-\documentclass[10pt, conference, compsocconf]{IEEEtran}
+\documentclass[10pt, peerreview, compsocconf]{IEEEtran}
%\usepackage{latex8}
%\usepackage{times}
\usepackage[utf8]{inputenc}
\newcommand{\tab}{\ \ \ }
-
\begin{document}
-\title{Using FPGAs for high speed and real time cantilever deflection estimation}
+\title{A new approach based on least square methods to estimate in real time cantilevers deflection with a FPGA}
\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}
-\maketitle
+%\maketitle
\thispagestyle{empty}
\begin{abstract}
-
+Atomics force microscope (AFM) provide high resolution images of surfaces. In
+this paper, we focus our attention on an interferometry method to estimate the
+cantilevers deflection. The initial method was based on splines to determine
+the phase of interference fringes, and thus the deflection. Computations were
+performed on a PC with LabView. Here, we propose a new approach based on the
+least square methods and its implementation that we have developed on a FPGA,
+using the pipelining technique. Simulations and real tests showed us that this
+implementation is very efficient and should allow us to control a cantilevers
+array in real time.
+
-{\it keywords}: FPGA, cantilever, interferometry.
\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.
+Cantilevers are used inside atomic force microscopes (AFM) which provide high
+resolution images of surfaces. Several techniques have been used to measure the
+displacement of cantilevers in literature. 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~\cite{CantiCapacitive03}, authors have presented a cantilever mechanism
+based on capacitive sensing. This kind of technique also involves to instrument
+the cantilever which results 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
+by an optic source. The interferometry produces fringes on each cantilever which
+enables us 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.
+then an estimation method is required to determine the displacement of each
+cantilever. In~\cite{AFMCSEM11}, 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 overall process gives accurate results but all the computations are
+performed on a standard computer using LabView. Consequently, the main drawback
+of this implementation is that the computer is a bottleneck. In this paper we
+propose to use a method based on least squares and to implement all the
+computation on a FPGA.
The remainder of the paper is organized as follows. Section~\ref{sec:measure}
describes more precisely the measurement process. Our solution based on the
-%% 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
+authors of ~\cite{AFMCSEM11} have developed a system based on
interferometry. In opposition to other optical based systems, using a laser beam
-deflection scheme and sentitive to the angular displacement of the cantilever,
+deflection scheme and sensitive 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 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
+The system built by these authors is based on a Linnick
+interferometer~\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 reach 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.
+fringes need to be analyzed. In~\cite{AFMCSEM11}, authors used a LabView program
+to compute the cantilevers' deflections from the fringes.
\begin{figure}
\begin{center}
\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.\\
+\begin{figure}
+\begin{center}
+\includegraphics[width=\columnwidth]{lever-xp}
+\end{center}
+\caption{Portion of an image picked by the camera}
+\label{fig:img-xp}
+\end{figure}
-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.
+As shown on image \ref{fig:img-xp}, each cantilever is covered by
+several interferometric fringes. The fringes will distort when
+cantilevers are deflected. Estimating the deflection is done by
+computing this distortion. For that, authors of \cite{AFMCSEM11}
+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 used to "unwrap" the phase at the tip and to determine the deflection.\\
-The pixels intensity $I$ (in gray level) of each profile is modelized by :
+More precisely, segments of pixels are extracted from images taken by
+the camera. These segments are large enough to cover several
+interferometric fringes. As said above, they 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}
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
+to determine the frequency $f$ of each profile with an algorithm based
+on spline interpolation (see section \ref{sec: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
+is the acquisition loop, during 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
+In fact, this timing is a very hard constraint. Let us consider a very
+small program that initializes twenty million of doubles in memory
+and then does 1,000,000 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 a few
+tenth of values.\\
+
+In order to evaluate the original algorithm, we translated it in C
+language. As stated before, for 20 pixels, it does about 1,550
+operations, thus an estimated execution time of $1,550/155
+=$10$\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 our 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 of transferring
+profile in GPU memory and of taking 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, a 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. In this 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{FPGA constraints}
+\subsection{FPGAs}
A field-programmable gate array (FPGA) is an integrated circuit designed to be
-configured by the customer. A hardware description language (HDL) is used to
-configure a FPGA. FGPAs are composed of programmable logic components, called
-logic blocks. These blocks can be configured to perform simple (AND, XOR, ...)
