inst/zernike_cartesian.m
new/optics-0.1.4/inst/zernike_cartesian.m
--- old/optics-0.1.3/inst/zernike_cartesian.m 2015-05-06 19:16:52.815576259
+0200
+++ new/optics-0.1.4/inst/zernike_cartesian.m 2019-09-20 15:58:52.218840227
+0200
@@ -61,6 +61,28 @@
%! title (zname)
%! endfor
+%!demo
+%! n=30;
+%! [x,y,z] = sombrero (n);
+%! z += 0.05 * peaks (x./4, y./4);
+%! z += 0.02 * x;
+%! figure
+%! surf (x, y, z)
+%! title ("Original sombrero + peaks + tilt");
+%!
+%! ## approximate sombrero+peaks
+%! d = 10;
+%! x /= d;
+%! y /= d;
+%! max_order = 50;
+%! Z = zernike_cartesian (x, y, max_order, false);
+%! f = Z\z(:);
+%! ## create approximated plot
+%! z_approx = reshape (Z * f, n, n);
+%! figure
+%! surf (10 * x, 10 * y, z_approx)
+%! title ("approximated");
+
%!assert (zernike_cartesian (0, 0, 1), 1)
%!error (zernike_cartesian (0, 0, 0))
%!error (zernike_cartesian (0, 0, pi))
diff -urN '--exclude=CVS' '--exclude=.cvsignore' '--exclude=.svn'
'--exclude=.svnignore' old/optics-0.1.3/inst/zernike_osa_ansi_to_mn.m
new/optics-0.1.4/inst/zernike_osa_ansi_to_mn.m
--- old/optics-0.1.3/inst/zernike_osa_ansi_to_mn.m 1970-01-01
01:00:00.000000000 +0100
+++ new/optics-0.1.4/inst/zernike_osa_ansi_to_mn.m 2019-09-20
15:58:52.218840227 +0200
@@ -0,0 +1,135 @@
+## Copyright (C) 2019 José Ramom Flores das Seixas
+##
<[email protected]>
+##
+## 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/>.
+
+## -*- texinfo -*-
+## @deftypefn {} {[@var{m}, @var{n}] =} zernike_osa_ansi_to_mn (@var{j})
+## Convert OSA/ANSI single-index @var{j} to double index @var{m} (Azimuthal
+## degree) and @var{n} (Radial degree).
+##
+## Example
+## @example
+## [m,n] = zernike_osa_ansi_to_mn(4)
+## @result{} [0, 2]
+## @end example
+##
+## References:
+##
+## @enumerate
+## @item Thibos, L.N, Applegate, R.A., Schwiegerling, J.T. & Webb, R.,
+## Standards for reporting the optical aberrations of eyes. Journal of
+## refractive surgery , 18 (5), S652-S660 (2002).
+## @item
@url{https://en.wikipedia.org/wiki/Zernike_polynomials#OSA/ANSI_standard_indices,
+## OSA/ANSI standard indices of Zernike polynomials}, last retrieved on
+## July 2109.
