X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/dmems12.git/blobdiff_plain/c6645d99e9614d4fe8c5d8699f94440ba339900f..85ccdc9bbf8d6b30f5737245a16d277a7713908c:/dmems12.tex diff --git a/dmems12.tex b/dmems12.tex index ad693f6..c5ee6cf 100644 --- a/dmems12.tex +++ b/dmems12.tex @@ -44,7 +44,7 @@ -\title{Using FPGAs for high speed and real time cantilever deflection estimation} +\title{A new approach based on least square methods to estimate in real time cantilevers deflection 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\\ \{raphael.couturier,stephane.domas\}@univ-fcomte.fr} @@ -63,7 +63,15 @@ \begin{abstract} - +Atomic force microscope (AFM) provides high resolution images of surfaces. We +focus our attention on an interferometry method to estimate the cantilevers +deflection. This method was based on the spline method to interpolate the +deflection and the computations were performed on a PC with LabView. In this +paper, we propose a new method based on the least square method and we present +the implementation that we developped on a FPGA. Our method can be pipelined on +a FPGA in order to manipulate image profiles very quickly. Simulations and real +tests we have performed showed us that this implementation is very efficient and +should allow us to control of a cantilevers array in real time. \end{abstract} @@ -90,19 +98,18 @@ the cantiliver which result in a complex fabrication process. In this paper our attention is focused on a method based on interferometry to measure cantilevers' displacements. In this method cantilevers are illuminated -by an optic source. The interferometry produces fringes on each cantilevers +by an optic source. The interferometry produces fringes on each cantilever which enables to compute the cantilever displacement. In order to analyze the fringes a high speed camera is used. Images need to be processed quickly and then a estimation method is required to determine the displacement of each -cantilever. In~\cite{AFMCSEM11}, the authors have used an algorithm based on +cantilever. In~\cite{AFMCSEM11}, authors have used an algorithm based on spline to estimate the cantilevers' positions. - The overall process gives -accurate results but all the computation are performed on a standard computer -using labview. Consequently, the main drawback of this implementation is that -the computer is a bootleneck in the overall process. In this paper we propose to -use a method based on least square and to implement all the computation on a -FGPA. +The overall process gives accurate results but all the computations +are performed on a standard computer using LabView. Consequently, the +main drawback of this implementation is that the computer is a +bootleneck. In this paper we propose to use a method based on least +square and to implement all the computation on a FGPA. The remainder of the paper is organized as follows. Section~\ref{sec:measure} describes more precisely the measurement process. Our solution based on the @@ -118,13 +125,6 @@ presented. \section{Measurement principles} \label{sec:measure} - - - - - - - \subsection{Architecture} \label{sec:archi} %% description de l'architecture générale de l'acquisition d'images @@ -138,24 +138,26 @@ deflection scheme and sentitive to the angular displacement of the cantilever, interferometry is sensitive to the optical path difference induced by the vertical displacement of the cantilever. -The system build by authors of~\cite{AFMCSEM11} has been developped based on a -Linnick interferomter~\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 that reachs the cantilever array. In order to -be able to move the cantilever array, it is mounted on a translation and -rotational hexapod stage with five degrees of freedom. The optical system is -also fixed to the stage. Thus, the cantilever array is centered in the optical -system which can be adjusted accurately. The beam illuminates the array by a -microscope objective and the light reflects on the cantilevers. Likewise the -reference beam reflects on a movable mirror. A CMOS camera chip records the -reference and sample beams which are recombined in the beam splitter and the -interferogram. At the beginning of each experiment, the movable mirror is -fitted manually in order to align the interferometric fringes approximately -parallel to the cantilevers. When cantilevers move due to the surface, the -bending of cantilevers produce movements in the fringes that can be detected -with the CMOS camera. Finally the fringes need to be -analyzed. In~\cite{AFMCSEM11}, the authors used a LabView program to compute the -cantilevers' movements from the fringes. +The system build by these authors