-or complex combinational functions. Logic blocks are interconnected by
-reconfigurable links. Modern FPGAs contains memory elements and multipliers
-which enables to simplify the design and increase the speed. As the most complex
-operation operation on FGPAs is the multiplier, design of FGPAs should not used
-complex operations. For example, a divider is not an available operation and it
-should be programmed using simple components.
-
-FGPAs programming is very different from classic processors programming. When
-logic block are programmed and linked to performed an operation, they cannot be
-reused anymore. FPGA are cadenced slowly than classic processors but they can
-performed pipelined as well as pipelined operations. A pipeline provides a way
-manipulate data quickly since at each clock top to handle a new data. However,
-using a pipeline consomes more logics and components since they are not
-reusable, nevertheless it is probably the most efficient technique on FPGA.
-Parallel operations can be used in order to manipulate several data
-simultaneously. When it is possible, using a pipeline is a good solution to
-manipulate new data at each clock top and using parallelism to handle
-simultaneously several data streams.
-
-%% contraintes imposées par le FPGA : algo pipeline/parallele, pas d'op math complexe, ...
-
+configured by the customer. FGPAs are composed of programmable logic components,
+called configurable logic blocks (CLB). These blocks mainly contain 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~\cite{ISE}. Such a software can synthetize a design written in a hardware
+description language (HDL), map it onto CLBs, place/route them for a specific
+FPGA, and finally produce a bitstream that is used to configure the FPGA. Thus,
+from the developer's point of view, the main difficulty is to translate an
+algorithm in HDL code, taking into 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. Moreover, HDL instructions can executed
+concurrently. Basic logic operations are used to aggregate signals to
+produce new states and assign it to another signal. States are mainly
+expressed as arrays of bits. Fortunately, 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 a 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. Whenever 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 company, 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, each slice
+containing 4 LUTs and 8 flip/flops. It is 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}
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
+classical least square method which supposes that the frequency is already
known.
-\subsubsection{Spline algorithm}
+\subsubsection{Spline algorithm (SPL)}
\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
+Let us consider a profile $P$, that is a segment of $M$ pixels with an
+intensity in gray levels. Let us 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
+\ldots,M-1$. A normalization allows to scale known intensities into
+$[-1,1]$. We compute splines that fit at best these normalized
intensities. Splines are used to interpolate $N = k\times M$ points
-(typically $k=3$ is sufficient), within $[0,M[$. Let call $x^s$ the
+(typically $k=4$ is sufficient), within $[0,M[$. Let $x^s$ be 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.
+computed. Finding intersections of $I^s$ and this line allows us to obtain
+the period and 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 only once, 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 would lead to a much faster
+ computation of $\theta$.
