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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        optic
interferometer~\cite{CantiOptic89},     pizeoresistor~\cite{CantiPiezzo01}    or
capacitive  sensing~\cite{CantiCapacitive03}.  In  this paper  our  attention is
focused   on  a  method   based  on   interferometry  to   measure  cantilevers'
displacements.   In  this  method   cantilevers  are  illiminated  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} {\bf verifier ref}, the authors have used an algorithm based
on spline  to estimate  the cantilevers' positions.   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.

%% 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}

%% 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 $k=4$ 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 :
\begin{itemize}
\item the frequency could also be obtained using the derivates of
  spline equations, which only implies to solve quadratic equations.
\item 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 $\theta$.
\end{itemize}

\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 $nb_s$ 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 $log_2(nb_s)$ 

\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$_s$[$i$] $\leftarrow sin \theta$\\
     lut$_c$[$i$] $\leftarrow cos \theta$\\
     lut$_A$[$i$] $\leftarrow cos 2 \theta \times s4i + sin 2 \theta \times c4i$\\
     lut$_{sfi}$[$i$] $\leftarrow sin (2\pi f.i)$\\
     lut$_{cfi}$[$i$] $\leftarrow cos (2\pi f.i)$\\
   }
\end{algorithm}

\begin{algorithm}[ht]
\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 $M-1$}{
     $I[i] \leftarrow I[i] - start - slope\times i$\\
   }
   
   $I_{max} \leftarrow max_i(I[i])$, $I_{min} \leftarrow min_i(I[i])$\\
   $amp \leftarrow \frac{I_{max}-I_{min}}{2}$\\

   $Is \leftarrow 0$, $Ic \leftarrow 0$\\
   \For{$i=0$ to $M-1$}{
     $Is \leftarrow Is + I[i]\times $ lut$_{sfi}$[$i$]\\
     $Ic \leftarrow Ic + I[i]\times $ lut$_{cfi}$[$i$]\\
   }

   $\delta \leftarrow \frac{nb_s}{2}$, $b_l \leftarrow 0$, $b_r \leftarrow \delta$\\
   $v_l \leftarrow -2.I_s - amp.$lut$_A$[$b_l$]\\

   \While{$\delta >= 1$}{

     $v_r \leftarrow 2.[ Is.$lut$_c$[$b_r$]$ + Ic.$lut$_s$[$b_r$]$ ] - amp.$lut$_A$[$b_r$]\\

     \If{$!(v_l < 0$ and $v_r >= 0)$}{
       $v_l \leftarrow v_r$ \\
       $b_l \leftarrow b_r$ \\
     }
     $\delta \leftarrow \frac{\delta}{2}$\\
     $b_r \leftarrow b_l + \delta$\\
   }
   \uIf{$!(v_l < 0$ and $v_r >= 0)$}{
     $v_l \leftarrow v_r$ \\
     $b_l \leftarrow b_r$ \\
     $b_r \leftarrow b_l + 1$\\
     $v_r \leftarrow 2.[ Is.$lut$_c$[$b_r$]$ + Ic.$lut$_s$[$b_r$]$ ] - amp.$lut$_A$[$b_r$]\\
   }
   \Else {
     $b_r \leftarrow b_l + 1$\\
   }

   \uIf{$ abs(v_l) < v_r$}{
     $b_{\theta} \leftarrow b_l$ \\
   }
   \Else {
     $b_{\theta} \leftarrow b_r$ \\
   }
   $\theta \leftarrow \pi\times \left[\frac{2.b_{ref}}{nb_s}-1\right]$\\

\end{algorithm}

\subsubsection{Comparison}

We compared the two algorithms on the base of three criterions :
\begin{itemize}
\item precision of results on a cosinus profile, distorted with noise,
\item number of operations,
\item complexity to implement an FPGA version.
\end{itemize}

For the first item, we produced a matlab version of each algorithm,
running with double precision values. The profile was generated for
about 34000 different values of period ($\in [3.1, 6.1]$, step = 0.1),
phase ($\in [-3.1 , 3.1]$, step = 0.062) and slope ($\in [-2 , 2]$,
step = 0.4). For LSQ, $nb_s = 1024$, which leads to a maximal error of
$\frac{\pi}{1024}$ on phase computation. Current A. Meister and
M. Favre experiments show a ratio of 50 between variation of phase and
the deflection of a lever. Thus, the maximal error due to
discretization correspond to an error of 0.15nm on the lever
deflection, which is smaller than the best precision they achieved,
i.e. 0.3nm.

