X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/dmems12.git/blobdiff_plain/79572a1f43f7dd9f07ba3d9a96f8a8958e267379..5bd43a57b7e5c885d77aac6dceea8cc19f2cfaea:/dmems12.tex

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-\documentclass{article}
+
+\documentclass[10pt, conference, compsocconf]{IEEEtran}
+%\usepackage{latex8}
+%\usepackage{times}
+\usepackage[utf8]{inputenc}
+%\usepackage[cyr]{aeguill}
+%\usepackage{pstricks,pst-node,pst-text,pst-3d}
+%\usepackage{babel}
+\usepackage{amsmath}
+\usepackage{url}
+\usepackage{graphicx}
+\usepackage{thumbpdf}
+\usepackage{color}
+\usepackage{moreverb}
+\usepackage{commath}
+\usepackage{subfigure}
+%\input{psfig.sty}
+\usepackage{fullpage}
+\usepackage{fancybox}
+
+\usepackage[ruled,lined,linesnumbered]{algorithm2e}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%% LyX specific LaTeX commands.
+\newcommand{\noun}[1]{\textsc{#1}}
+
+\newcommand{\tab}{\ \ \ }
+
+
+
 \begin{document}
-\abstract {
-In this paper we describe....
+
+
+%% \author{\IEEEauthorblockN{Authors Name/s per 1st Affiliation (Author)}
+%% \IEEEauthorblockA{line 1 (of Affiliation): dept. name of organization\\
+%% line 2: name of organization, acronyms acceptable\\
+%% line 3: City, Country\\
+%% line 4: Email: name@xyz.com}
+%% \and
+%% \IEEEauthorblockN{Authors Name/s per 2nd Affiliation (Author)}
+%% \IEEEauthorblockA{line 1 (of Affiliation): dept. name of organization\\
+%% line 2: name of organization, acronyms acceptable\\
+%% line 3: City, Country\\
+%% line 4: Email: name@xyz.com}
+%% }
+
+
+
+\title{Using FPGAs for high speed and real time cantilever deflection estimation}
+\author{\IEEEauthorblockN{RaphaÃ«l Couturier\IEEEauthorrefmark{1}, StÃ©phane Domas\IEEEauthorrefmark{1}, GwenhaÃ«l Goavec-Merou\IEEEauthorrefmark{2} and Michel Lenczner\IEEEauthorrefmark{2}}
+\IEEEauthorblockA{\IEEEauthorrefmark{1}FEMTO-ST, DISC, University of Franche-Comte, Belfort, France\\
+\{raphael.couturier,stephane.domas\}@univ-fcomte.fr}
+\IEEEauthorblockA{\IEEEauthorrefmark{2}FEMTO-ST, Time-Frequency, University of Franche-Comte, BesanÃ§on, France\\
+\{michel.lenczner@utbm.fr,gwenhael.goavec@trabucayre.com}
 }
 
 
+
+
+
+
+\maketitle
+
+\thispagestyle{empty}
+
+\begin{abstract}
+
+  
+
+{\it keywords}: FPGA, cantilever, interferometry.
+\end{abstract}
+
 \section{Introduction}
 
-\section{Conclusion}
+%% blabla +
+%% quelques ref commentÃ©es sur les calculs basÃ©s sur l'interfÃ©romÃ©trie
+
+\section{Measurement principles}
+\label{sec:measure}
+
+\subsection{Architecture}
+\label{sec:archi}
+%% description de l'architecture gÃ©nÃ©rale de l'acquisition d'images
+%% avec au milieu une unitÃ© de traitement dont on ne prÃ©cise pas ce
+%% qu'elle est.
+
+%% image tirÃ©e des expÃ©riences.