+\end{itemize}
-\subsubsection{Least square algorithm}
+\subsubsection{Least square algorithm (LSQ)}
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
+Fortunately, 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 derivating this expression and to solving 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{eqnarray*}{l}
+$$2\left[ cos\theta \sum_{i=0}^{M-1} I^{corr}(i).sin(2\pi f.i) \right.$$
+$$\left. + sin\theta \sum_{i=0}^{M-1} I^{corr}(i).cos(2\pi f.i)\right]- $$
+$$ A\left[ cos2\theta \sum_{i=0}^{M-1} sin(4\pi f.i) + sin2\theta \sum_{i=0}^{M-1} cos(4\pi f.i)\right] = 0$$
+%\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 $\theta$ :
+ 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
- result closest to zero. By the way, three other lookup tables can
- also be computed before the loop :
+ $[-\pi,\pi]$ in $nb_s$ steps, and to search which step leads to the
+ result closest to zero. Hence, three other lookup tables can
+ also be computed before the loop:
-\[ sin \theta, cos \theta, \left[ cos 2\theta \sum_{i=0}^{M-1} sin(4\pi f.i) + sin 2\theta \sum_{i=0}^{M-1} cos(4\pi f.i)\right] \]
+\[ sin \theta, cos \theta, \]
-\item This search can be very fast using a dichotomous process in $log_2(N)$
+\[ \left[ cos 2\theta \sum_{i=0}^{M-1} sin(4\pi f.i) + sin 2\theta \sum_{i=0}^{M-1} cos(4\pi f.i)\right] \]
+
+\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]
+Finally, this is synthetized in an algorithm (called LSQ in the following) in two parts, one before and one during the acquisition loop:
+\begin{algorithm}[htbp]
\caption{LSQ algorithm - before acquisition loop.}
\label{alg:lsq-before}
$M \leftarrow $ number of pixels of the profile\\
- I[] $\leftarrow $ intensities of pixels\\
+ I[] $\leftarrow $ intensity of pixels\\
$f \leftarrow $ frequency of the profile\\
$s4i \leftarrow \sum_{i=0}^{M-1} sin(4\pi f.i)$\\
$c4i \leftarrow \sum_{i=0}^{M-1} cos(4\pi f.i)$\\
\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}[htbp]
\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{$\delta >= 1$}{
- \lIf{$val_1 < 0$ et $val_2 >= 0$}{
- $\theta_s \leftarrow \theta - \left[ \frac{val_2}{val_2-val_1}\times \frac{\pi}{n_s} \right]$\\
+ $v_r \leftarrow 2.[ Is.$lut$_c$[$b_r$]$ + Ic.$lut$_s$[$b_r$]$ ] - amp.$lut$_A$[$b_r$]\\
+
+ \If{$!(v_l < 0$ and $v_r >= 0)$}{
+ $v_l \leftarrow v_r$ \\
+ $b_l \leftarrow b_r$ \\
}
- $val_1 \leftarrow val_2$\\
+ $\delta \leftarrow \frac{\delta}{2}$\\
+ $b_r \leftarrow b_l + \delta$\\
+ }
+ \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 cosines profile, distorted by noise,
+\item number of operations,
+\item complexity of implementating 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's experiments show a ratio of 50 between the variation of a phase and
+the deflection of a lever. Thus, the maximal error due to
+discretization corresponds 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 percentage. That is: $err =
+100\times \frac{|\theta_{ref} - \theta_{comp}|}{2\pi}$.
+
+Table \ref{tab:algo_prec} gives the maximum and average error for both
+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 cosines 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. Furthermore, both behave very well against
+noise. Assuming the experimental ratio of 50 (see above), an error of
+1 percent on the phase corresponds 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 experiments. Figure \ref{fig:noise60}
+shows a sample of worst profile for $N=30$. It is completely 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 the 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 into account. Translating both algorithms in C code, we
+obtain about 430 operations for LSQ and 1,550 (plus a 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 a 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 quite a 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 fall into that category and, moreover,
+they need a varying number of clock cycles to complete. Even multiplications can
+be a problem: a DSP48 takes 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 implementation will be efficient only if
+they can be fully (or near) pipelined. Thus, the choice is quickly made: only a
+small part of SPL can be pipelined. 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 latest 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 at 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. Thus, 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}
+
+%In this section we explain what we have done yet. Until now, we could not perform
+%real experiments since we just have received the FGPA board. Nevertheless, we
+%will include real experiments in the final version of this paper.
