X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/dmems12.git/blobdiff_plain/68ef101fa4f71c2911e9ffa93ceb5e07afb4af88..HEAD:/dmems12.tex?ds=inline diff --git a/dmems12.tex b/dmems12.tex index 51d7344..331dbab 100644 --- a/dmems12.tex +++ b/dmems12.tex @@ -1,10 +1,15 @@ -\documentclass[12pt]{article} %\usepackage{latex8} %\usepackage{times} -\usepackage[latin1]{inputenc} -\usepackage[cyr]{aeguill} +%\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} @@ -13,250 +18,486 @@ \usepackage{moreverb} \usepackage{commath} \usepackage{subfigure} -%\input{psfig.sty} \usepackage{fullpage} \usepackage{fancybox} - \usepackage[ruled,lined,linesnumbered]{algorithm2e} -%%%%%%%%%%%%%%%%%%%%%%%%%%%% LyX specific LaTeX commands. -\newcommand{\noun}[1]{\textsc{#1}} +\setcounter{MaxMatrixCols}{10} +%TCIDATA{OutputFilter=LATEX.DLL} +%TCIDATA{Version=5.50.0.2953} +%TCIDATA{} +%TCIDATA{BibliographyScheme=BibTeX} +%TCIDATA{LastRevised=Wednesday, October 26, 2011 09:49:54} +%TCIDATA{} +\newcommand{\noun}[1]{\textsc{#1}} \newcommand{\tab}{\ \ \ } -%%%%%%%%%%%%%%%%%%%%%%%%%%%% my bib path. - -\title{Using FPGAs for high speed and real time cantilever deflection estimation} - -\author{ Raphaël COUTURIER\\ -Laboratoire d'Informatique -de l'Universit\'e de Franche-Comt\'e, \\ -BP 527, \\ -90016~Belfort CEDEX, France\\ - \and Stéphane Domas\\ -Laboratoire d'Informatique -de l'Universit\'e de Franche-Comt\'e, \\ -BP 527, \\ -90016~Belfort CEDEX, France\\ - \and Gwenhaël Goavec\\ -?? -?? \\ -??, \\ -??\\} +\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}, 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{document} +\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} -\maketitle +%% \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{abstract} - - +\begin{IEEEkeywords} +FPGA, cantilever arrays, interferometry. +\end{IEEEkeywords} -{\it keywords}: FPGA, cantilever, interferometry. -\end{abstract} +\IEEEpeerreviewmaketitle +%\maketitle \section{Introduction} -%% blabla + -%% quelques ref commentées sur les calculs basés sur l'interférométrie +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} -\section{Measurement principles} \label{sec:measure} -\subsection{Architecture} -\label{sec:archi} -%% description de l'architecture générale de l'acquisition d'images -%% avec au milieu une unité de traitement dont on ne précise pas ce -%% qu'elle est. - -%% image tirée des expériences. +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{Cantilever deflection estimation} -\label{sec:deflest} +\subsection{Experimental setup} -As shown on image \ref{img:img-xp}, each cantilever is covered by -interferometric fringes. The fringes will distort when cantilevers are -deflected. Estimating the deflection is done by computing this -distortion. For that, (ref A. Meister + M Favre) proposed a method -based on computing the phase of the fringes, at the base of each -cantilever, near the tip, and on the base of the array. They assume -that a linear relation binds these phases, which can be use to -"unwrap" the phase at the tip and to determine the deflection.\\ +\label{sec:archi} -More precisely, segment of pixels are extracted from images taken by a -high-speed camera. These segments are large enough to cover several -interferometric fringes and are placed at the base and near the tip of -the cantilevers. They are called base profile and tip profile in the -following. Furthermore, a reference profile is taken on the base of -the cantilever array. +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} -The pixels intensity $I$ (in gray level) of each profile is modelized by : +%% image tirée des expériences. -\begin{equation} -\label{equ:profile} -I(x) = ax+b+A.cos(2\pi f.x + \theta) -\end{equation} +\subsection{Inteferometric based cantilever deflection estimation} -where $x$ is the position of a pixel in its associated segment. +\label{sec:deflest} -The global method consists in two main sequences. The first one aims -to determin the frequency $f$ of each profile with an algorithm based -on spline interpolation (see section \ref{algo-spline}). It