is based on a Linnick +interferomter~\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 that reachs the cantilever +array. In order to be able to move the cantilever array, it is +mounted on a translation and rotational hexapod stage with five +degrees of freedom. The optical system is also fixed to the stage. +Thus, the cantilever array is centered in the optical system which can +be adjusted accurately. The beam illuminates the array by a +microscope objective and the light reflects on the cantilevers. +Likewise the reference beam reflects on a movable mirror. A CMOS +camera chip records the reference and sample beams which are +recombined in the beam splitter and the interferogram. At the +beginning of each experiment, the movable mirror is fitted manually in +order to align the interferometric fringes approximately parallel to +the cantilevers. When cantilevers move due to the surface, the +bending of cantilevers produce movements in the fringes that can be +detected with the CMOS camera. Finally the fringes need to be +analyzed. In~\cite{AFMCSEM11}, authors used a LabView program to +compute the cantilevers' deflections from the fringes. \begin{figure} \begin{center} @@ -171,21 +173,29 @@ cantilevers' movements from the fringes. \subsection{Cantilever deflection estimation} \label{sec:deflest} -As shown on image \ref{img:img-xp}, each cantilever is covered by -interferometric fringes. The fringes will distort when cantilevers are -deflected. Estimating the deflection is done by computing this -distortion. For that, (ref A. Meister + M Favre) proposed a method -based on computing the phase of the fringes, at the base of each -cantilever, near the tip, and on the base of the array. They assume -that a linear relation binds these phases, which can be use to -"unwrap" the phase at the tip and to determine the deflection.\\ - -More precisely, segment of pixels are extracted from images taken by a -high-speed camera. These segments are large enough to cover several -interferometric fringes and are placed at the base and near the tip of -the cantilevers. They are called base profile and tip profile in the -following. Furthermore, a reference profile is taken on the base of -the cantilever array. +\begin{figure} +\begin{center} +\includegraphics[width=\columnwidth]{lever-xp} +\end{center} +\caption{Portion of an image picked by the camera} +\label{fig:img-xp} +\end{figure} + +As shown on image \ref{fig:img-xp}, each cantilever is covered by +several interferometric fringes. The fringes will distort when +cantilevers are deflected. Estimating the deflection is done by +computing this distortion. For that, authors of \cite{AFMCSEM11} +proposed a method based on computing the phase of the fringes, at the +base of each cantilever, near the tip, and on the base of the +array. They assume that a linear relation binds these phases, which +can be use to "unwrap" the phase at the tip and to determine the deflection.\\ + +More precisely, segment of pixels are extracted from images taken by +the camera. These segments are large enough to cover several +interferometric fringes. As said above, they 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. The pixels intensity $I$ (in gray level) of each profile is modelized by: @@ -344,10 +354,11 @@ that communicate between i.MX and Spartan6, using Spartan3 as a tunnel. By default, the WEIM interface provides a clock signal at 100MHz that is connected to dedicated FPGA pins. -The Spartan6 is an LX100 version. It has 15822 slices, equivalent to -101261 logic cells. There are 268 internal block RAM of 18Kbits, and -180 dedicated multiply-adders (named DSP48), which is largely enough -for our project. +The Spartan6 is an LX100 version. It has 15822 slices, each slice +containing 4 LUTs and 8 flip/flops. It is equivalent to 101261 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 as user wants. For the project, they will be @@ -362,18 +373,18 @@ 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} +\subsubsection{Spline algorithm (SPL)} \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 (SPL in the -following) are used to interpolate $N = k\times M$ points (typically $k=4$ is -sufficient), within $[0,M[$. Let call $x^s$ the coordinates of these $N$ points - and $I^s$ their intensities. +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 $k=4$ 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 @@ -394,7 +405,7 @@ Two things can be noticed: computation of $\theta$. \end{itemize} -\subsubsection{Least square algorithm} +\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 @@ -460,7 +471,7 @@ computed. \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] +\begin{algorithm}[htbp] \caption{LSQ algorithm - before acquisition loop.