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\begin{document}
+\title{A new approach based on a least square method for real-time estimation of cantilever array deflections 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 \and
+\{raphael.couturier,stephane.domas\}@univ-fcomte.fr}
+\IEEEauthorblockA{\IEEEauthorrefmark{2}FEMTO-ST, Time-Frequency, University of Franche-Comte, Besançon, France \and
+\{michel.lenczner@utbm.fr,gwenhael.goavec@trabucayre.com} }
+
+\begin{abstract}
+Atomic force microscopes (AFM) provide high resolution images of surfaces.
+In this paper, we focus our attention on an interferometry method for
+deflection estimation of cantilever arrays in quasi-static regime. In its
+original form, spline interpolation was used to determine interference
+fringe phase, and thus the deflections. Computations were performed on a PC.
+Here, we propose a new complete solution with a least square based algorithm
+and an optimized FPGA implementation. Simulations and real tests showed very
+good results and open perspective for real-time estimation and control of
+cantilever arrays in the dynamic regime.
+\end{abstract}
%% \author{\IEEEauthorblockN{Authors Name/s per 1st Affiliation (Author)}
%% \IEEEauthorblockA{line 1 (of Affiliation): dept. name of organization\\
%% line 4: Email: name@xyz.com}
%% }
-
-
-\title{Using FPGAs for high speed and real time cantilever deflection estimation}
-\author{\IEEEauthorblockN{Raphaël Couturier\IEEEauthorrefmark{1}, Stéphane Domas\IEEEauthorrefmark{1}, Gwenhaël Goavec-Merou\IEEEauthorrefmark{2} and Michel Lenczner\IEEEauthorrefmark{2}}
-\IEEEauthorblockA{\IEEEauthorrefmark{1}FEMTO-ST, DISC, University of Franche-Comte, Belfort, France\\
-\{raphael.couturier,stephane.domas\}@univ-fcomte.fr}
-\IEEEauthorblockA{\IEEEauthorrefmark{2}FEMTO-ST, Time-Frequency, University of Franche-Comte, Besançon, France\\
-\{michel.lenczner@utbm.fr,gwenhael.goavec@trabucayre.com}
-}
-
-
-
-
-
-
-\maketitle
+%\maketitle
\thispagestyle{empty}
-\begin{abstract}
-
-
+\begin{IEEEkeywords}
+FPGA, cantilever arrays, interferometry.
+\end{IEEEkeywords}
-{\it keywords}: FPGA, cantilever, interferometry.
-\end{abstract}
+\IEEEpeerreviewmaketitle
\section{Introduction}
-Cantilevers are used inside atomic force microscope which provides high
-resolution images of surfaces. Several technics have been used to measure the
-displacement of cantilevers in litterature. For example, it is possible to
-determine accurately the deflection with different mechanisms.
-In~\cite{CantiPiezzo01}, authors used piezoresistor integrated into the
-cantilever. Nevertheless this approach suffers from the complexity of the
-microfabrication process needed to implement the sensor in the cantilever.
-In~\cite{CantiCapacitive03}, authors have presented an cantilever mechanism
-based on capacitive sensing. This kind of technic also involves to instrument
-the cantiliver which result in a complex fabrication process.
-
-In this paper our attention is focused on a method based on interferometry to
-measure cantilevers' displacements. In this method cantilevers are illuminated
-by an optic source. The interferometry produces fringes on each cantilevers
-which enables to compute the cantilever displacement. In order to analyze the
-fringes a high speed camera is used. Images need to be processed quickly and
-then a estimation method is required to determine the displacement of each
-cantilever. In~\cite{AFMCSEM11}, the authors have used an algorithm based on
-spline to estimate the cantilevers' positions.
-%%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
-the computer is a bootleneck in the overall process. In this paper we propose to
-use a method based on least square and to implement all the computation on a
-FGPA.
-
-The remainder of the paper is organized as follows. Section~\ref{sec:measure}
-describes more precisely the measurement process. Our solution based on the
-least square method and the implementation on FPGA is presented in
-Section~\ref{sec:solus}. Experimentations are described in
-Section~\ref{sec:results}. Finally a conclusion and some perspectives are
-presented.