For each test, we add some noise to the profile : each group of two
pixels has its intensity added to a random number picked in $[-N,N]$
(NB: it should be noticed that picking a new value for each pixel does
not distort enough the profile). The absolute error on the result is
evaluated by comparing the difference between the reference and
computed phase, out of $2\pi$, expressed in percents. That is : $err =
100\times \frac{|\theta_{ref} - \theta_{comp}|}{2\pi}$.

Table \ref{tab:algo_prec} gives the maximum and average error for the two algorithms and increasing values of $N$.

\begin{table}[ht]
  \begin{center}
    \begin{tabular}{|c|c|c|c|c|}
      \hline
  & \multicolumn{2}{c|}{SPL} & \multicolumn{2}{c|}{LSQ} \\ \cline{2-5}
  noise & max. err. & aver. err. & max. err. & aver. err. \\ \hline
  0 & 2.46 & 0.58 & 0.49 & 0.1 \\ \hline
  2.5 & 2.75 & 0.62 & 1.16 & 0.22 \\ \hline
  5 & 3.77 & 0.72 & 2.47 & 0.41 \\ \hline
  7.5 & 4.72 & 0.86 & 3.33 & 0.62 \\ \hline
  10 & 5.62 & 1.03 & 4.29 & 0.81 \\ \hline
  15 & 7.96 & 1.38 & 6.35 & 1.21 \\ \hline
  30 & 17.06 & 2.6 & 13.94 & 2.45 \\ \hline

\end{tabular}
\caption{Error (in \%) for cosinus profiles, with noise.}
\label{tab:algo_prec}
\end{center}
\end{table}

These results show that the two algorithms are very close, with a
slight advantage for LSQ. Furthemore, both behave very well against
noise. Assuming the experimental ratio of 50 (see above), an error of
1 percent on phase correspond to an error of 0.5nm on the lever
deflection, which is very close to the best precision.

Obviously, it is very hard to predict which level of noise will be
present in real experiments and how it will distort the
profiles. Nevertheless, we can see on figure \ref{fig:noise20} the
profile with $N=10$ that leads to the biggest error. It is a bit
distorted, with pikes and straight/rounded portions, and relatively
close to most of that come from experiments. Figure \ref{fig:noise60}
shows a sample of worst profile for $N=30$. It is completly distorted,
largely beyond the worst experimental ones. 

\begin{figure}[ht]
\begin{center}
  \includegraphics[width=9cm]{intens-noise20-spl}
\end{center}
\caption{Sample of worst profile for N=10}
\label{fig:noise20}
\end{figure}

\begin{figure}[ht]
\begin{center}
  \includegraphics[width=9cm]{intens-noise60-lsq}
\end{center}
\caption{Sample of worst profile for N=30}
\label{fig:noise60}
\end{figure}

The second criterion is relatively easy to estimate for LSQ and harder
for SPL because of $atan$ operation. In both cases, it is proportional
to numbers of pixels $M$. For LSQ, it also depends on $nb_s$ and for
SPL on $N = k\times M$, i.e. the number of interpolated points. 

We assume that $M=20$, $nb_s=1024$, $k=4$, all possible parts are
already in lookup tables and only arithmetic operations (+, -, *, /)
are taken account. Translating the two algorithms in C code, we obtain
about 400 operations for LSQ and 1340 (plus the unknown for $atan$)
for SPL. Even if the result is largely in favor of LSQ, we can notice
that executing the C code (compiled with \tt{-O3}) of SPL on an
2.33GHz Core 2 Duo only takes 6.5µs in average, which is under our
desing goals. The final decision is thus driven by the third criterion.\\

The Spartan 6 used in our architecture has hard constraint : it has no
floating point units. Thus, all computations have to be done with
integers. 



\subsection{VHDL design paradigms}

\subsection{VHDL implementation}

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




\section{Conclusion and perspectives}


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\end{document}