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[dmems12.git] / dmems12.tex

diff --git a/dmems12.tex b/dmems12.tex
index f559661e358c7e6f2041166b10f9a7a69daf7081..701ce926d15b7d884e07ce022bb967fbce6a029a 100644 (file)
--- a/dmems12.tex
+++ b/dmems12.tex
@@ -1,5 +1,5 @@
 
 

-\documentclass[10pt, conference, compsocconf]{IEEEtran}
+\documentclass[10pt, peerreview, compsocconf]{IEEEtran}

 %\usepackage{latex8}
 %\usepackage{times}
 \usepackage[utf8]{inputenc}
 %\usepackage{latex8}
 %\usepackage{times}
 \usepackage[utf8]{inputenc}


@@ -26,7 +26,6 @@
 \newcommand{\tab}{\ \ \ }
 
 
 \newcommand{\tab}{\ \ \ }
 
 

-

 \begin{document}
 
 
 \begin{document}
 
 


@@ -58,7 +57,7 @@
 
 
 
 
 
 

-\maketitle
+%\maketitle

 
 \thispagestyle{empty}
 
 
 \thispagestyle{empty}
 


@@ -66,25 +65,39 @@
 
   
 
 
   
 

-{\it keywords}: FPGA, cantilever, interferometry.
+

 \end{abstract}
 
 \end{abstract}
 

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

 \section{Introduction}
 
 \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
 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
+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
 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


@@ -105,12 +118,54 @@ presented.
 \section{Measurement principles}
 \label{sec:measure}
 
 \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.
 
 \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}
 %% image tirée des expériences.
 
 \subsection{Cantilever deflection estimation}


@@ -132,7 +187,7 @@ 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.
 
 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}
 
 \begin{equation}
 \label{equ:profile}


@@ -147,45 +202,158 @@ 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
 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}
 
 
 \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
 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
 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}
 
 
 
 \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}
 
 
 \subsection{Considered algorithms}
 


@@ -202,23 +370,23 @@ 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 $x
 \in [0,M[$. 
 
 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.
+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.
 
 
 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}
 
 \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 :
+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.
 \begin{itemize}
 \item the frequency could also be obtained using the derivates of
   spline equations, which only implies to solve quadratic equations.


@@ -231,9 +399,9 @@ Two things can be noticed :
 \subsubsection{Least square algorithm}
 
 Assuming that we compute the phase during the acquisition loop,
 \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
 $\theta$, $f$ 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.
 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.


@@ -241,16 +409,16 @@ 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
 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
 
 \[I^{corr}(x^p) = I(x^p) - a.x^p - b\]
 
 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 $b$:

 
 \[a = \frac{covar(x^p,I(x^p))}{var(x^p)} \]
 
 
 \[a = \frac{covar(x^p,I(x^p))}{var(x^p)} \]
 

-Assuming an overlined symbol means an average, then :
+Assuming an overlined symbol means an average, then:

 
 \[b = \overline{I(x^p)} - a.\overline{{x^p}}\]
 
 
 \[b = \overline{I(x^p)} - a.\overline{{x^p}}\]
 


@@ -258,22 +426,22 @@ Let $A$ be the amplitude of $I^{corr}$, i.e.
 
 \[A = \frac{max(I^{corr}) - min(I^{corr})}{2}\]
 
 
 \[A = \frac{max(I^{corr}) - min(I^{corr})}{2}\]
 

-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:

 
 \[\sum_{i=0}^{M-1} \left[ cos(2\pi f.i + \theta) - \frac{I^{corr}(i)}{A} \right]^2\]
 
 
 \[\sum_{i=0}^{M-1} \left[ cos(2\pi f.i + \theta) - \frac{I^{corr}(i)}{A} \right]^2\]
 

-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*}
 
 
 \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
 \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 $\theta$:

 
 \[ \sum_{i=0}^{M-1} sin(4\pi f.i), \sum_{i=0}^{M-1} cos(4\pi f.i) \] 
 
 
 \[ \sum_{i=0}^{M-1} sin(4\pi f.i), \sum_{i=0}^{M-1} cos(4\pi f.i) \] 
 


@@ -283,7 +451,7 @@ 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
 \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 :
+  also be computed before the loop:

 
 \[ sin \theta, cos \theta, \]
 
 
 \[ sin \theta, cos \theta, \]
 


@@ -293,7 +461,7 @@ computed.
 
 \end{itemize}
 
 
 \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}
 \begin{algorithm}[h]
 \caption{LSQ algorithm - before acquisition loop.}
 \label{alg:lsq-before}


@@ -380,7 +548,7 @@ Finally, the whole summarizes in an algorithm (called LSQ in the following) in t
 
 \subsubsection{Comparison}
 
 
 \subsubsection{Comparison}
 

-We compared the two algorithms on the base of three criterions :
+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,
 \begin{itemize}
 \item precision of results on a cosinus profile, distorted with noise,
 \item number of operations,


@@ -399,12 +567,12 @@ 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.
 
 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
+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
 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 =
+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 $N$.
 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 $N$.


@@ -446,7 +614,7 @@ largely beyond the worst experimental ones.
 
 \begin{figure}[ht]
 \begin{center}
 
 \begin{figure}[ht]
 \begin{center}

-  \includegraphics[width=9cm]{intens-noise20-spl}
+  \includegraphics[width=\columnwidth]{intens-noise20}

 \end{center}
 \caption{Sample of worst profile for N=10}
 \label{fig:noise20}
 \end{center}
 \caption{Sample of worst profile for N=10}
 \label{fig:noise20}


@@ -454,7 +622,7 @@ largely beyond the worst experimental ones.
 
 \begin{figure}[ht]
 \begin{center}
 
 \begin{figure}[ht]
 \begin{center}

-  \includegraphics[width=9cm]{intens-noise60-lsq}
+  \includegraphics[width=\columnwidth]{intens-noise60}

 \end{center}
 \caption{Sample of worst profile for N=30}
 \label{fig:noise60}
 \end{center}
 \caption{Sample of worst profile for N=30}
 \label{fig:noise60}


@@ -466,25 +634,76 @@ 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
 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. 
+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}

 
 

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

 
 

-\subsection{VHDL implementation}
+% pas fait mais prévision d'une sortie tous les 480ns avec une latence de 1120

 
 

-\section{Experimental results}

 \label{sec:results}
 
 
 \label{sec:results}