-
-
-
-%% quelques ref commentées sur les calculs basés sur l'interférométrie
-
-\section{Measurement principles}
-\label{sec:measure}
-
-
-
-
+Cantilevers are used in atomic force microscopes (AFM) which provide high
+resolution surface images. Several techniques have been reported in
+literature for cantilever displacement measurement. In~\cite{CantiPiezzo01},
+authors have shown how a piezoresistor can be integrated into a cantilever
+for deflection measurement. Nevertheless this approach suffers from the
+complexity of the microfabrication process needed to implement the sensor.
+In~\cite{CantiCapacitive03}, authors have presented a cantilever mechanism
+based on capacitive sensing. These techniques require cantilever
+instrumentation resulting in\ complex fabrication processes.
+
+In this paper our attention is focused on a method based on interferometry for
+cantilever displacement measurement in quasi-static regime. Cantilevers are
+illuminated by an optical source. Interferometry produces fringes enabling
+cantilever displacement computation. A high speed camera is used to analyze the
+fringes. In view of real time applications, images need to be processed quickly
+and then a fast estimation method is required to determine the displacement of
+each cantilever. In~\cite{AFMCSEM11}, an algorithm based on spline has been
+introduced for cantilever position estimation. The overall process gives
+accurate results but computations are performed on a standard computer using
+LabView \textsuperscript{\textregistered} \textsuperscript{\copyright}.
+Consequently, the main drawback of this implementation is that the computer is a
+bottleneck. In this paper we pose the problem of real-time cantilever position
+estimation and bring a hardware/software solution. It includes a fast method
+based on least squares and its FPGA implementation.
+
+The remainder of the paper is organized as follows. Section~\ref{sec:measure}
+describes the measurement process. Our solution based on the least square
+method and its implementation on a FPGA is presented in Section~\ref{sec:solus}. Numerical experimentations are described in Section~\ref{sec:results}. Finally a conclusion and some perspectives are drawn.
+
+\section{Architecture and goals}
+\label{sec:measure}
+In order to build simple, cost effective and user-friendly cantilever
+arrays, authors of ~\cite{AFMCSEM11} have developed a system based on
+interferometry.
+\subsection{Experimental setup}
-\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.
-%%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.
-
+In opposition to other optical based system\textbf{s u}sing a laser beam
+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 is based on a Linnick interferometer~\cite{Sinclair:05}.
+It is illustrated in Figure~\ref{fig:AFM} \footnote{by courtesy of
+ CSEM}. A laser diode is first split (by the splitter) into a
+reference beam and a sample beam both reaching the cantilever array.
+The complete system including a cantilever array\ and the optical
+system can be moved thanks to a translation and rotational hexapod
+stage with five degrees of freedom. 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. Then, cantilever motion in
+the transverse direction produces movements in the fringes. They are
+detected with the CMOS camera which images are analyzed by a Labview
+program to recover the cantilever deflections.
+
+\begin{figure}[tbp]
+\begin{center}
+\includegraphics[width=\columnwidth]{AFM}
+\end{center}
+\caption{AFM Setup}
+\label{fig:AFM}
+\end{figure}
%% image tirée des expériences.
-\subsection{Cantilever deflection estimation}
-\label{sec:deflest}
-
-As shown on image \ref{img:img-xp}, each cantilever is covered by
-interferometric fringes. The fringes will distort when cantilevers are
-deflected. Estimating the deflection is done by computing this
-distortion. For that, (ref A. Meister + M Favre) proposed a method
-based on computing the phase of the fringes, at the base of each
-cantilever, near the tip, and on the base of the array. They assume
-that a linear relation binds these phases, which can be use to
-"unwrap" the phase at the tip and to determine the deflection.\\
-
-More precisely, segment of pixels are extracted from images taken by a
-high-speed camera. These segments are large enough to cover several
-interferometric fringes and are placed at the base and near the tip of
-the cantilevers. They are called base profile and tip profile in the
-following. Furthermore, a reference profile is taken on the base of
-the cantilever array.