+
+\subsection{Cantilever deflection estimation}
+\label{sec:deflest}
+
+As shown on image \ref{img:img-xp}, each cantilever is covered by
+interferometric fringes. The fringes will distort when cantilevers are
+deflected. Estimating the deflection is done by computing this
+distortion. For that, (ref A. Meister + M Favre) proposed a method
+based on computing the phase of the fringes, at the base of each
+cantilever, near the tip, and on the base of the array. They assume
+that a linear relation binds these phases, which can be use to
+"unwrap" the phase at the tip and to determine the deflection.\\
+
+More precisely, segment of pixels are extracted from images taken by a
+high-speed camera. These segments are large enough to cover several
+interferometric fringes and are placed at the base and near the tip of
+the cantilevers. They are called base profile and tip profile in the
+following. Furthermore, a reference profile is taken on the base of
+the cantilever array.
+
+The pixels intensity $I$ (in gray level) of each profile is modelized by :
+
+\begin{equation}
+\label{equ:profile}
+I(x) = ax+b+A.cos(2\pi f.x + \theta)
+\end{equation}
+
+where $x$ is the position of a pixel in its associated segment.
+
+The global method consists in two main sequences. The first one aims
+to determin the frequency $f$ of each profile with an algorithm based
+on spline interpolation (see section \ref{algo-spline}). It also
+computes the coefficient used for unwrapping the phase. The second one
+is the acquisition loop, while which images are taken at regular time
+steps. For each image, the phase $\theta$ of all profiles is computed
+to obtain, after unwrapping, the deflection of cantilevers.
+
+\subsection{Design goals}
+\label{sec:goals}
+
+If we put aside some hardware issues like the speed of the link
+between the camera and the computation unit, the time to deserialize
+pixels and to store them in memory, ... the phase computation is
+obviously the bottle-neck of the whole process. For example, if we
+consider the camera actually in use, an exposition time of 2.5ms for
+$1024\times 1204$ pixels seems the minimum that can be reached. For a
+$10\times 10$ cantilever array, if we neglect the time to extract
+pixels, it implies that computing the deflection of a single
+cantilever should take less than 25$\mu$s, thus 12.5$\mu$s by phase.\\
+
+In fact, this timing is a very hard constraint. Let consider a very
+small programm that initializes twenty million of doubles in memory
+and then does 1000000 cumulated sums on 20 contiguous values
+(experimental profiles have about this size). On an intel Core 2 Duo
+E6650 at 2.33GHz, this program reaches an average of 155Mflops. It
+implies that the phase computation algorithm should not take more than
+$240\times 12.5 = 1937$ floating operations. For integers, it gives
+$3000$ operations.
+
+%% to be continued ...
+
+%% ï¿½ faire : timing de l'algo spline en C avec atan et tout le bordel.
+
+
+
+
+\section{Proposed solution}
+\label{sec:solus}
+
+
+\subsection{FPGA constraints}
+
+%% contraintes imposÃ©es par le FPGA : algo pipeline/parallele, pas d'op math complexe, ...
+
+
+\subsection{Considered algorithms}
+
+Two solutions have been studied to achieve phase computation. The
+original one, proposed by A. Meister and M. Favre, is based on
+interpolation by splines. It allows to compute frequency and
+phase. The second one, detailed in this article, is based on a
+classical least square method but suppose that frequency is already
+known.
+
+\subsubsection{Spline algorithm}
+\label{sec:algo-spline}
+Let consider a profile $P$, that is a segment of $M$ pixels with an
+intensity in gray levels. Let call $I(x)$ the intensity of profile in $x
+\in [0,M[$. 
+
+At first, only $M$ values of $I$ are known, for $x = 0, 1,
+\ldots,M-1$. A normalisation allows to scale known intensities into
+$[-1,1]$. We compute splines that fit at best these normalised
+intensities. Splines are used to interpolate $N = k\times M$ points
+(typically $k=3$ is sufficient), within $[0,M[$. Let call $x^s$ the
+coordinates of these $N$ points and $I^s$ their intensities.
+
+In order to have the frequency, the mean line $a.x+b$ (see equation \ref{equ:profile}) of $I^s$ is
+computed. Finding intersections of $I^s$ and this line allow to obtain
+the period thus the frequency.