+## @end enumerate
+##
+## @seealso{zernike_noll_to_mn, zernikes_and_derivatives_cartesian_OSA,
zernike_cartesian, zernike_name, zernike_polar, zernike_R_poly}
+##
+## @end deftypefn
+
+function [m, n] = zernike_osa_ansi_to_mn (j)
+ if (nargin != 1 || nargout != 2)
+ print_usage ();
+ elseif (any (j(:) < 0 | j(:) != fix (j(:))) || ! isvector (j))
+ error ("zernike_osa_ansi_to_mn: j has to be a vector with integers >=0");
+ endif
+ n = ceil((-3+sqrt(9+8*j))/2);
+ m = 2*j - n .* (n+2);
+endfunction
+
+%!test
+%! [m,n]=zernike_osa_ansi_to_mn(0);
+%! assert([m n],[0 0])
+%! [m,n]=zernike_osa_ansi_to_mn(1);
+%! assert([m n],[-1 1])
+%! [m,n]=zernike_osa_ansi_to_mn(2);
+%! assert([m n],[1 1])
+%! [m,n]=zernike_osa_ansi_to_mn(3);
+%! assert([m n],[-2 2])
+%! [m,n]=zernike_osa_ansi_to_mn(4);
+%! assert([m n],[0 2])
+%! [m,n]=zernike_osa_ansi_to_mn(5);
+%! assert([m n],[2 2])
+%! [m,n]=zernike_osa_ansi_to_mn(6);
+%! assert([m n],[-3 3])
+%! [m,n]=zernike_osa_ansi_to_mn(7);
+%! assert([m n],[-1 3])
+%! [m,n]=zernike_osa_ansi_to_mn(8);
+%! assert([m n],[1 3])
+%! [m,n]=zernike_osa_ansi_to_mn(9);
+%! assert([m n],[3 3])
+%! [m,n]=zernike_osa_ansi_to_mn(10);
+%! assert([m n],[-4 4])
+%! [m,n]=zernike_osa_ansi_to_mn(11);
+%! assert([m n],[-2 4])
+%! [m,n]=zernike_osa_ansi_to_mn(12);
+%! assert([m n],[0 4])
+%! [m,n]=zernike_osa_ansi_to_mn(13);
+%! assert([m n],[2 4])
+%! [m,n]=zernike_osa_ansi_to_mn(14);
+%! assert([m n],[4 4])
+%! [m,n]=zernike_osa_ansi_to_mn(15);
+%! assert([m n],[-5 5])
+%! [m,n]=zernike_osa_ansi_to_mn(16);
+%! assert([m n],[-3 5])
+%! [m,n]=zernike_osa_ansi_to_mn(17);
+%! assert([m n],[-1 5])
+%! [m,n]=zernike_osa_ansi_to_mn(18);
+%! assert([m n],[1 5])
+%! [m,n]=zernike_osa_ansi_to_mn(19);
+%! assert([m n],[3 5])
+%! [m,n]=zernike_osa_ansi_to_mn(20);
+%! assert([m n],[5 5])
+%! [m,n]=zernike_osa_ansi_to_mn(21);
+%! assert([m n],[-6 6])
+%! [m,n]=zernike_osa_ansi_to_mn(22);
+%! assert([m n],[-4 6])
+%! [m,n]=zernike_osa_ansi_to_mn(23);
+%! assert([m n],[-2 6])
+%! [m,n]=zernike_osa_ansi_to_mn(24);
+%! assert([m n],[0 6])
+%! [m,n]=zernike_osa_ansi_to_mn(25);
+%! assert([m n],[2 6])
+%! [m,n]=zernike_osa_ansi_to_mn(26);
+%! assert([m n],[4 6])
+%! [m,n]=zernike_osa_ansi_to_mn(27);
+%! assert([m n],[6 6])
+%! [m,n]=zernike_osa_ansi_to_mn(28);
+%! assert([m n],[-7 7])
+%! [m,n]=zernike_osa_ansi_to_mn(29);
+%! assert([m n],[-5 7])
+%! [m,n]=zernike_osa_ansi_to_mn(30);
+%! assert([m n],[-3 7])
+%! [m,n]=zernike_osa_ansi_to_mn(31);
+%! assert([m n],[-1 7])
+%! [m,n]=zernike_osa_ansi_to_mn(32);
+%! assert([m n],[1 7])
+%! [m,n]=zernike_osa_ansi_to_mn(33);
+%! assert([m n],[3 7])
+%! [m,n]=zernike_osa_ansi_to_mn(34);
+%! assert([m n],[5 7])
+%! [m,n]=zernike_osa_ansi_to_mn(35);
+%! assert([m n],[7 7])
+
+## vector test
+%!test
+%! [m,n]=zernike_osa_ansi_to_mn([2 7 15 35]);
+%! assert(m,[1 -1 -5 7])
+%! assert(n,[1 3 5 7])
+
+
+
+
diff -urN '--exclude=CVS' '--exclude=.cvsignore' '--exclude=.svn'
'--exclude=.svnignore'
old/optics-0.1.3/inst/zernikes_and_derivatives_cartesian_OSA.m
new/optics-0.1.4/inst/zernikes_and_derivatives_cartesian_OSA.m
--- old/optics-0.1.3/inst/zernikes_and_derivatives_cartesian_OSA.m
1970-01-01 01:00:00.000000000 +0100
+++ new/optics-0.1.4/inst/zernikes_and_derivatives_cartesian_OSA.m
2019-09-20 15:58:52.222840305 +0200
@@ -0,0 +1,214 @@
+## Copyright (C) 2019 José Ramom Flores das Seixas
+##
<[email protected]>
+##
+## 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/>.