+\subsection{Inteferometric based cantilever deflection estimation}
-The pixels intensity $I$ (in gray level) of each profile is modelized by :
+\label{sec:deflest}
+\begin{figure}[tbp]
+\begin{center}
+\includegraphics[width=\columnwidth]{lever-xp}
+\end{center}
+\caption{Portion of a camera image showing moving interferometric fringes in
+cantilevers}
+\label{fig:img-xp}
+\end{figure}
+
+As shown in Figure \ref{fig:img-xp} \footnote{by courtesy of CSEM}, each
+cantilever is covered by several interferometric fringes. The fringes
+distort when cantilevers are deflected. In \cite{AFMCSEM11}, a novel
+method for interferometric based cantilever deflection measurement was
+reported. For each cantilever, the method uses three segments of pixels,
+parallel to its section, to determine phase shifts. The first is
+located just above the AFM tip (tip profile), it provides the phase
+shift modulo $2\pi $. The second one is close to the base junction
+(base profile) and is used to determine the exact multiple of $2\pi $
+through an operation called unwrapping where it is assumed that the
+deflection means along the two measurement segments are linearly
+dependent. The third is on the base and provides a reference for
+noise suppression. Finally, deflections are simply derived from phase
+shifts.
+
+The pixel gray-level intensity $I$ of each profile is modelized by%
\begin{equation}
-\label{equ:profile}
-I(x) = ax+b+A.cos(2\pi f.x + \theta)
-\end{equation}
-
-where $x$ is the position of a pixel in its associated segment.
+I(x)=A\text{ }\cos (2\pi fx+\theta )+ax+b \label{equ:profile}
+\end{equation}%
+where $x$ denotes the position of a pixel in a segment, $A$, $f$ and $\theta
+$ are the amplitude, the frequency and the phase of the light signal when
+the affine function $ax+b$ corresponds to the cantilever array surface tilt
+with respect to the light source.
+
+The method consists in two main sequences. In the first one
+corresponding to precomputation, the frequency $f$ of each profile is
+determined using a spline interpolation (see section \ref%
+{sec:algo-spline}) and the coefficient used for phase unwrapping is
+computed. The second one, that we call the \textit{acquisition loop,}
+is done after images have been taken at regular time steps. For each
+image, the phase $\theta $ of all profiles is computed to obtain,
+after unwrapping, the cantilever deflection. The phase determination
+in \cite{AFMCSEM11} is achieved by a spline based algorithm which is
+the most consuming part of the computation. In this article, we
+propose an alternate version based on the least square method which is
+faster and better suited for FPGA implementation.
+
+\subsection{Computation design goals}
-The global method consists in two main sequences. The first one aims
-to determin the frequency $f$ of each profile with an algorithm based
-on spline interpolation (see section \ref{algo-spline}). It also
-computes the coefficient used for unwrapping the phase. The second one
-is the acquisition loop, while which images are taken at regular time
-steps. For each image, the phase $\theta$ of all profiles is computed
-to obtain, after unwrapping, the deflection of cantilevers.
-
-\subsection{Design goals}
\label{sec:goals}
-If we put aside some hardware issues like the speed of the link
+To evaluate the solution performances, we choose a goal which consists
+in designing a computing unit able to estimate the deflections of
+a $10\times 10$%
+-cantilever array, faster than the camera image stream. In addition,
+the result accuracy must be close to 0.3nm, the maximum precision
+reached in \cite{AFMCSEM11}. Finally, the latency between the entrance
+of the first pixel of an image and the end of deflection computation
+must be as small as possible. All these requirement are
+stated in the perspective of implementing real-time active control for
+each cantilever, see~\cite{LencznerChap10,Hui11}.
+
+If we put aside other 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
-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.
+pixels and to store them in memory, the phase computation is the
+bottleneck of the whole process. For example, the camera in the setup
+of \cite{AFMCSEM11} provides $%
+1024\times 1204$ pixels with an exposition time of 2.5ms. Thus, if we
+the pixel extraction time is neglected, each phase calculation of a
+100-cantilever array should take no more than 12.5$\mu$s.
+
+In fact, this timing is a very hard constraint. To illustrate this point, we
+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.
+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 is using 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. Solutions would be to use a real-time operating system or
+to search for a more efficient algorithm.
+
+However, the main drawback is the latency of such a solution because each
+profile must be treated one after another and the deflection of 100
+cantilevers takes about $200\times 10.5=2.1$ms. This would be inadequate
+for real-time requirements as for individual cantilever active 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.
+
+We remark that when possible, it is more efficient to pipeline the
+computation. For example, supposing that 200 profiles of 20 pixels
+could be pushed sequentially in a 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. Such a solution would be meeting our requirements and
+would be 50 times faster than our C code, and even more compared to
+the LabView version use in \cite{AFMCSEM11}. FPGAs are appropriate for
+such implementation, so they turn out to be the computation units of
+choice to reach our performance requirements. Nevertheless, passing
+from a C code to a pipelined version in VHDL is not obvious at all. It
+can even be impossible because of FPGA hardware constraints. All these
+points are discussed in the following sections.
