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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

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\begin{abstract}

  


\end{abstract}

\begin{IEEEkeywords}
FPGA, cantilever, interferometry.
\end{IEEEkeywords}


\IEEEpeerreviewmaketitle

\section{Introduction}

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. 
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.

   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.

The system build  by authors of~\cite{AFMCSEM11} has been  developped based on a
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.

\begin{figure}    
\begin{center}
\includegraphics[width=\columnwidth]{AFM}
\end{center}
\caption{schema of the AFM}
\label{fig:AFM}   
\end{figure}


%% 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. Originally, this computation was also done with an
algorithm based on spline. This article proposes a new version based
on a least square method.

\subsection{Design goals}
\label{sec:goals}

The main goal is to implement a computing unit to estimate the
deflection of about $10\times10$ cantilevers, faster than the stream of
images coming from the camera. The accuracy of results must be close
to the maximum precision ever obtained experimentally on the
architecture, i.e. 0.3nm. Finally, the latency between an image
entering in the unit and the deflections must be as small as possible
(NB: future works plan to add some control on the cantilevers).\\

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
100 cantilevers, 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. 

%%Itimplies that the phase computation algorithm should not take more than
%%$155\times 12.5 = 1937$ floating operations. For integers, it gives $3000$ operations. 

Obviously, some cache effects and optimizations on
huge amount of computations can drastically increase these
performances: peak efficiency is about 2.5Gflops for the considered
CPU. But this is not the case for phase computation that used only few
tenth of values.\\

In order to evaluate the original algorithm, we translated it in C
language. As said further, for 20 pixels, it does about 1550
operations, thus an estimated execution time of $1550/155
=$10$\mu$s. For a more realistic evaluation, we constructed a file of
1Mo containing 200 profiles of 20 pixels, equally scattered. This file
is equivalent to an image stored in a device file representing the
camera. We obtained an average of 10.5$\mu$s by profile (including I/O
accesses). It is under are requirements but close to the limit. In
case of an occasional load of the system, it could be largely
overtaken. A solution would be to use a real-time operating system but
another one to search for a more efficient algorithm.

But the main drawback is the latency of such a solution: since each
profile must be treated one after another, the deflection of 100
cantilevers takes about $200\times 10.5 = 2.1$ms, which is inadequate
for an efficient control. An obvious solution is to parallelize the
computations, for example on a GPU. Nevertheless, the cost to transfer
profile in GPU memory and to take back results would be prohibitive
compared to computation time. It is certainly more efficient to
pipeline the computation. For example, supposing that 200 profiles of
20 pixels can be pushed sequentially in the pipelined unit cadenced at
a 100MHz (i.e. a pixel enters in the unit each 10ns), all profiles
would be treated in $200\times 20\times 10.10^{-9} =$ 40$\mu$s plus
the latency of the pipeline. This is about 500 times faster than
actual results.\\

For these reasons, an FPGA as the computation unit is the best choice
to achieve the required performance. Nevertheless, passing from
a C code to a pipelined version in VHDL is not obvious at all. As
explained in the next section, it can even be impossible because of
some hardware constraints specific to FPGAs.


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

Project Oscar aims  to provide a hardware and  software architecture to estimate
and  control the  deflection of  cantilevers. The  hardware part  consists  in a
high-speed camera,  linked on an embedded  board hosting FPGAs. By  the way, the
camera output stream can be pushed  directly into the FPGA. The software part is
mostly the VHDL  code that deserializes the camera  stream, extracts profile and
computes  the deflection. Before  focusing on  our work  to implement  the phase
computation, we give some general information about FPGAs and the board we use.

\subsection{FPGAs}

A field-programmable gate array (FPGA) is an integrated circuit
designed to be configured by the customer. FGPAs are composed of
programmable logic components, called configurable logic blocks
(CLB). These blocks mainly contains look-up tables (LUT), flip/flops
(F/F) and latches, organized in one or more slices connected
together. Each CLB can be configured to perform simple (AND, XOR, ...)
or complex combinational functions. They are interconnected by
reconfigurable links. Modern FPGAs contain memory elements and
multipliers which enable to simplify the design and to increase the
performance. Nevertheless, all other complex operations, like
division, trigonometric functions, $\ldots$ are not available and must
be done by configuring a set of CLBs. Since this configuration is not
obvious at all, it can be done via a framework, like ISE. Such a
software can synthetize a design written in an hardware description
language (HDL), map it onto CLBs, place/route them for a specific
FPGA, and finally produce a bitstream that is used to configre the
FPGA. Thus, from the developper point of view, the main difficulty is
to translate an algorithm in HDL code, taking account FPGA resources
and constraints like clock signals and I/O values that drive the FPGA.

Indeed, HDL programming is very different from classic languages like
C. A program can be seen as a state-machine, manipulating signals that
evolve from state to state. By the way, HDL instructions can execute
concurrently. Basic logic operations are used to agregate signals to
produce new states and assign it to another signal. States are mainly
expressed as arrays of bits. Fortunaltely, libraries propose some
higher levels representations like signed integers, and arithmetic
operations.