\subsection{VHDL implementation}
-\section{Experimental results}
+From the LSQ algorithm, we have written a C program that uses only
+integer values. We use a very simple quantization by multiplying
+double precision values by a power of two, keeping the integer
+part. For example, all values stored in lut$_s$, lut$_c$, $\ldots$ are
+scaled by 1,024. Since LSQ also computes average, variance, ... to
+remove the slope, the result of implied Euclidean divisions may be
+relatively wrong. To avoid that, we also scale the pixel intensities
+by a power of two. Furthermore, assuming $nb_s$ is fixed, these
+divisions have a known denominator. Thus, they can be replaced by
+their multiplication/shift counterpart. Finally, all other
+multiplications or divisions by a power of two have been replaced by
+left or right bit shifts. Thus, the code only contains
+additions, subtractions and multiplications of signed integers, which
+are perfectly adapted to FGPAs.
+
+As mentioned above, hardware constraints have a great influence on the VHDL
+implementation. Consequently, we searched the maximum value of each variable as
+a function of the different scale factors and the size of profiles, which gives
+their maximum size in bits. That size determines the maximum scale factors that
+allow to use the least possible RAMs and DSPs. Actually, we implemented our
+algorithm with this maximum size but current works study the impact of
+quantization on the results precision and design complexity. We have compared
+the result of the LSQ version using integers and doubles and observed that the
+precision of both were similar.
+
+Then we built two versions of VHDL codes: one directly by hand coding
+and the other with Matlab using the Simulink HDL coder
+feature~\cite{HDLCoder}. Although the approach is completely different
+we obtained VHDL codes that are quite comparable. Each approach has
+advantages and drawbacks. Roughly speaking, hand coding provides
+beautiful and much better structured code while Simulink enables us to
+produce a code faster. In terms of throughput and latency,
+simulations show that the two approaches are close with a slight
+advantage for hand coding. We hope that real experiments will confirm
+that.
+
+\subsection{Simulation}
+
+Before experimental tests on the board, we simulated our two VHDL
+codes with GHDL and GTKWave (two free tools with linux). For that, we
+built a testbench based on profiles taken from experimentations and
+compared the results to values given by the SPL algorithm. Both
+versions lead to correct results.
+
+Our first codes were highly optimized : the pipeline could compute a
+new phase each 33 cycles and its latency was equal to 95 cycles. Since
+the Spartan6 is clocked at 100MHz, it implies that estimating the
+deflection of 100 cantilevers would take about $(95 + 200\times 33).10
+= 66.95\mu$s, i.e. nearly 15,000 estimations by second.
+
+\subsection{Bitstream creation}
+
+In order to test our code on the SP Vision board, the design was
+extended with a component that keeps profiles in RAM, flushes them in
+the phase computation component and stores its output in another
+RAM. We also added a wishbone : a component that can "drive" signals
+to communicate between i.MX and other components. It is mainly used
+to start to flush profiles and to retrieve the computed phases in RAM.
+
+Unfortunately, the first designs could not be placed and route with ISE on the
+Spartan6 with a 100MHz clock. The main problems came from routing values from
+RAMs to DSPs and obtaining a result under 10ns. So, we needed to decompose some
+parts of the pipeline, which adds some cycles. For example, some delays have
+been introduced between RAMs output and DSPs. Finally, we obtained a bitstream
+that has a latency of 112 cycles and computes a new phase every 40 cycles. For
+100 cantilevers, it takes $(112 + 200\times 40).10 = 81.12\mu$s to compute their
+deflection.
+
+This bitstream has been successfully tested on the board.
+
+
+
\label{sec:results}
\section{Conclusion and perspectives}
-
+In this paper we have presented a new method to estimate the
+cantilevers deflection in an AFM. This method is based on least
+square methods. We have used quantization to produce an algorithm
+based exclusively on integer values, which is adapted to a FPGA
+implementation. We obtained a precision on results similar to the
+initial version based on splines. Our solution has been implemented
+with a pipeline technique. Consequently, it enables to handle a new
+profile image very quickly. Currently we have performed simulations
+and real tests on a Spartan6 FPGA.
+
+In future work, we plan to study the quantization. Then we want to couple our
+algorithm with a high speed camera and we plan to control the whole AFM system.
\bibliographystyle{plain}
\bibliography{biblio}