+
+\section{An hardware/software solution}
-%% to be continued ...
-
-%% � faire : timing de l'algo spline en C avec atan et tout le bordel.
-
-
-
-
-\section{Proposed solution}
\label{sec:solus}
+In this section we present parts of the computing solution to the above
+requirements. The hardware part consists in a high-speed camera, linked on an
+embedded board hosting two 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 profiles and computes the deflection.
+
+We first give some general information about FPGAs, then we
+describe the FPGA board we use for implementation and finally the two
+algorithms for phase computation are detailed. Presentation of VHDL
+implementations is postponned until Section \ref{Experimental tests}.
+
+
+
+\subsection{Elements of FPGA architecture and programming}
+
+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
+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 and
+other functions like trigonometric functions are not available and must be
+built by configuring a set of CLBs. Since this is not an obvious task at
+all, tools like ISE~\cite{ISE} have been built to do this operation. Such a
+software can synthetize a design written in a hardware description language
+(HDL), maps it onto CLBs, place/route them for a specific FPGA, and finally
+produces 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
+into 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 be executed
+concurrently. Signals may be combined with basic logic operations to
+produce new states that are assigned 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 pipelines 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. We observe that the
+components of a task are not reusable by another one. Nevertheless, this is
+the most efficient technique on FPGAs. Because of their architecture, it is
+also very easy to process several data concurrently. Finally, the best
+performance can be reached when several pipelines are operating on multiple
+data streams in parallel.
+
+\subsection{The FPGA board}
+
+The architecture we use is designed by the Armadeus Systems
+company. It consists in a development board called APF27 \textsuperscript{\textregistered}, hosting a
+i.MX27 ARM processor (from Freescale) and a Spartan3A (from
+XIlinx). This board includes all classical connectors as USB and
+Ethernet for instance. A Flash memory contains a Linux kernel that can
+be launched after booting the board via u-Boot. The processor is
+directly connected to the Spartan3A via its special interface called
+WEIM. The Spartan3A is itself connected to an extension board called
+SP Vision \textsuperscript{\textregistered}, that hosts a Spartan6 FPGA. Thus, it is
+possible to develop programs that communicate between i.MX and
+Spartan6, using Spartan3 as a tunnel. A clock signal at 100MHz (by
+default) is delivered to dedicated FPGA pins. The Spartan6 of our
+board is an LX100 version. It has 15,822 slices, each slice containing
+4 LUTs and 8 flip/flops. It is equivalent to 101,261 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 for any purpose as to be connected to the
+interface of a camera.
+
+\subsection{Two algorithms for phase computation}
+
+In \cite{AFMCSEM11}, $f$ the frequency and $\theta $\ the phase of the
+light wave are computed thanks to spline interpolation. As said in
+section \ref{sec:deflest}, $f$ is computed only once time but the
+phase needs to be computed for each image. This is why, in this paper,
+we focus on its computation.
+
+\subsubsection{Spline algorithm (SPL)}
-\subsection{FPGA constraints}
-
-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, ...
-
-
-\subsection{Considered algorithms}
-
-Two solutions have been studied to achieve phase computation. The
-original one, proposed by A. Meister and M. Favre, is based on
-interpolation by splines. It allows to compute frequency and
-phase. The second one, detailed in this article, is based on a
-classical least square method but suppose that frequency is already
-known.
-
-\subsubsection{Spline algorithm}
\label{sec:algo-spline}
-Let consider a profile $P$, that is a segment of $M$ pixels with an
-intensity in gray levels. Let call $I(x)$ the intensity of profile in $x
-\in [0,M[$.
-
-At first, only $M$ values of $I$ are known, for $x = 0, 1,
-\ldots,M-1$. A normalisation allows to scale known intensities into
-$[-1,1]$. We compute splines that fit at best these normalised
-intensities. Splines 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.
-
-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 :
+
+We denote by $M$ the number of pixels in a segment used for phase
+computation. For the sake of simplicity of the notations, we consider
+the light intensity $I$ to be a mapping of the physical segment in the
+interval $[0,M[$. The pixels are assumed to be regularly spaced and
+centered at the positions $x^{p}\in\{0,1,\ldots,M-1\}.$ We use the simplest
+definition of a pixel, namely the value of $I$ at its center. The
+pixel intensities are considered as pre-normalized so that their
+minimum and maximum have been resized to $-1$ and $1$.
