-\documentclass{article}
+%\usepackage{latex8}
+%\usepackage{times}
+%\usepackage[cyr]{aeguill}
+%\usepackage{pstricks,pst-node,pst-text,pst-3d}
+%\usepackage{babel}
+%\input{psfig.sty}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%% LyX specific LaTeX commands.
+
+
+\documentclass[10pt, peerreview, compsocconf]{IEEEtran}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\usepackage[utf8]{inputenc}
+\usepackage{amsmath}
+\usepackage{url}
+\usepackage{graphicx}
+\usepackage{thumbpdf}
+\usepackage{color}
+\usepackage{moreverb}
+\usepackage{commath}
+\usepackage{subfigure}
+\usepackage{fullpage}
+\usepackage{fancybox}
+\usepackage[ruled,lined,linesnumbered]{algorithm2e}
+
+\setcounter{MaxMatrixCols}{10}
+%TCIDATA{OutputFilter=LATEX.DLL}
+%TCIDATA{Version=5.50.0.2953}
+%TCIDATA{<META NAME="SaveForMode" CONTENT="1">}
+%TCIDATA{BibliographyScheme=BibTeX}
+%TCIDATA{LastRevised=Wednesday, October 26, 2011 09:49:54}
+%TCIDATA{<META NAME="GraphicsSave" CONTENT="32">}
+
+\newcommand{\noun}[1]{\textsc{#1}}
+\newcommand{\tab}{\ \ \ }
+
+
\begin{document}
-\abstract {
-In this paper we describe....
-}
+\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 2: name of organization, acronyms acceptable\\
+%% line 3: City, Country\\
+%% line 4: Email: name@xyz.com}
+%% \and
+%% \IEEEauthorblockN{Authors Name/s per 2nd Affiliation (Author)}
+%% \IEEEauthorblockA{line 1 (of Affiliation): dept. name of organization\\
+%% line 2: name of organization, acronyms acceptable\\
+%% line 3: City, Country\\
+%% line 4: Email: name@xyz.com}
+%% }
+
+%\maketitle
+
+\thispagestyle{empty}
+
+\begin{IEEEkeywords}
+FPGA, cantilever arrays, interferometry.
+\end{IEEEkeywords}
+
+\IEEEpeerreviewmaketitle
\section{Introduction}
-\section{Conclusion}
+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}
+
+\label{sec:archi}
+
+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{Inteferometric based cantilever deflection estimation}
+
+\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}
+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}
+
+\label{sec:goals}
+
+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 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}
+
+\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)}
+
+\label{sec:algo-spline}
+
+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}\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}
+
+\textit{Remarks: }
+
+\begin{itemize}
+\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 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}
+
+\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}
+
+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 $ intensity of pixels\\
+ $f \leftarrow $ frequency of the profile\\
+ $s4i \leftarrow \sum_{i=0}^{M-1} sin(4\pi f.i)$\\
+ $c4i \leftarrow \sum_{i=0}^{M-1} cos(4\pi f.i)$\\
+ $nb_s \leftarrow $ number of discretization steps of $[-\pi,\pi]$\\
+
+ \For{$i=0$ to $nb_s $}{
+ $\theta \leftarrow -\pi + 2\pi\times \frac{i}{nb_s}$\\
+ lut$_s$[$i$] $\leftarrow sin \theta$\\
+ lut$_c$[$i$] $\leftarrow cos \theta$\\
+ lut$_A$[$i$] $\leftarrow cos 2 \theta \times s4i + sin 2 \theta \times c4i$\\
+ lut$_{sfi}$[$i$] $\leftarrow sin (2\pi f.i)$\\
+ lut$_{cfi}$[$i$] $\leftarrow cos (2\pi f.i)$\\
+ }
+\end{algorithm}
+
+\begin{algorithm}[htbp]
+\caption{LSQ algorithm - during acquisition loop.}
+\label{alg:lsq-during}
+
+ $\bar{x} \leftarrow \frac{M-1}{2}$\\
+ $\bar{y} \leftarrow 0$, $x_{var} \leftarrow 0$, $xy_{covar} \leftarrow 0$\\
+ \For{$i=0$ to $M-1$}{
+ $\bar{y} \leftarrow \bar{y} + $ I[$i$]\\
+ $x_{var} \leftarrow x_{var} + (i-\bar{x})^2$\\
+ }
+ $\bar{y} \leftarrow \frac{\bar{y}}{M}$\\
+ \For{$i=0$ to $M-1$}{
+ $xy_{covar} \leftarrow xy_{covar} + (i-\bar{x}) \times (I[i]-\bar{y})$\\
+ }
+ $slope \leftarrow \frac{xy_{covar}}{x_{var}}$\\
+ $start \leftarrow y_{moy} - slope\times \bar{x}$\\
+ \For{$i=0$ to $M-1$}{
+ $I[i] \leftarrow I[i] - start - slope\times i$\\
+ }
+
+ $I_{max} \leftarrow max_i(I[i])$, $I_{min} \leftarrow min_i(I[i])$\\
+ $amp \leftarrow \frac{I_{max}-I_{min}}{2}$\\
+
+ $Is \leftarrow 0$, $Ic \leftarrow 0$\\
+ \For{$i=0$ to $M-1$}{
+ $Is \leftarrow Is + I[i]\times $ lut$_{sfi}$[$i$]\\
+ $Ic \leftarrow Ic + I[i]\times $ lut$_{cfi}$[$i$]\\
+ }
+
+ $\delta \leftarrow \frac{nb_s}{2}$, $b_l \leftarrow 0$, $b_r \leftarrow \delta$\\
+ $v_l \leftarrow -2.I_s - amp.$lut$_A$[$b_l$]\\
+
+ \While{$\delta >= 1$}{
+
+ $v_r \leftarrow 2.[ Is.$lut$_c$[$b_r$]$ + Ic.$lut$_s$[$b_r$]$ ] - amp.$lut$_A$[$b_r$]\\
+
+ \If{$!(v_l < 0$ and $v_r >= 0)$}{
+ $v_l \leftarrow v_r$ \\
+ $b_l \leftarrow b_r$ \\
+ }
+ $\delta \leftarrow \frac{\delta}{2}$\\
+ $b_r \leftarrow b_l + \delta$\\
+ }
+ \uIf{$!(v_l < 0$ and $v_r >= 0)$}{
+ $v_l \leftarrow v_r$ \\
+ $b_l \leftarrow b_r$ \\
+ $b_r \leftarrow b_l + 1$\\
+ $v_r \leftarrow 2.[ Is.$lut$_c$[$b_r$]$ + Ic.$lut$_s$[$b_r$]$ ] - amp.$lut$_A$[$b_r$]\\
+ }
+ \Else {
+ $b_r \leftarrow b_l + 1$\\
+ }
+
+ \uIf{$ abs(v_l) < v_r$}{
+ $b_{\theta} \leftarrow b_l$ \\
+ }
+ \Else {
+ $b_{\theta} \leftarrow b_r$ \\
+ }
+ $\theta \leftarrow \pi\times \left[\frac{2.b_{ref}}{nb_s}-1\right]$\\
+
+\end{algorithm}
+
+\subsubsection{Algorithm comparison}
+
+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 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.
+
+\bibliographystyle{plain}
+\bibliography{biblio}
\end{document}