+
+The phase is computed via the equation :
+\begin{equation}
+\theta = atan \left[ \frac{\sum_{i=0}^{N-1} sin(2\pi f x^s_i) \times I^s(x^s_i)}{\sum_{i=0}^{N-1} cos(2\pi f x^s_i) \times I^s(x^s_i)} \right]
+\end{equation}
+
+Two things can be noticed. Firstly, the frequency could also be
+obtained using the derivates of spline equations, which only implies
+to solve quadratic equations. Secondly, frequency of each profile is
+computed a single time, before the acquisition loop. Thus, $sin(2\pi f
+x^s_i)$ and $cos(2\pi f x^s_i)$ could also be computed before the loop, which leads to a
+much faster computation of $\theta$.
+
+\subsubsection{Least square algorithm}
+
+Assuming that we compute the phase during the acquisition loop,
+equation \ref{equ:profile} has only 4 parameters :$a, b, A$, and
+$\theta$, $f$ and $x$ being already known. Since $I$ is non-linear, a
+least square method based an Gauss-newton algorithm must be used to
+determine these four parameters. Since it is an iterative process
+ending with a convergence criterion, it is obvious that it is not
+particularly adapted to our design goals.
+
+Fortunatly, it is quite simple to reduce the number of parameters to
+only $\theta$. Let $x^p$ be the coordinates of pixels in a segment of
+size $M$. Thus, $x^p = 0, 1, \ldots, M-1$. Let $I(x^p)$ be their
+intensity. Firstly, we "remove" the slope by computing :
+
+\[I^{corr}(x^p) = I(x^p) - a.x^p - b\]
+
+Since linear equation coefficients are searched, a classical least
+square method can be used to determine $a$ and $b$ :
+
+\[a = \frac{covar(x^p,I(x^p))}{var(x^p)} \]
+
+Assuming an overlined symbol means an average, then :
+
+\[b = \overline{I(x^p)} - a.\overline{{x^p}}\]
+
+Let $A$ be the amplitude of $I^{corr}$, i.e. 
+
+\[A = \frac{max(I^{corr}) - min(I^{corr})}{2}\]
+
+Then, the least square method to find $\theta$ is reduced to search the minimum of :
+
+\[\sum_{i=0}^{M-1} \left[ cos(2\pi f.i + \theta) - \frac{I^{corr}(i)}{A} \right]^2\]
+
+It is equivalent to derivate this expression and to solve the following equation :
+
+\begin{eqnarray*}
+2\left[ cos\theta \sum_{i=0}^{M-1} I^{corr}(i).sin(2\pi f.i) + sin\theta \sum_{i=0}^{M-1} I^{corr}(i).cos(2\pi f.i)\right] \\
+- A\left[ cos2\theta \sum_{i=0}^{M-1} sin(4\pi f.i) + sin2\theta \sum_{i=0}^{M-1} cos(4\pi f.i)\right]   = 0
+\end{eqnarray*}
+
+Several points can be noticed :
+\begin{itemize}
+\item As in the spline method, some parts of this equation can be
+  computed before the acquisition loop. It is the case of sums that do
+  not depend on $\theta$ :
+
+\[ \sum_{i=0}^{M-1} sin(4\pi f.i), \sum_{i=0}^{M-1} cos(4\pi f.i) \] 
+
+\item Lookup tables for $sin(2\pi f.i)$ and $cos(2\pi f.i)$ can also be
+computed.