+
+The first step consists in computing the cubic spline interpolation of
+the intensities. This allows for interpolating $I$ at a larger number
+$L=k\times M$ of points (typically $k=4$ is sufficient) $%
+x^{s}$ in the interval $[0,M[$. During the precomputation sequence,
+the second step is to determin the afine part $a.x+b$ of $I$. It is
+found with an ordinary least square method, taking account the $L$
+points. Values of $I$ in $x^s$ are used to compute its intersections
+with $a.x+b$. The period of $I$ (and thus its frequency) is deduced
+from the number of intersections and the distance between the first
+and last.
+
+During the acquisition loop, the second step is the phase computation, with
\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]
+\theta =atan\left[ \frac{\sum_{i=0}^{N-1}\text{sin}(2\pi fx_{i}^{s})\times
+I(x_{i}^{s})}{\sum_{i=0}^{N-1}\text{cos}(2\pi fx_{i}^{s})\times I(x_{i}^{s})}%
+\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 $\theta$.
-
-\subsubsection{Least square algorithm}
-
-Assuming that we compute the phase during the acquisition loop,
-equation \ref{equ:profile} has only 4 parameters :$a, b, A$, and
-$\theta$, $f$ and $x$ being already known. Since $I$ is non-linear, a
-least square method based an Gauss-newton algorithm must be used to
-determine these four parameters. Since it is an iterative process
-ending with a convergence criterion, it is obvious that it is not
-particularly adapted to our design goals.
-
-Fortunatly, it is quite simple to reduce the number of parameters to
-only $\theta$. Let $x^p$ be the coordinates of pixels in a segment of
-size $M$. Thus, $x^p = 0, 1, \ldots, M-1$. Let $I(x^p)$ be their
-intensity. Firstly, we "remove" the slope by computing :
-
-\[I^{corr}(x^p) = I(x^p) - a.x^p - b\]
-
-Since linear equation coefficients are searched, a classical least
-square method can be used to determine $a$ and $b$ :
+\textit{Remarks: }
-\[a = \frac{covar(x^p,I(x^p))}{var(x^p)} \]
-
-Assuming an overlined symbol means an average, then :
-
-\[b = \overline{I(x^p)} - a.\overline{{x^p}}\]
-
-Let $A$ be the amplitude of $I^{corr}$, i.e.
-
-\[A = \frac{max(I^{corr}) - min(I^{corr})}{2}\]
-
-Then, the least square method to find $\theta$ is reduced to search the minimum of :
-
-\[\sum_{i=0}^{M-1} \left[ cos(2\pi f.i + \theta) - \frac{I^{corr}(i)}{A} \right]^2\]
-
-It is equivalent to derivate this expression and to solve the following equation :
-
-\begin{eqnarray*}
-2\left[ cos\theta \sum_{i=0}^{M-1} I^{corr}(i).sin(2\pi f.i) + sin\theta \sum_{i=0}^{M-1} I^{corr}(i).cos(2\pi f.i)\right] \\
-- A\left[ cos2\theta \sum_{i=0}^{M-1} sin(4\pi f.i) + sin2\theta \sum_{i=0}^{M-1} cos(4\pi f.i)\right] = 0
-\end{eqnarray*}
-
-Several points can be noticed :
\begin{itemize}
-\item As in the spline method, some parts of this equation can be
- computed before the acquisition loop. It is the case of sums that do
- not depend on $\theta$ :
-
-\[ \sum_{i=0}^{M-1} sin(4\pi f.i), \sum_{i=0}^{M-1} cos(4\pi f.i) \]
+\item The frequency could also be obtained using the derivates of spline
+equations, which only implies to solve quadratic equations but certainly
+yields higher errors.
-\item Lookup tables for $sin(2\pi f.i)$ and $cos(2\pi f.i)$ can also be
-computed.
-
-\item The simplest method to find the good $\theta$ is to discretize
- $[-\pi,\pi]$ in $N$ 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 :
-
-\[ sin \theta, cos \theta, \left[ cos 2\theta \sum_{i=0}^{M-1} sin(4\pi f.i) + sin 2\theta \sum_{i=0}^{M-1} cos(4\pi f.i)\right] \]
+\item Profile frequency are computed during the precomputation step,
+ thus the values sin$(2\pi fx_{i}^{s})$ and cos$(2\pi fx_{i}^{s})$
+ can be determined once for all.