+
+## -*- texinfo -*-
+## @deftypefn {} {[@var{Z}, @var{dZx}, @var{dZy}] =}
zernikes_and_derivarives_cartesian_OSA (@var{x}, @var{y}, @var{n})
+## @deftypefnx {} {[@var{Z}, @var{dZx}, @var{dZy}] =}
zernikes_and_derivarives_cartesian_OSA (@var{x}, @var{y}, @var{n},
@var{nan_zero})
+## Return the cartesian Zernike's pollynomials and its partial
+## derivatives up to radial degree @var{n}, i.e. until Z[@var{n},@var{n}]
+##
+## @var{x} is a matrix with the X coordinates of the points where the Zernike's
+## polynomials and its derivatives are computed.
+## @var{y} is a matrix with the Y coordinates of the same points.
+## @var{n} is an integer with the maximum radial degree desired.
+## @var{nan_zero} is a string that determines the values of polynomial
+## and derived values outside the radio unit circle.
+##
+## Strictly, the polynoms are only defined for 0 <= X²+Y² <= 1.
+## If variable @var{nan_zero} = 'nan', the values of the polynomials for which
+## it is verified that (X²+Y²)>1 are set = NaN.
+## If variable @var{nan_zero} = 'zero', the values of the polynomials for which
+## it is verified that (X²+Y²)>1 are set = 0.
+##
+## @var{Z} is a 3D matrix. Each page contains a 2D matrix with the values of a
+## Zernike's polynomial at the points defined by @var{x} and @var{y}.
+##
+## @var{dZx} is a 3D matrix. Each page contains the values of the
+## partial derivative in x.
+##
+## @var{dZy} is a 3D matrix. Each page contains the values of the
+## partial derivative in y.
+##
+## It should be noted that in standard OSA/ANSI the simple-index j starts
+## at 0, but in octave the indices of the vectors and matrices start at 1.
+## So that page 1 of the 3D Z, dZx and dZy matrices corresponds to the
+## single-index j = 0, and therefore to the double-index m = 0 and n = 0.
+## Page 2 corresponds to j = 1, page 3 --> j = 2, etc.
+##
+## Example
+## @example
+## x = linspace(-1,1,101);
+## [X,Y] = meshgrid(x,x);
+## [Z,dZx,dZy] = zernikes_and_derivatives_cartesian_OSA (X,Y,7,'zero');
+## Z_00 = Z(:,:,1);
+## # Z_00 is a 2D matrix with the values of Zernike's polynomial
+## # with simple-index j = 0, and double-index m = 0 & n = 0.
+## dZx_-24 = dZx(:,:,11);
+## # Z_-44 is a 2D matrix with the values of the partial
+## # derivative in x of Zernike's polynomial with
+## # simple-index j = 10, and double-index m = -4 & n = 4.
+## @end example
+##
+## Run the demo to see a more complete example.
+##
+##
+## Size of @var{x} must be equal size of @var{y}.
+##
+## References:
+##
+## @enumerate
+## @item Andersen T.B., @url{https://doi.org/10.1364/OE.26.018878, "Efficient
+## and robust recurrence relations for the Zernike circle polynomials and
+## their derivatives in Cartesian coordinates"}. Optic Express 26(15),
+## 18878-18896 (2018).
+## @item Thibos, L.N, Applegate, R.A., Schwiegerling, J.T. & Webb, R.,
+## Standards for reporting the optical aberrations of eyes. Journal of
+## refractive surgery, 18(5), S652-S660 (2002).
