-\documentclass[10pt, conference, compsocconf]{IEEEtran}
+\documentclass[10pt, peerreview, compsocconf]{IEEEtran}
%\usepackage{latex8}
%\usepackage{times}
\usepackage[utf8]{inputenc}
\newcommand{\tab}{\ \ \ }
-
\begin{document}
-\maketitle
+%\maketitle
\thispagestyle{empty}
-{\it keywords}: FPGA, cantilever, interferometry.
+
\end{abstract}
+\begin{IEEEkeywords}
+FPGA, cantilever, interferometry.
+\end{IEEEkeywords}
+
+
+\IEEEpeerreviewmaketitle
+
\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
\section{Measurement principles}
\label{sec:measure}
-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.
%% 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}
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 :
+The pixels intensity $I$ (in gray level) of each profile is modelized by:
\begin{equation}
\label{equ:profile}
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.
+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 a
-$10\times 10$ cantilever array, if we neglect the time to extract
-pixels, it implies that computing the deflection of a single
+$1024\times 1204$ 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. 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.
-
-
+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}
-
-\subsection{FPGA constraints}
-
-%% contraintes imposées par le FPGA : algo pipeline/parallele, pas d'op math complexe, ...
-
+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}
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.
+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 :
+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 $\theta$.
+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
+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
+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 :
+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 $b$ :
+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 :
+Assuming an overlined symbol means an average, then:
\[b = \overline{I(x^p)} - a.\overline{{x^p}}\]
\[A = \frac{max(I^{corr}) - min(I^{corr})}{2}\]
-Then, the least square method to find $\theta$ is reduced to search the minimum of :
+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 :
+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 :
+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 $\theta$ :
+ not depend on $\theta$:
\[ \sum_{i=0}^{M-1} sin(4\pi f.i), \sum_{i=0}^{M-1} cos(4\pi f.i) \]
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 :
+ also be computed before the loop:
+
+\[ sin \theta, cos \theta, \]
-\[ sin \theta, cos \theta, \left[ cos 2\theta \sum_{i=0}^{M-1} sin(4\pi f.i) + sin 2\theta \sum_{i=0}^{M-1} cos(4\pi f.i)\right] \]
+\[ \left[ cos 2\theta \sum_{i=0}^{M-1} sin(4\pi f.i) + sin 2\theta \sum_{i=0}^{M-1} cos(4\pi f.i)\right] \]
-\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}
-Finally, the whole summarizes in an algorithm (called LSQ in the following) in two parts, one before and one during the acquisition loop :
+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}
\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}
-\subsection{VHDL design paradigms}
+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}
-\section{Experimental results}
+% - 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}