+\end{itemize}
-\item This search can be very fast using a dichotomous process in $log_2(N)$
+\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 $\theta $, $f$ and $x$
+being already known. Since $I$ is non-linear, a least square method based on
+a Gauss-newton algorithm can be used to determine these four parameters.
+This kind of iterative process ends with a convergence criterion, so it is
+not suited to our design goals. Fortunately, it is quite simple to reduce
+the number of parameters to only $\theta $. Firstly, the afine part $ax+b$
+is estimated from the $M$ values $I(x^{p})$ to determine the rectified
+intensities,%
+\begin{equation*}
+I^{corr}(x^{p})\approx I(x^{p})-a.x^{p}-b.
+\end{equation*}%
+To find $a$ and $b$ we apply an ordinary least square method (as in SPL but on $M$ points)%
+\begin{equation*}
+a=\frac{covar(x^{p},I(x^{p}))}{\text{var}(x^{p})}\text{ and }b=\overline{%
+I(x^{p})}-a.\overline{{x^{p}}}
+\end{equation*}%
+where overlined symbols represent average. Then the amplitude $A$ is
+approximated by%
+\begin{equation*}
+A\approx \frac{\text{max}(I^{corr})-\text{min}(I^{corr})}{2}.
+\end{equation*}%
+Finally, the problem of approximating $\theta $ is reduced to minimizing%
+\begin{equation*}
+\min_{\theta \in \lbrack -\pi ,\pi ]}\sum_{i=0}^{M-1}\left[ \text{cos}(2\pi
+f.i+\theta )-\frac{I^{corr}(i)}{A}\right] ^{2}.
+\end{equation*}%
+An optimal value $\theta ^{\ast }$ of the minimization problem is a zero of
+the first derivative of the above argument,%\begin{eqnarray*}{l}
+\begin{equation*}
+2\left[ \text{cos}\theta ^{\ast }\sum_{i=0}^{M-1}I^{corr}(i).\text{sin}(2\pi
+f.i)\right.
+\end{equation*}%
+\begin{equation*}
+\left. +\text{sin}\theta ^{\ast }\sum_{i=0}^{M-1}I^{corr}(i).\text{cos}(2\pi
+f.i)\right] -
+\end{equation*}%
+\begin{equation*}
+A\left[ \text{cos}2\theta ^{\ast }\sum_{i=0}^{M-1}\sin (4\pi f.i)+\text{sin}%
+2\theta ^{\ast }\sum_{i=0}^{M-1}\cos (4\pi f.i)\right] =0
+\end{equation*}%
+%
+%\end{eqnarray*}
+
+Several points can be noticed:
+\begin{itemize}
+\item The terms $\sum_{i=0}^{M-1}$sin$(4\pi f.i)$ and$\sum_{i=0}^{M-1}$cos$%
+(4\pi f.i)$ are independent of $\theta $, they can be precomputed.
+
+\item Lookup tables (namely lut$_{sfi}$ and lut$_{cfi}$ in the following algorithms) can be
+ set with the $2.M$ values $\sin (2\pi f.i)$ and $\cos (2\pi f.i)$.
+
+\item A simple method to find a zero $\theta ^{\ast }$ of the optimality
+condition is to discretize the range $[-\pi ,\pi ]$ with a large number $%
+nb_{s}$ of nodes and to find which one is a minimizer in the absolute value
+sense. Hence, three other lookup tables (lut$_{s}$, lut$_{c}$ and lut$_{A}$) can be set with the $%
+3\times nb_{s}$ values $\sin \theta ,$ $\cos \theta ,$
+\begin{equation*}
+\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] .