also -computes the coefficient used for unwrapping the phase. The second one -is the acquisition loop, while which images are taken at regular time -steps. For each image, the phase $\theta$ of all profiles is computed -to obtain, after unwrapping, the deflection of cantilevers. +\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} -\subsection{Design goals} \label{sec:goals} -If we put aside some hardware issues like the speed of the link +To evaluate the solution performances, we choose a goal which consists +in designing a computing unit able to estimate the deflections of a +$10\times 10$-cantilever array, faster than the camera image +stream. In addition, the result accuracy must be close to 0.3nm, the +maximum precision reached in~\cite{AFMCSEM11}. Finally, the latency +between the entrance of the first pixel of an image and the end of +deflection computation must be as small as possible. All these +requirement are stated in the perspective of implementing real-time +active control for each cantilever, see~\cite{LencznerChap10,Hui11}. + +If we put aside other hardware issues like the speed of the link between the camera and the computation unit, the time to deserialize -pixels and to store them in memory, ... the phase computation is -obviously the bottle-neck of the whole process. For example, if we -consider the camera actually in use, an exposition time of 2.5ms for -$1024\times 1204$ pixels seems the minimum that can be reached. For a -$10\times 10$ cantilever array, if we neglect the time to extract -pixels, it implies that computing the deflection of a single -cantilever should take less than 25$µ$s, thus 12.5$µ$s by phase.\\ - -In fact, this timing is a very hard constraint. Let consider a very -small programm that initializes twenty million of doubles in memory -and then does 1000000 cumulated sums on 20 contiguous values -(experimental profiles have about this size). On an intel Core 2 Duo -E6650 at 2.33GHz, this program reaches an average of 155Mflops. It -implies that the phase computation algorithm should not take more than -$240\times 12.5 = 1937$ floating operations. For integers, it gives -$3000$ operations. +pixels and to store them in memory, the phase computation is the +bottleneck of the whole process. For example, the camera in the setup +of \cite{AFMCSEM11} provides $1024\times 1204$ pixels with an +exposition time of 2.5ms. Thus, if 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} -%% to be continued ... - -%% à faire : timing de l'algo spline en C avec atan et tout le bordel. - - - - -\section{Proposed solution} \label{sec:solus} +In this section we present parts of the computing solution to the above +requirements. The hardware part consists in a high-speed camera linked on an +embedded board hosting two FPGAs. In this way, the camera output stream can be +pushed directly into the FPGA. The software part is mostly the VHDL code that +deserializes the camera stream, extracts profiles and computes the deflection. + +We first give some general information about FPGAs, then we +describe the FPGA board we use for implementation and finally the two +algorithms for phase computation are detailed. Presentation of VHDL +implementations is postponned until Section \ref{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)} -\subsection{FPGA constraints} - -%% contraintes imposées par le FPGA : algo pipeline/parallele, pas d'op math complexe, ... - - -\subsection{Considered algorithms} - -Two solutions have been studied to achieve phase computation. The -original one, proposed by A. Meister and M. Favre, is based on -interpolation by splines. It allows to compute frequency and -phase. The second one, detailed in this article, is based on a -classical least square method but suppose that frequency is already -known. - -\subsubsection{Spline algorithm} \label{sec:algo-spline} -Let consider a profile $P$, that is a segment of $M$ pixels with an -intensity in gray levels. Let call $I(x)$ the intensity of profile in $x -\in [0,M[$. - -At first, only $M$ values of $I$ are known, for $x = 0, 1, -\ldots,M-1$. A normalisation allows to scale known intensities into -$[-1,1]$. We compute splines that fit at best these normalised -intensities. Splines are used to interpolate $N = k\times M$ points -(typically $k=3$ is sufficient), within $[0,M[$. Let call $x^s$ the -coordinates of these $N$ points and $I^s$ their intensities. - -In order to have the frequency, the mean line $a.x+b$ (see equation \ref{equ:profile}) of $I^s$ is -computed. Finding intersections of $I^s$ and this line allow to obtain -the period thus the frequency. - -The phase is computed via the equation : + +We denote by $M$ the number of pixels in a segment used for phase +computation. For the sake of simplicity of the notations, we consider +the light intensity $I$ 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} sin(2\pi f x^s_i) \times I^s(x^s_i)}{\sum_{i=0}^{N-1} cos(2\pi f x^s_i) \times I^s(x^s_i)} \right] +\theta =atan\left[ \frac{\sum_{i=0}^{N-1}\text{sin}(2\pi fx_{i}^{s})\times +I(x_{i}^{s})}{\sum_{i=0}^{N-1}\text{cos}(2\pi fx_{i}^{s})\times I(x_{i}^{s})}% +\right] . \end{equation} -Two things can be noticed. Firstly, the frequency could also be -obtained using the derivates of spline equations, which only implies -to solve quadratic equations. Secondly, frequency of each profile is -computed a single time, before the acquisition loop. Thus, $sin(2\pi f -x^s_i)$ and $cos(2\pi f x^s_i)$ could also be computed before the loop, which leads to a -much faster computation of $\theta$. - -\subsubsection{Least square algorithm} - -Assuming that we compute the phase during the acquisition loop, -equation \ref{equ:profile} has only 4 parameters :$a, b, A$, and -$\theta$, $f$ and $x$ being already known. Since $I$ is non-linear, a -least square method based an Gauss-newton algorithm must be used to -determine these four parameters. Since it is an iterative process -ending with a convergence criterion, it is obvious that it is not -particularly adapted to our design goals. - -Fortunatly, it is quite simple to reduce the number of parameters to -only $\theta$. Let $x^p$ be the coordinates of pixels in a segment of -size $M$. Thus, $x^p = 0, 1, \ldots, M-1$. Let $I(x^p)$ be their -intensity. Firstly, we "remove" the slope by computing : - -\[I^{corr}(x^p) = I(x^p) - a.x^p - b\] - -Since linear equation coefficients are searched, a classical least -square method can be used to determine $a$ and $b$ : - -\[a = \frac{covar(x^p,I(x^p))}{var(x^p)} \] - -Assuming an overlined symbol means an average, then : +\textit{Remarks: } -\[b = \overline{I(x^p)} - a.\overline{{x^p}}\] - -Let $A$ be the amplitude of $I^{corr}$, i.e. - -\[A = \frac{max(I^{corr}) - min(I^{corr})}{2}\] - -Then, the least square method to find $\theta$ is reduced to search the minimum of : - -\[\sum_{i=0}^{M-1} \left[ cos(2\pi f.i + \theta) - \frac{I^{corr}(i)}{A} \right]^2\] - -It is equivalent to derivate this expression and to solve the following equation : - -\begin{eqnarray*} -2\left[ cos\theta \sum_{i=0}^{M-1} I^{corr}(i).sin(2\pi f.i) + sin\theta \sum_{i=0}^{M-1} I^{corr}(i).cos(2\pi f.i)\right] \\ -- A\left[ cos2\theta \sum_{i=0}^{M-1} sin(4\pi f.i) + sin2\theta \sum_{i=0}^{M-1} cos(4\pi f.i)\right] = 0 -\end{eqnarray*} - -Several points can be noticed : \begin{itemize} -\item As in the spline method, some parts of this equation can be - computed before the acquisition loop. It is the case of sums that do - not depend on $\theta$ : - -\[ \sum_{i=0}^{M-1} sin(4\pi f.i), \sum_{i=0}^{M-1} cos(4\pi f.i) \] - -\item Lookup tables for $sin(2\pi f.i)$ and $cos(2\pi f.i)$ can also be -computed. +\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 The simplest method to find the good $\theta$ is to discretize - $[-\pi,\pi]$ in $N$ steps, and to search which step leads to the - result closest to zero. By the way, three other lookup tables can - also be computed before the loop : - -\[ sin \theta, cos \theta, \left[ cos 2\theta \sum_{i=0}^{M-1} sin(4\pi f.i) + sin 2\theta \sum_{i=0}^{M-1} cos(4\pi f.i)\right] \] +\item Profile frequency are computed during the precomputation