} \label{alg:lsq-before} @@ -481,7 +492,7 @@ Finally, the whole summarizes in an algorithm (called LSQ in the following) in t } \end{algorithm} -\begin{algorithm}[ht] +\begin{algorithm}[htbp] \caption{LSQ algorithm - during acquisition loop.} \label{alg:lsq-during} @@ -694,24 +705,42 @@ will include real experiments in the final version of this paper. % - en parallèle : simulink et VHDL à la main -From the LSQ algorithm, we have written a C program which uses only integer -values that have been previously scaled. The quantization of doubles into -integers has been performed in order to obtain a good trade-off between the -number of bits used and the precision. We have compared the result of -the LSQ version using integers and doubles. We have observed that the results of -both versions were similar. - -Then we have built two versions of VHDL codes: one directly by hand coding and -the other with Matlab using the Simulink HDL coder -feature~\cite{HDLCoder}. Although the approach is completely different we have -obtain VHDL codes that are quite comparable. Each approach has advantages and -drawbacks. Roughly speaking, hand coding provides beautiful and much better -structured code while HDL coder provides code faster. In terms of speed of -code, we think that both approaches will be quite comparable with a slightly -advantage for hand coding. We hope that real experiments will confirm that. In -the LSQ algorithm, we have replaced all the divisions by multiplications by -constants since divisions are performed with constants depending of the number -of pixels in the profile (i.e. $M$). +From the LSQ algorithm, we have written a C program that uses only +integer values. We use a very simple quantization by multiplying +double precision values by a power of two, keeping the integer +part. For example, all values stored in lut$_s$, lut$_c$, $\ldots$ are +scaled by 1024. Since LSQ also computes average, variance, ... to +remove the slope, the result of implied euclidian divisions may be +relatively wrong. To avoid that, we also scale the pixel intensities +by a power of two. Futhermore, assuming $nb_s$ is fixed, these +divisions have a knonw denominator. Thus, they can be replaced by +their multiplication/shift counterpart. Finally, all other +multiplications or divisions by a power of two have been replaced by +left or right bit shifts. By the way, the code only contains +additions, substractions and multiplications of signed integers, which +is perfectly adapted to FGPAs. + +As said above, hardware constraints have a great influence on the VHDL +implementation. Consequently, we searched the maximum value of each +variable as a function of the different scale factors and the size of +profiles, which gives their maximum size in bits. That size determines +the maximum scale factors that allow to use the least possible RAMs +and DSPs. Actually, we implemented our algorithm with this maximum +size but current works study the impact of quantization on the results +precision and design complexity. We have compared the result of the +LSQ version using integers and doubles and observed that the precision +of both were similar. + +Then we built two versions of VHDL codes: one directly by hand coding +and the other with Matlab using the Simulink HDL coder +feature~\cite{HDLCoder}. Although the approach is completely different +we obtained VHDL codes that are quite comparable. Each approach has +advantages and drawbacks. Roughly speaking, hand coding provides +beautiful and much better structured code while Simulink allows to +produce a code faster. In terms of throughput and latency, +simulations shows that the two approaches are close with a slight +advantage for hand coding. We hope that real experiments will confirm +that. \subsection{Simulation} @@ -740,7 +769,16 @@ could accept new profiles of pixel each 48 cycles, i.e. 480$n$s. \section{Conclusion and perspectives} - +In this paper we have presented a new method to estimate the cantilevers +deflection in an AFM. This method is based on least square methods. We have +studied the quantization of this algorithm and have implemented it on a +FPGA. Our method gives comparable results compared to the initial version based +on splines. Our solution has been be implemented with a pipeline technique. +Consequently, it enables to handle a new profile image very quickly. Currently +we have performed simulations and real tests on a Spartan6 FPGA. + +In future work, we want to couple our algorithm with a high speed camera +and we plan to control the whole AFM system. \bibliographystyle{plain} \bibliography{biblio}