+\end{equation*}
+
+\item The search algorithm 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]
+The overall method is synthetized in an algorithm (called LSQ in the
+following) divided into the precomputing part and 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}\\
+ $I[i] \leftarrow I[i] - start - slope\times i$\\
}
-
+
$I_{max} \leftarrow max_i(I[i])$, $I_{min} \leftarrow min_i(I[i])$\\
$amp \leftarrow \frac{I_{max}-I_{min}}{2}$\\
$Is \leftarrow 0$, $Ic \leftarrow 0$\\
\For{$i=0$ to $M-1$}{
- $Is \leftarrow Is + I[i]\times $ lut\_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 {
+ $b_r \leftarrow b_l + 1$\\
}
+ \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{Algorithm comparison}
-\subsubsection{Comparison}
+We compared the two algorithms on the base of three criteria:
-\subsection{VHDL design paradigms}
+\begin{itemize}
+\item precision of results on a cosines profile, distorted by noise,
-\subsection{VHDL implementation}
+\item number of operations,
-\section{Experimental results}
-\label{sec:results}
+\item complexity of FPGA implementation
+\end{itemize}
+For the first item, we produced a matlab version of each algorithm,
+running in double precision. The profile was generated for about
+34,000 different quadruplets of periods ($\in \lbrack 3.1,6.1]$, step
+= 0.1), phases ($\in \lbrack -3.1,3.1]$, steps = 0.062) and slope
+($\in \lbrack -2,2]$, step = 0.4). Obviously, the discretization of
+$[-\pi ,\pi ]$ introduces an error in the phase estimation. It is at
+most equal to $\frac{\pi}{nb_s}$. From some experiments on a $17\times
+4$ array, authors of \cite{AFMCSEM11} noticed a average ratio of 50
+between phase variation in radians and lever end position in
+nanometers. Assuming such a ratio and $nb_s = 1024$, the maximum lever
+deflection error would be 0.15nm which is smaller than 0.3nm, the best
+precision achieved with the setup used in \cite{AFMCSEM11}.
+
+Moreover, pixels have been paired and the paired intensities have been
+perturbed by addition of a random number uniformly picked in
+$[-N,N]$. Notice that we have observed that perturbing each pixel
+independently yields too weak profile distortion. We report
+percentages of errors between the reference and the computed phases
+out of $2\pi ,$%
+\begin{equation*}
+err=100\times \frac{|\theta _{ref}-\theta _{comp}|}{2\pi }.
+\end{equation*}%
+Table \ref{tab:algo_prec} gives the maximum and the average errors for both
+algorithms and for increasing values of $N$ the noise parameter.
+
+\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 (N)& 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}%
+\end{center}
+\caption{Error (in \%) for cosines profiles, with noise.}
+\label{tab:algo_prec}
+\end{table}
+
+The results show that the two algorithms yield close results, with a slight
+advantage for LSQ. Furthermore, both behave very well against noise.
+Assuming an average 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.
+
+It is very hard to predict which level of noise will be present in
+real experiments and how it will distort the profiles. Authors of
+\cite{AFMCSEM11} gave us the authorization to exploit some of their
+results on a $17\times 4$ array. It allowed us to compare experimental
+profiles to simulated ones. 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. In fact, it is
+very close to some of the worst experimental profiles. Figure
+\ref{fig:noise60} shows a sample of worst profile for $N=30$. It is
+completely distorted, largely beyond any experimental ones. Obviously,
+these comparisons are a bit subjectives and experimental profiles
+could also be completly distorted on other experiments. Nevertheless,
+they give an idea about the possible error.
+
+\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 the use of the arctangent function. In both cases, the number
+of operation is proportional to $M$ the numbers of pixels. For LSQ, it also
+depends on $nb_{s}$ and for SPL on $L=k\times M$ the number of interpolated
+points. We assume that $M=20$, $nb_{s}=1024$ and $k=4$, that all possible
+parts are already in lookup tables and that 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 fully relevant for FPGA
+implementation which time and space consumption depends not only on the type
+of operations but also of their ordering. The final evaluation is thus very
+much driven by the third criterion.
+
+The Spartan 6 used in our architecture has a hard constraint since it
+has no built-in floating point units. Obviously, it is possible to use
+some existing "black-boxes" for double precision operations. But they
+require a lot of clock cycles to complete. It is much simpler to
+exclusively use integers, with a quantization of all double precision
+values. It should be chosen in a manner that does not alterate result
+precision. 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 since a DSP48 takes inputs of 18 bits
+maximum. So, for larger multiplications, several DSP must be combined
+which increases the overall latency.
+
+In the present algorithms, the hardest constraint does not come from the
+FPGA characteristics but from the algorithms. Their VHDL implementation can
+be efficient only if they can be fully (or near) pipelined. We observe that
+only a small part of SPL can be pipelined, indeed, the computation of spline
+coefficients implies to solve a linear tridiagonal system which matrix and
+right-hand side are 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 since all parts, except the dichotomic search, work
+on the same amount of data, i.e. the profile size. Furthermore, LSQ requires
+less operations than SPL, implying a smaller output latency. In total, LSQ
+turns out to be the best candidate for phase computation on any architecture
+including FPGA.