+
+\item The simplest method to find the good $\theta$ is to discretize
+  $[-\pi,\pi]$ in $N$ steps, and to search which step leads to the
+  result closest to zero. By the way, three other lookup tables can
+  also be computed before the loop :
+
+\[ sin \theta, cos \theta, \left[ cos 2\theta \sum_{i=0}^{M-1} sin(4\pi f.i) + sin 2\theta \sum_{i=0}^{M-1} cos(4\pi f.i)\right] \]
+
+\item This search can be very fast using a dichotomous process in $log_2(N)$ 
+
+\end{itemize}
+
+Finally, the whole summarizes in an algorithm (called LSQ in the following) in two parts, one before and one during the acquisition loop :
+\begin{algorithm}[h]
+\caption{LSQ algorithm - before acquisition loop.}
+\label{alg:lsq-before}
+
+   $M \leftarrow $ number of pixels of the profile\\
+   I[] $\leftarrow $ intensities of pixels\\
+   $f \leftarrow $ frequency of the profile\\
+   $s4i \leftarrow \sum_{i=0}^{M-1} sin(4\pi f.i)$\\
+   $c4i \leftarrow \sum_{i=0}^{M-1} cos(4\pi f.i)$\\
+   $nb_s \leftarrow $ number of discretization steps of $[-\pi,\pi]$\\
+
+   \For{$i=0$ to $nb_s $}{
+     $\theta  \leftarrow -\pi + 2\pi\times \frac{i}{nb_s}$\\
+     lut\_sin[$i$] $\leftarrow sin \theta$\\
+     lut\_cos[$i$] $\leftarrow cos \theta$\\
+     lut\_A[$i$] $\leftarrow cos 2 \theta \times s4i + sin 2 \theta \times c4i$\\
+     lut\_sinfi[$i$] $\leftarrow sin (2\pi f.i)$\\
+     lut\_cosfi[$i$] $\leftarrow cos (2\pi f.i)$\\
+   }
+\end{algorithm}
+
+\begin{algorithm}[h]
+\caption{LSQ algorithm - during acquisition loop.}
+\label{alg:lsq-during}
+
+   $\bar{x} \leftarrow \frac{M-1}{2}$\\
+   $\bar{y} \leftarrow 0$, $x_{var} \leftarrow 0$, $xy_{covar} \leftarrow 0$\\
+   \For{$i=0$ to $M-1$}{
+     $\bar{y} \leftarrow \bar{y} + $ I[$i$]\\
+     $x_{var} \leftarrow x_{var} + (i-\bar{x})^2$\\
+   }
+   $\bar{y} \leftarrow \frac{\bar{y}}{M}$\\
+   \For{$i=0$ to $M-1$}{
+     $xy_{covar} \leftarrow xy_{covar} + (i-\bar{x}) \times (I[i]-\bar{y})$\\
+   }
+   $slope \leftarrow \frac{xy_{covar}}{x_{var}}$\\
+   $start \leftarrow y_{moy} - slope\times \bar{x}$\\
+   \For{$i=0$ to $M-1$}{
+     $I[i] \leftarrow I[i] - start - slope\times i$\tcc*[f]{slope removal}\\
+   }
+   
+   $I_{max} \leftarrow max_i(I[i])$, $I_{min} \leftarrow min_i(I[i])$\\
+   $amp \leftarrow \frac{I_{max}-I_{min}}{2}$\\
+
+   $Is \leftarrow 0$, $Ic \leftarrow 0$\\
+   \For{$i=0$ to $M-1$}{
+     $Is \leftarrow Is + I[i]\times $ lut\_sinfi[$i$]\\
+     $Ic \leftarrow Ic + I[i]\times $ lut\_cosfi[$i$]\\
+   }
+
+   $\theta \leftarrow -\pi$\\
+   $val_1 \leftarrow 2\times \left[ Is.\cos(\theta) + Ic.\sin(\theta) \right] - amp\times \left[ c4i.\sin(2\theta) + s4i.\cos(2\theta) \right]$\\
+   \For{$i=1-n_s$ to $n_s$}{
+     $\theta \leftarrow \frac{i.\pi}{n_s}$\\
+     $val_2 \leftarrow 2\times \left[ Is.\cos(\theta) + Ic.\sin(\theta) \right] - amp\times \left[ c4i.\sin(2\theta) + s4i.\cos(2\theta) \right]$\\
+
+     \lIf{$val_1 < 0$ et $val_2 >= 0$}{
+       $\theta_s \leftarrow \theta - \left[ \frac{val_2}{val_2-val_1}\times \frac{\pi}{n_s} \right]$\\
+     }
+     $val_1 \leftarrow val_2$\\
+   }
+
+\end{algorithm}
+
+
+\subsubsection{Comparison}
+
+\subsection{VDHL design paradigms}
+
+\subsection{VDHL implementation}
+
+\section{Experimental results}
+\label{sec:results}
+
+
+
+
+\section{Conclusion and perspectives}
+
+
+\bibliographystyle{plain}
+\bibliography{biblio}
 
 \end{document}