+## @end enumerate
+##
+## @seealso{zernike_osa_ansi_to_nm, zernike_cartesian, zernike_name,
+## zernike_polar, zernike_R_poly}
+## @end deftypefn
+
+function [Z,dZx,dZy] = zernikes_and_derivatives_cartesian_OSA (x, y, N,
nan_zero = 'NaN');
+ if (nargin < 3 || nargin > 4)
+ print_usage ();
+ elseif (! isscalar (N) || N < 1 || N != fix (N))
+ error ("zernike_and_derivatives_cartesian_OA: Maximum radial degree must
be a integer >=1");
+ elseif (any (size (x) != size (y)))
+ error ("zernike_and_derivatives_cartesian_OA: X and Y must have the same
size");
+ endif
+ r2 = x.^2 + y.^2; out_p = find(r2>1);
+ MaxJ = N*(N+3)/2;
+ U1 = ones(size(x));
+ switch(nan_zero)
+ case "zero"
+ x(out_p) = 0;
+ y(out_p) = 0;
+ U1(out_p) = 0;
+ case "NaN"
+ x(out_p) = NaN;
+ y(out_p) = NaN;
+ U1(out_p) = NaN;
+ endswitch
+ U = zeros ([size(x) (MaxJ+1)]); dUx = U; dUy = U;
+ Z = U; dZx = U; dZy = U;
+ U(:,:,1) = U1; dUx(:,:,1) = U1*0; dUy(:,:,1) = U1*0;
+ Z(:,:,1) = U1; dZx(:,:,1) = U1*0; dZy(:,:,1) = U1*0;
+ U(:,:,2) = y; dUx(:,:,2) = U1*0; dUy(:,:,2) = U1;
+ Z(:,:,2) = 2*y; dZx(:,:,2) = U1*0; dZy(:,:,2) = 2*U1;
+ U(:,:,3) = x; dUx(:,:,3) = U1; dUy(:,:,3) = U1*0;
+ Z(:,:,3) = 2*x; dZx(:,:,3) = 2*U1; dZy(:,:,3) = U1*0;
+ for k = 4:(MaxJ+1)
+ [m, n] = zernike_osa_ansi_to_mn (k-1);
+ switch(m)
+ case -n
+ U(:,:,k) = x.*U(:,:,(k-n))+y.*U(:,:,(k-1));
+ dUx(:,:,k) = n*U(:,:,(k-n));
+ dUy(:,:,k) = n*U(:,:,(k-1));
+ case n
+ U(:,:,k) = x.*U(:,:,(k-(n+1)))-y.*U(:,:,(k-2*n));
+ dUx(:,:,k) = n*U(:,:,(k-(n+1)));
+ dUy(:,:,k) = -n*U(:,:,(k-2*n));
+ case -1
+ U(:,:,k) =
x.*U(:,:,(k-(n+1)))+y.*U(:,:,(k-n))-y.*U(:,:,(k-(n-1)))-...
+ U(:,:,(k-2*n));
+ dUx(:,:,k) = n*U(:,:,(k-(n+1)))+dUx(:,:,(k-2*n));
+ dUy(:,:,k) = n*U(:,:,(k-n))-n*U(:,:,(k-(n-1)))+dUy(:,:,(k-2*n));
+ case 1
+ U(:,:,k) =
x.*U(:,:,(k-n))+x.*U(:,:,(k-(n+1)))+y.*U(:,:,(k-(n+2)))-...
+ U(:,:,(k-2*n));
+ dUx(:,:,k) = n*U(:,:,(k-n))+n*U(:,:,(k-(n+1)))+dUx(:,:,(k-2*n));
+ dUy(:,:,k) = n*U(:,:,(k-(n+2)))+dUy(:,:,(k-2*n));
+ case 0
+ U(:,:,k) = 2*x.*U(:,:,(k-n))+2*y.*U(:,:,(k-(n+1)))-U(:,:,(k-2*n));
+ dUx(:,:,k) = 2*n*U(:,:,(k-n))+dUx(:,:,(k-2*n));
+ dUy(:,:,k) = 2*n*U(:,:,(k-(n+1)))+dUy(:,:,(k-2*n));
+ otherwise
+ U(:,:,k) =
x.*U(:,:,(k-n))+y.*U(:,:,(k-(n+m+1)))+x.*U(:,:,(k-(n+1)))-...
