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\begin{document}
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+
+\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}, Mélanie Favre\IEEEauthorrefmark{3},
+Michel Lenczner\IEEEauthorrefmark{2} and André Meister\IEEEauthorrefmark{3}} \\
+\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}\\
+\IEEEauthorblockA{\IEEEauthorrefmark{3}CSEM, Centre Suisse d’Electronique et de Microtechnique, Neuchatel, Switzerland \and
+\{melanie.favre,andre.meister\}@csem.ch}
+ }
+
+\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
+%\maketitle
+
+\section{Introduction}
+
+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}. 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:xp-test}. 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, we use a system based on interferometry. The two following
+sections summarize the original characteristics of its architecture
+and computation method.
+
+\subsection{Experimental setup}
+
+\label{sec:archi}
+
+In opposition to other optical based systems using 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}. 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}, each cantilever is covered by
+several interferometric fringes. The fringes distort when cantilevers
+are deflected. 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 coefficients used for phase unwrapping are
+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
+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. Moreover, it can be used in real-time,
+i.e. after each image is picked by the camera.
+
+\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 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 in section \ref{sec:algo-comp}, 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.
+
+It should be noticed 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. 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{sec:xp-test}.
+
+
+
+\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 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. The drawback is 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}
+
+As said in section \ref{sec:deflest}, $f$ is computed only once but
+the phase needs to be computed for each image. This is why, in this
+paper, we focus on its computation. The next section describes the
+original method, based on spline interpolation, and section
+\ref{sec:algo-square} presents the new one based on least
+squares. Finally, in section \ref{sec:algo-comp}, we compare the two
+algorithms from their FPGA implementation point of view.
+
+\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 $I$ a function on the interval [0,M] which itself
+is the range of a one-to-one mapping defined on the physical
+segment. 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 determine the affine 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 derivative 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)}
+\label{sec:algo-square}
+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. 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 $\theta$ only. Firstly, the affine 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
+ $\sin \theta$, $\cos \theta$, and
+\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 $M-1$}{
+ $xy_{covar} \leftarrow xy_{covar} + (i-\bar{x}) \times (I[i]-\bar{y})$\\
+ }
+ $slope \leftarrow \frac{xy_{covar}}{x_{var}}$\\
+ $start \leftarrow \bar{y} - slope\times \bar{x}$\\
+ \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}
+\label{sec:algo-comp}
+We compared the two algorithms regarding 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 slopes
+($\in \lbrack -2,2]$, 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, we noticed an 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.
+
+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. Results on
+a $17\times 4$ array 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 subjective and experimental profiles could also
+be more 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 number 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 for 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.
+
+Nevertheless, in the present algorithms, the hardest constraint does
+not come from the FPGA characteristics but from the algorithms
+themselves. 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{sec:xp-test}
+
+\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 determine 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 width 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 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 with covariance)
+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 divisor 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 Simulink HDL coder allows 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 components that implement the wishbone protocol,
+in order to "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 arithmetic
+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 it computes a new phase every 40
+cycles. For 100 cantilevers, it takes $(112+200\times 40)\times 10ns =81.12\mu
+$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.
+
+\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}