Furthermore, even if FPGAs are cadenced more slowly than classic
processors, they can perform pipeline as well as parallel
operations. A pipeline consists in cutting a process in sequence of
small tasks, taking the same execution time. It accepts a new data at
each clock top, thus, after a known latency, it also provides a result
at each clock top. However, using a pipeline consumes more logics
since the components of a task are not reusable by another
one. Nevertheless it is probably the most efficient technique on
FPGA. Because of its architecture, it is also very easy to process
several data concurrently. When it is possible, the best performance
is reached using parallelism to handle simultaneously several
pipelines in order to handle multiple data streams.

\subsection{The board}

The board we use is designed by the Armadeus compagny, under the name
SP Vision. It consists in a development board hosting a i.MX27 ARM
processor (from Freescale). The board includes all classical
connectors: USB, Ethernet, ... A Flash memory contains a Linux kernel
that can be launched after booting the board via u-Boot.

The processor is directly connected to a Spartan3A FPGA (from Xilinx)
via its special interface called WEIM. The Spartan3A is itself
connected to a Spartan6 FPGA. Thus, it is possible to develop programs
that communicate between i.MX and Spartan6, using Spartan3 as a
tunnel. By default, the WEIM interface provides a clock signal at
100MHz that is connected to dedicated FPGA pins.

The Spartan6 is an LX100 version. It has 15822 slices, equivalent to
101261 logic cells. There are 268 internal block RAM of 18Kbits, and
180 dedicated multiply-adders (named DSP48), which is largely enough
for our project.

Some I/O pins of Spartan6 are connected to two $2\times 17$ headers
that can be used as user wants. For the project, they will be
connected to the interface card of the camera.

\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  (SPL  in the
following) 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 on a Gauss-newton algorithm can 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 criteria:
\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=\columnwidth]{intens-noise20}
\end{center}
\caption{Sample of worst profile for N=10}
\label{fig:noise20}
\end{figure}

\begin{figure}[ht]
\begin{center}
  \includegraphics[width=\columnwidth]{intens-noise60}
\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 a limited set of operations (+, -, *, /,
$<$, $>$) is taken account. Translating the two algorithms in C code, we
obtain about 430 operations for LSQ and 1550 (plus few tenth for
$atan$) for SPL. This result is largely in favor of LSQ. Nevertheless,
considering the total number of operations is not really pertinent for
an FPGA implementation: it mainly depends on the type of operations
and their
ordering. The final decision is thus driven by the third criterion.\\

The Spartan 6 used in our architecture has hard constraint: it has no
built-in floating point units. Obviously, it is possible to use some
existing "black-boxes" for double precision operations. But they have
a quite long latency. It is much simpler to exclusively use integers,
with a quantization of all double precision values. Obviously, this
quantization should not decrease too much the precision of
results. Furthermore, it should not lead to a design with a huge
latency because of operations that could not complete during a single
or few clock cycles. Divisions are in this case and, moreover, they
need an varying number of clock cycles to complete. Even
multiplications can be a problem: DSP48 take inputs of 18 bits
maximum. For larger multiplications, several DSP must be combined,
increasing the latency.

Nevertheless, the hardest constraint does not come from the FPGA
characteristics but from the algorithms. Their VHDL implentation will
be efficient only if they can be fully (or near) pipelined. By the
way, the choice is quickly done: only a small part of SPL can be.
Indeed, the computation of spline coefficients implies to solve a
tridiagonal system $A.m = b$. Values in $A$ and $b$ can be computed
from incoming pixels intensity but after, the back-solve starts with
the lastest values, which breaks the pipeline. Moreover, SPL relies on
interpolating far more points than profile size. Thus, the end
of SPL works on a larger amount of data than the beginning, which
also breaks the pipeline.

LSQ has not this problem: all parts except the dichotomial search
work on the same amount of data, i.e. the profile size. Furthermore,
LSQ needs less operations than SPL, implying a smaller output
latency. Consequently, it is the best candidate for phase
computation. Nevertheless, obtaining a fully pipelined version
supposes that operations of different parts complete in a single clock
cycle. It is the case for simulations but it completely fails when
mapping and routing the design on the Spartan6. By the way,
extra-latency is generated and there must be idle times between two
profiles entering into the pipeline.

%%Before obtaining the least bitstream, the crucial question is: how to
%%translate the C code the LSQ into VHDL ?


%\subsection{VHDL design paradigms}

\section{Experimental tests}

\subsection{VHDL implementation}

% - ecriture d'un code en C avec integer
% - calcul de la taille max en bit de chaque variable en fonction de la quantization.
% - tests de quantization : équilibre entre précision et contraintes FPGA
% - en parallèle : simulink et VHDL à la main
%
\subsection{Simulation}

% ghdl + gtkwave
% au mieux : une phase tous les 33 cycles, latence de 95 cycles.
% mais routage/placement impossible.
\subsection{Bitstream creation}

% pas fait mais prévision d'une sortie tous les 480ns avec une latence de 1120

\label{sec:results}




\section{Conclusion and perspectives}


\bibliographystyle{plain}
\bibliography{biblio}

\end{document}