-\documentclass[12pt]{article}
+
+\documentclass[10pt, conference, compsocconf]{IEEEtran}
%\usepackage{latex8}
%\usepackage{times}
-\usepackage[latin1]{inputenc}
+\usepackage[utf8]{inputenc}
+%\usepackage[cyr]{aeguill}
%\usepackage{pstricks,pst-node,pst-text,pst-3d}
%\usepackage{babel}
\usepackage{amsmath}
\usepackage{fullpage}
\usepackage{fancybox}
+\usepackage[ruled,lined,linesnumbered]{algorithm2e}
+
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\newcommand{\noun}[1]{\textsc{#1}}
\newcommand{\tab}{\ \ \ }
-%%%%%%%%%%%%%%%%%%%%%%%%%%%% my bib path.
+
+
+\begin{document}
+
+
+%% \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{Raphaël Couturier\IEEEauthorrefmark{1}, Stéphane Domas\IEEEauthorrefmark{1}, Gwenhaël 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, Besançon, France\\
+\{michel.lenczner@utbm.fr,gwenhael.goavec@trabucayre.com}
+}
+
+
-\author{ Raphaël COUTURIER\\
-Laboratoire d'Informatique
-de l'Universit\'e de Franche-Comt\'e, \\
-BP 527, \\
-90016~Belfort CEDEX, France\\
- \and Stéphane Domas\\
-Laboratoire d'Informatique
-de l'Universit\'e de Franche-Comt\'e, \\
-BP 527, \\
-90016~Belfort CEDEX, France\\
- \and Gwenhaël Goavec\\
-??
-?? \\
-??, \\
-??\\}
-\begin{document}
\maketitle
\section{Introduction}
-%% blabla +
+Cantilevers are used inside atomic force microscope which provides high
+resolution images of surfaces. Several technics have been used to measure the
+displacement of cantilevers in litterature. For example, it is possible to
+determine accurately the deflection with different mechanisms.
+In~\cite{CantiPiezzo01}, authors used piezoresistor integrated into the
+cantilever. Nevertheless this approach suffers from the complexity of the
+microfabrication process needed to implement the sensor in the cantilever.
+In~\cite{CantiCapacitive03}, authors have presented an cantilever mechanism
+based on capacitive sensing. This kind of technic also involves to instrument
+the cantiliver which result in a complex fabrication process.
+
+In this paper our attention is focused on a method based on interferometry to
+measure cantilevers' displacements. In this method cantilevers are illuminated
+by an optic source. The interferometry produces fringes on each cantilevers
+which enables to compute the cantilever displacement. In order to analyze the
+fringes a high speed camera is used. Images need to be processed quickly and
+then a estimation method is required to determine the displacement of each
+cantilever. In~\cite{AFMCSEM11}, the authors have used an algorithm based on
+spline to estimate the cantilevers' positions.
+%%RAPH : ce qui est génant c'est qu'ils ne parlent pas de spline dans ce papier...
+ The overall process gives
+accurate results but all the computation are performed on a standard computer
+using labview. Consequently, the main drawback of this implementation is that
+the computer is a bootleneck in the overall process. In this paper we propose to
+use a method based on least square and to implement all the computation on a
+FGPA.
+
+The remainder of the paper is organized as follows. Section~\ref{sec:measure}
+describes more precisely the measurement process. Our solution based on the
+least square method and the implementation on FPGA is presented in
+Section~\ref{sec:solus}. Experimentations are described in
+Section~\ref{sec:results}. Finally a conclusion and some perspectives are
+presented.
+
+
+
%% quelques ref commentées sur les calculs basés sur l'interférométrie
-\section{Measurement architecture}
-\label{sec:measure-archi}
+\section{Measurement principles}
+\label{sec:measure}
-%% description de l'architecture générale de l'acquisition d'images
-%% avec au milieu une unité de traitement dont on ne précise pas ce
-%% qu'elle est.
+In order to develop simple, cost effective and user-friendly cantilever arrays,
+authors of ~\cite{AFMCSEM11} have developped a system based of
+interferometry. In opposition to other optical based systems, using a laser beam
+deflection scheme and sentitive to the angular displacement of the cantilever,
+interferometry is sensitive to the optical path difference induced by the
+vertical displacement of the cantilever.
+%%RAPH : est ce qu'on pique une image? génant ou non?
+The system build by authors of~\cite{AFMCSEM11} has been developped based on a
+Linnick interferomter~\cite{Sinclair:05}. A laser beam is first split (by the
+splitter) into a reference beam and a sample beam that reachs the cantilever
+array. In order to be able to move the cantilever array, it is mounted on a
+translation and rotational stage with five degrees of freedom. The optical
+system is also fixed to the stage. Thus, the cantilever array is centered in the
+optical system which can be adjusted accurately. The beam illuminates the array
+by a microscope objective and the light reflects on the cantilevers. Likewise
+the reference beam reflects on a movable mirror. A CMOS camera chip records the
+reference and sample beams which are recombined in the beam splitter and the
+interferogram. At the beginning of each experiment, the movable mirror is fitted
+manually in order to align the interferometric fringes approximately parallel to
+the cantilevers. When cantilevers move due to the surface, the bending of
+cantilevers produce movements in the fringes that can be detected with the CMOS
+camera. Finally the fringes need to be analyzed. In~\cite{AFMCSEM11}, the
+authors used a LabView program to compute the cantilevers' movements from the
+fringes.
-%% image tirée des expériences.
-\section{Design goals}
-\label{sec:goals}
-%% objectifs en terme de rapidité et de précision, avec en vue l'ajout
-%% du contrôle => l'unité de traitement qui s'impose est un FPGA =>
-%% algo adapté au FPGA.
-%% peut etre que cette section peut être déplacée en intro ... à voir.
-\section{Proposed solution}
-\label{sec:solus}
+
+
+\subsection{Architecture}
+\label{sec:archi}
+%% description de l'architecture générale de l'acquisition d'images
+%% avec au milieu une unité de traitement dont on ne précise pas ce
+%% qu'elle est.
+
+%% image tirée des expériences.
\subsection{Cantilever deflection estimation}
+\label{sec:deflest}
-%% => faire de l'interpolation de signal sinusoidal
-%% descriptif rapide des deux méthodes : splines et moindres carrés
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
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 below). 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.
-
-This 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$µ$s, which is
-quite small.
+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 ...
+
+%% � faire : timing de l'algo spline en C avec atan et tout le bordel.
+
+
+
+
+\section{Proposed solution}
+\label{sec:solus}
+
+
+\subsection{FPGA constraints}
+
+%% contraintes imposées par le FPGA : algo pipeline/parallele, pas d'op math complexe, ...
+
\subsection{Considered algorithms}
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 $x
\in [0,M[$.
\end{itemize}
-\subsubsection{Comparison}
-
-\subsection{FPGA constraints}
+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}
-%% contraintes imposées par le FPGA : algo pipeline/parallele, pas d'op math complexe, ...
-
-\subsection{Least square algorithm}
-%% description précise
-%% avantage sur FPGA
+\subsubsection{Comparison}
-\subsection{VDHL design paradigms}
+\subsection{VHDL design paradigms}
-\subsection{VDHL implementation}
+\subsection{VHDL implementation}
\section{Experimental results}
\label{sec:results}