+
+\section{VHDL implementation and experimental tests}
+
+\label{Experimental tests}
+\subsection{VHDL implementation}
+From the LSQ algorithm, we have written a C program that uses only
+integer values. We used a very simple quantization which consists in
+multiplying each double precision value by a factor power of two and
+by keeping the integer part. For an accurate evaluation of the
+division in the computation of $a$ the slope coefficient, we also
+scaled the pixel intensities by another power of two. The main problem
+was to determin these factors. Most of the time, they are chosen to
+minimize the error induced by the quantization. But in our case, we
+also have some hardware constraints, for example the size and depth of
+RAMs or the input size of DSPs. Thus, having a maximum of values that
+fit in these sizes is a very important criterion to choose the scaling
+factors.
+
+Consequently, we have determined the maximum value of each variable as
+a function of the scale factors and the profile size involved in the
+algorithm. It gave us the the maximum number of bits necessary to code
+them. We have chosen the scale factors so that any variable (except
+the covariance) fits in 18 bits, which is the maximum input size of
+DSPs. In this way, all multiplications, except one, could be done with
+a single DSP, in a single clock cycle. Moreover, assuming that $nb_s =
+1024$, all LUTs could fit in the 18Kbits RAMs. Finally, we compared
+the double and integer versions of LSQ and found a nearly perfect
+agreement between their results.
+
+As mentionned above, some operations like divisions must be
+avoided. But when the denominator is fixed, a division can be replaced
+by its multiplication/shift counterpart. This is always the case in
+LSQ. For example, assuming that $M$ is fixed, $x_{var}$ is known and
+fixed. Thus, $\frac{xy_{covar}}{x_{var}}$ can be replaced by
+
+\[ (xy_{covar}\times \left \lfloor\frac{2^n}{x_{var}} \right \rfloor) \gg n\]
+
+where $n$ depends on the desired precision (in our case $n=24$).
+
+Obviously, multiplications and divisions by a power of two can be
+replaced by left or right bit shifts. Finally, the code only contains
+shifts, additions, subtractions and multiplications of signed integers, which
+are perfectly adapted to FGPAs.
+
+
+We built two versions of VHDL codes, namely one directly by hand
+coding and the other with Matlab using the Simulink HDL coder feature~\cite%
+{HDLCoder}. Although the approaches are completely different we obtained
+quite comparable VHDL codes. Each approach has advantages and drawbacks.
+Roughly speaking, hand coding provides beautiful and much better structured
+code while Simulind HDL coder produces allows for fast code production. In
+terms of throughput and latency, simulations show that the two approaches
+yield close results with a slight advantage for hand coding.
+
+\subsection{Simulation}
+
+Before experimental tests on the FPGA board, we simulated our two VHDL
+codes with GHDL and GTKWave (two free tools with linux). We built a
+testbench based on experimental profiles and compared the results to
+values given by the SPL algorithm. Both versions lead to correct
+results. Our first codes were highly optimized, indeed 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, 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 routed
+with ISE on the Spartan6 with a 100MHz clock. The main problems were
+encountered with series of arthmetic operations and more especially
+with RAM outputs used in DSPs. So, we needed to decompose some parts
+of the pipeline, which added few clock cycles. Finally, we obtained a
+bitstream that has been successfully tested on the board.
+
+Its latency is 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. It corresponds to about 12300 images
+per second, which is largely beyond the camera capacities and the
+possibility to extract a new profile from an image every 40
+cycles. Nevertheless, it also largely fits our design goals.
+
+\label{sec:results}
\section{Conclusion and perspectives}
+In this paper we have presented a full hardware/software solution for
+real-time cantilever deflection computation from interferometry images.
+Phases are computed thanks to a new algorithm based on the least square
+method. It has been quantized and pipelined to be mapped into a FPGA, the
+architecture of our solution. Performances have been analyzed through
+simulations and real experiments on a Spartan6 FPGA. The results meet our
+initial requirements. In future work, the algorithm quantization will be
+better analyzed and an high speed camera will be introduced in the
+processing chain so that to process real images. Finally, we will address
+real-time filtering and control problems for AFM arrays in dynamic regime.
+
+\section{Acknowledgments}
+We would like to thank A. Meister and M. Favre, from CSEM, for sharing all the
+material we used to write this article and for the time they spent to
+explain us their approach.
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