+ y.*U(:,:,(k-(n+m)))-U(:,:,(k-2*n));
+ dUx(:,:,k) = n*U(:,:,(k-n))+n*U(:,:,(k-(n+1)))+dUx(:,:,(k-2*n));
+ dUy(:,:,k) =
n*U(:,:,(k-(n+m+1)))-n*U(:,:,(k-(n+m)))+dUy(:,:,(k-2*n));
+ endswitch
+ Nnm = sqrt(2*(n+1));
+ if m == 0
+ Nnm = Nnm/sqrt(2);
+ endif
+ Z(:,:,k) = Nnm * U(:,:,k);
+ dZx(:,:,k) = Nnm * dUx(:,:,k);
+ dZy(:,:,k) = Nnm * dUy(:,:,k);
+ endfor
+endfunction
+
+%!demo
+%! Nmgr = 7; # Maximum radial degree number, default = 7
+%! Nmodos = ((Nmgr+3)*Nmgr/2)+1; # Mode number, default = 36
+%!
+%! pr = 20; # Pupil radius, mm
+%! # Let us suppose that we obtain the heights of a biconical surface
+%! # for a series of points in the xy plane
+%! Rx = 0.002; # mm⁻¹
+%! px = 0.8;
+%! Ry = 0.001; # mm⁻¹
+%! py = 0.5;
+%! x = pr*(2*rand(200,1) - 1); # mm
+%! y = pr*(2*rand(200,1) - 1); # mm
+%! surface = (Rx*x.^2 + Ry*y.^2)/(1+sqrt(1-(px*(Rx*x).^2+py*(Ry*y).^2)));
+%! sag = max(max(surface));
+%! surface = sag - surface; # surface height, mm
+%!
+%! # Coordinates normalization
+%! xnorm=x/pr;
+%! ynorm=y/pr;
+%! usefuldata = find(sqrt(x.^2+y.^2)<pr); # data inside the pupil circle
+%! ixnorm = xnorm(usefuldata);
+%! iynorm = ynorm(usefuldata);
+%! isurface = surface(usefuldata);
+%!
+%! # Fitting the surface Note that ixnomr and iynorm are
vectors
+%! [Z,dZx,dZy] = zernikes_and_derivatives_cartesian_OSA (ixnorm, iynorm, Nmgr);
+%! Mrec = inv(Z'*Z)*Z'; # LSQ fitting matrix
+%! coefZ = Mrec*isurface; # Fitting Zernike's coefficients
+%!
+%! # Reconstructed surface and partial derivatives
+%! nx = linspace(-1,1,129);
+%! [Xn,Yn] = meshgrid(nx,nx);
+%! [Z,dZx,dZy] = zernikes_and_derivatives_cartesian_OSA (Xn, Yn, Nmgr);
+%! S = zeros(size(Xn)); # Note that Xn and Yn are square
matrices
+%! dSx = zeros(size(Xn));
+%! dSy = zeros(size(Xn));
+%! for m = 1:Nmodos
+%! S = S + Z(:,:,m)*coefZ(m); # Reconstructed surface
+%! dSx = dSx + dZx(:,:,m) * coefZ(m); # Derivative of S with respect
to x
+%! dSy = dSy + dZy(:,:,m) * coefZ(m); # Derivative of S with respect
to y
+%! endfor
+%!
+%! # Plotting the reconstructed surface and their derivatives
+%! nx = linspace(-pr,pr,129);
+%! [Xn,Yn] = meshgrid(nx,nx);
+%! colormap ("default");
+%! subplot (3,2,1:4)
+%! mesh(Xn,Yn,S);
+%! title("Reconstructed surface");
+%! xlabel("x (mm)"); ylabel("y (mm)"); zlabel("z (mm)");
+%! subplot(3,2,5)
+%! imagesc(nx,nx,dSx);
+%! axis square; axis xy; colorbar;
+%! title("Derivative of Surface with respect to x");
+%! subplot(3,2,6)
+%! imagesc(nx,nx,dSy);
+%! axis square; axis xy; colorbar;
+%! title("Derivative of Surface with respect to y");
