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\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}
}






\maketitle

\thispagestyle{empty}

\begin{abstract}

  

{\it keywords}: FPGA, cantilever, interferometry.
\end{abstract}

\section{Introduction}

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 principles}
\label{sec:measure}








\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.

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.

\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 ...

%% � faire : timing de l'algo spline en C avec atan et tout le bordel.




\section{Proposed solution}
\label{sec:solus}


\subsection{FPGA constraints}

A field-programmable gate  array (FPGA) is an integrated  circuit designed to be
configured by  the customer.  A hardware  description language (HDL)  is used to
configure a  FPGA. FGPAs are  composed of programmable logic  components, called
logic blocks.  These blocks can be  configured to perform simple (AND, XOR, ...)
or  complex  combinational  functions.    Logic  blocks  are  interconnected  by
reconfigurable  links. Modern  FPGAs  contains memory  elements and  multipliers
which enables to simplify the design and increase the speed. As the most complex
operation operation on FGPAs is the  multiplier, design of FGPAs should not used
complex operations. For example, a divider  is not an available operation and it
should be programmed using simple components.

FGPAs programming  is very different  from classic processors  programming. When
logic block are programmed and linked  to performed an operation, they cannot be
reused anymore.  FPGA  are cadenced slowly than classic  processors but they can
performed pipelined as  well as pipelined operations. A  pipeline provides a way
manipulate data quickly  since at each clock top to handle  a new data. However,
using  a  pipeline  consomes more  logics  and  components  since they  are  not
reusable,  nevertheless it  is probably  the most  efficient technique  on FPGA.
Parallel  operations   can  be  used   in  order  to  manipulate   several  data
simultaneously. When  it is  possible, using  a pipeline is  a good  solution to
manipulate  new  data  at  each  clock  top  and  using  parallelism  to  handle
simultaneously several data streams.

%% contraintes imposées 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{VHDL design paradigms}

\subsection{VHDL implementation}

\section{Experimental results}
\label{sec:results}




\section{Conclusion and perspectives}


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\end{document}
