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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}

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%\maketitle

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\begin{IEEEkeywords}
FPGA, cantilever arrays, interferometry.
\end{IEEEkeywords}

\IEEEpeerreviewmaketitle

\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: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 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} \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 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
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. Moreover, it can be
used in real-time, i.e. after each image 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 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. 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}

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 a function on the interval [0,M] which 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 determin 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 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 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 $\sin \theta ,$ $\cos \theta ,$ 
\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 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 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, authors of \cite{AFMCSEM11} 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 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 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.

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 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.

\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}