step, + thus the values sin$(2\pi fx_{i}^{s})$ and cos$(2\pi fx_{i}^{s})$ + can be determined once for all. +\end{itemize} -\item This search can be very fast using a dichotomous process in $log_2(N)$ +\subsubsection{Least square algorithm (LSQ)} +\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} -Finally, the whole summarizes in an algorithm (called LSQ in the following) in two parts, one before and one during the acquisition loop : -\begin{algorithm}[h] +The overall method is synthetized in an algorithm (called LSQ in the +following) divided into the precomputing part and the acquisition loop. + +\begin{algorithm}[htbp] \caption{LSQ algorithm - before acquisition loop.} \label{alg:lsq-before} $M \leftarrow $ number of pixels of the profile\\ - I[] $\leftarrow $ intensities of pixels\\ + I[] $\leftarrow $ intensity of pixels\\ $f \leftarrow $ frequency of the profile\\ $s4i \leftarrow \sum_{i=0}^{M-1} sin(4\pi f.i)$\\ $c4i \leftarrow \sum_{i=0}^{M-1} cos(4\pi f.i)$\\ @@ -264,15 +505,15 @@ Finally, the whole summarizes in an algorithm (called LSQ in the following) in t \For{$i=0$ to $nb_s $}{ $\theta \leftarrow -\pi + 2\pi\times \frac{i}{nb_s}$\\ - lut\_sin[$i$] $\leftarrow sin \theta$\\ - lut\_cos[$i$] $\leftarrow cos \theta$\\ - lut\_A[$i$] $\leftarrow cos 2 \theta \times s4i + sin 2 \theta \times c4i$\\ - lut\_sinfi[$i$] $\leftarrow sin (2\pi f.i)$\\ - lut\_cosfi[$i$] $\leftarrow cos (2\pi f.i)$\\ + lut$_s$[$i$] $\leftarrow sin \theta$\\ + lut$_c$[$i$] $\leftarrow cos \theta$\\ + lut$_A$[$i$] $\leftarrow cos 2 \theta \times s4i + sin 2 \theta \times c4i$\\ + lut$_{sfi}$[$i$] $\leftarrow sin (2\pi f.i)$\\ + lut$_{cfi}$[$i$] $\leftarrow cos (2\pi f.i)$\\ } \end{algorithm} -\begin{algorithm}[h] +\begin{algorithm}[htbp] \caption{LSQ algorithm - during acquisition loop.} \label{alg:lsq-during} @@ -287,49 +528,299 @@ Finally, the whole summarizes in an algorithm (called LSQ in the following) in t $xy_{covar} \leftarrow xy_{covar} + (i-\bar{x}) \times (I[i]-\bar{y})$\\ } $slope \leftarrow \frac{xy_{covar}}{x_{var}}$\\ - $start \leftarrow y_{moy} - slope\times \bar{x}$\\ + $start \leftarrow \bar{y} - slope\times \bar{x}$\\ \For{$i=0$ to $M-1$}{ - $I[i] \leftarrow I[i] - start - slope\times i$\tcc*[f]{slope removal}\\ + $I[i] \leftarrow I[i] - start - slope\times i$\\ } - + $I_{max} \leftarrow max_i(I[i])$, $I_{min} \leftarrow min_i(I[i])$\\ $amp \leftarrow \frac{I_{max}-I_{min}}{2}$\\ $Is \leftarrow 0$, $Ic \leftarrow 0$\\ \For{$i=0$ to $M-1$}{ - $Is \leftarrow Is + I[i]\times $ lut\_sinfi[$i$]\\ - $Ic \leftarrow Ic + I[i]\times $ lut\_cosfi[$i$]\\ + $Is \leftarrow Is + I[i]\times $ lut$_{sfi}$[$i$]\\ + $Ic \leftarrow Ic + I[i]\times $ lut$_{cfi}$[$i$]\\ } - $\theta \leftarrow -\pi$\\ - $val_1 \leftarrow 2\times \left[ Is.\cos(\theta) + Ic.\sin(\theta) \right] - amp\times \left[ c4i.\sin(2\theta) + s4i.\cos(2\theta) \right]$\\ - \For{$i=1-n_s$ to $n_s$}{ - $\theta \leftarrow \frac{i.\pi}{n_s}$\\ - $val_2 \leftarrow 2\times \left[ Is.\cos(\theta) + Ic.\sin(\theta) \right] - amp\times \left[ c4i.\sin(2\theta) + s4i.\cos(2\theta) \right]$\\ + $\delta \leftarrow \frac{nb_s}{2}$, $b_l \leftarrow 0$, $b_r \leftarrow \delta$\\ + $v_l \leftarrow -2.I_s - amp.$lut$_A$[$b_l$]\\ - \lIf{$val_1 < 0$ et $val_2 >= 0$}{ - $\theta_s \leftarrow \theta - \left[ \frac{val_2}{val_2-val_1}\times \frac{\pi}{n_s} \right]$\\ - } - $val_1 \leftarrow val_2$\\ - } + \While{$\delta >= 1$}{ -\end{algorithm} + $v_r \leftarrow 2.[ Is.$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 { + $b_r \leftarrow b_l + 1$\\ + } -\subsubsection{Comparison} + \uIf{$ abs(v_l) < v_r$}{ + $b_{\theta} \leftarrow b_l$ \\ + } + \Else { + $b_{\theta} \leftarrow b_r$ \\ + } + $\theta \leftarrow \pi\times \left[\frac{2.b_{ref}}{nb_s}-1\right]$\\ -\subsection{VDHL design paradigms} +\end{algorithm} -\subsection{VDHL implementation} +\subsubsection{Algorithm comparison} +\label{sec:algo-comp} +We compared the two algorithms regarding three criteria: -\section{Experimental results} -\label{sec:results} +\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}