\section{Introduction}
-Cantilevers are used inside atomic force microscope which provides high
+Cantilevers are used inside atomic force microscope (AFM) 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.
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
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
+Linnick interferomter~\cite{Sinclair:05}. It is illustrated in
+Figure~\ref{fig:AFM}. A laser diode 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 hexapod 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.
-
+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.
+
+\begin{figure}
+\begin{center}
+\includegraphics[width=\columnwidth]{AFM}
+\end{center}
+\caption{schema of the AFM}
+\label{fig:AFM}
+\end{figure}
%% image tirée des expériences.
\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
+(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
\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 $\theta$.
+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}
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
+ $[-\pi,\pi]$ 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 :
-\[ sin \theta, cos \theta, \left[ cos 2\theta \sum_{i=0}^{M-1} sin(4\pi f.i) + sin 2\theta \sum_{i=0}^{M-1} cos(4\pi f.i)\right] \]
+\[ sin \theta, cos \theta, \]
+
+\[ \left[ cos 2\theta \sum_{i=0}^{M-1} sin(4\pi f.i) + sin 2\theta \sum_{i=0}^{M-1} cos(4\pi f.i)\right] \]
-\item This search can be very fast using a dichotomous process in $log_2(N)$
+\item This search can be very fast using a dichotomous process in $log_2(nb_s)$
\end{itemize}
\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[$i$] $\leftarrow cos (2\pi f.i)$\\
+ 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}[h]
+\begin{algorithm}[ht]
\caption{LSQ algorithm - during acquisition loop.}
\label{alg:lsq-during}
$slope \leftarrow \frac{xy_{covar}}{x_{var}}$\\
$start \leftarrow y_{moy} - slope\times \bar{x}$\\
\For{$i=0$ to $M-1$}{
- $I[i] \leftarrow I[i] - start - slope\times i$\tcc*[f]{slope removal}\\
+ $I[i] \leftarrow I[i] - start - slope\times i$\\
}
$I_{max} \leftarrow max_i(I[i])$, $I_{min} \leftarrow min_i(I[i])$\\
$Is \leftarrow 0$, $Ic \leftarrow 0$\\
\For{$i=0$ to $M-1$}{
- $Is \leftarrow Is + I[i]\times $ lut\_sinfi[$i$]\\
- $Ic \leftarrow Ic + I[i]\times $ lut\_cosfi[$i$]\\
+ $Is \leftarrow Is + I[i]\times $ lut$_{sfi}$[$i$]\\
+ $Ic \leftarrow Ic + I[i]\times $ lut$_{cfi}$[$i$]\\
}
- $\theta \leftarrow -\pi$\\
- $val_1 \leftarrow 2\times \left[ Is.\cos(\theta) + Ic.\sin(\theta) \right] - amp\times \left[ c4i.\sin(2\theta) + s4i.\cos(2\theta) \right]$\\
- \For{$i=1-n_s$ to $n_s$}{
- $\theta \leftarrow \frac{i.\pi}{n_s}$\\
- $val_2 \leftarrow 2\times \left[ Is.\cos(\theta) + Ic.\sin(\theta) \right] - amp\times \left[ c4i.\sin(2\theta) + s4i.\cos(2\theta) \right]$\\
+ $\delta \leftarrow \frac{nb_s}{2}$, $b_l \leftarrow 0$, $b_r \leftarrow \delta$\\
+ $v_l \leftarrow -2.I_s - amp.$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$]\\
- \lIf{$val_1 < 0$ et $val_2 >= 0$}{
- $\theta_s \leftarrow \theta - \left[ \frac{val_2}{val_2-val_1}\times \frac{\pi}{n_s} \right]$\\
+ \If{$!(v_l < 0$ and $v_r >= 0)$}{
+ $v_l \leftarrow v_r$ \\
+ $b_l \leftarrow b_r$ \\
}
- $val_1 \leftarrow val_2$\\
+ $\delta \leftarrow \frac{\delta}{2}$\\
+ $b_r \leftarrow b_l + \delta$\\
+ }
+ \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 {
+ $b_r \leftarrow b_l + 1$\\
}
-\end{algorithm}
+ \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}
