From: Stéphane Domas Date: Mon, 17 Oct 2011 17:03:37 +0000 (+0200) Subject: 3ème commit : X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/dmems12.git/commitdiff_plain/e73cda1143a6ba97c0904b1811130d22ba7daaa8 3ème commit : - fin de l'algo LSQ écrite - comparaison des deux algo quasi faite --- diff --git a/dmems12.tex b/dmems12.tex index 6a7f646..f559661 100644 --- a/dmems12.tex +++ b/dmems12.tex @@ -206,7 +206,7 @@ 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 +(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 @@ -218,12 +218,15 @@ The phase is computed via the 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] \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 $\theta$. +Two things can be noticed : +\begin{itemize} +\item the frequency could also be obtained using the derivates of + spline equations, which only implies to solve quadratic equations. +\item 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 $\theta$. +\end{itemize} \subsubsection{Least square algorithm} @@ -278,13 +281,15 @@ Several points can be noticed : computed. \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 + $[-\pi,\pi]$ in $nb_s$ 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 : -\[ sin \theta, cos \theta, \left[ cos 2\theta \sum_{i=0}^{M-1} sin(4\pi f.i) + sin 2\theta \sum_{i=0}^{M-1} cos(4\pi f.i)\right] \] +\[ sin \theta, cos \theta, \] + +\[ \left[ cos 2\theta \sum_{i=0}^{M-1} sin(4\pi f.i) + sin 2\theta \sum_{i=0}^{M-1} cos(4\pi f.i)\right] \] -\item This search can be very fast using a dichotomous process in $log_2(N)$ +\item This search can be very fast using a dichotomous process in $log_2(nb_s)$ \end{itemize} @@ -302,15 +307,15 @@ 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}[ht] \caption{LSQ algorithm - during acquisition loop.} \label{alg:lsq-during} @@ -327,7 +332,7 @@ Finally, the whole summarizes in an algorithm (called LSQ in the following) in t $slope \leftarrow \frac{xy_{covar}}{x_{var}}$\\ $start \leftarrow y_{moy} - slope\times \bar{x}$\\ \For{$i=0$ to $M-1$}{ - $I[i] \leftarrow I[i] - start - slope\times i$\tcc*[f]{slope removal}\\ + $I[i] \leftarrow I[i] - start - slope\times i$\\ } $I_{max} \leftarrow max_i(I[i])$, $I_{min} \leftarrow min_i(I[i])$\\ @@ -335,27 +340,146 @@ Finally, the whole summarizes in an algorithm (called LSQ in the following) in t $Is \leftarrow 0$, $Ic \leftarrow 0$\\ \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$_A$[$b_l$]\\ + + \While{$\delta >= 1$}{ - \lIf{$val_1 < 0$ et $val_2 >= 0$}{ - $\theta_s \leftarrow \theta - \left[ \frac{val_2}{val_2-val_1}\times \frac{\pi}{n_s} \right]$\\ + $v_r \leftarrow 2.[ Is.$lut$_c$[$b_r$]$ + Ic.$lut$_s$[$b_r$]$ ] - amp.$lut$_A$[$b_r$]\\ + + \If{$!(v_l < 0$ and $v_r >= 0)$}{ + $v_l \leftarrow v_r$ \\ + $b_l \leftarrow b_r$ \\ } - $val_1 \leftarrow val_2$\\ + $\delta \leftarrow \frac{\delta}{2}$\\ + $b_r \leftarrow b_l + \delta$\\ + } + \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 { + $b_r \leftarrow b_l + 1$\\ } -\end{algorithm} + \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{Comparison} +We compared the two algorithms on the base of three criterions : +\begin{itemize} +\item precision of results on a cosinus profile, distorted with noise, +\item number of operations, +\item complexity to implement an FPGA version. +\end{itemize} + +For the first item, we produced a matlab version of each algorithm, +running with double precision values. The profile was generated for +about 34000 different values of period ($\in [3.1, 6.1]$, step = 0.1), +phase ($\in [-3.1 , 3.1]$, step = 0.062) and slope ($\in [-2 , 2]$, +step = 0.4). For LSQ, $nb_s = 1024$, which leads to a maximal error of +$\frac{\pi}{1024}$ on phase computation. Current A. Meister and +M. Favre experiments show a ratio of 50 between variation of phase and +the deflection of a lever. Thus, the maximal error due to +discretization correspond to an error of 0.15nm on the lever +deflection, which is smaller than the best precision they achieved, +i.e. 0.3nm. + +For each test, we add some noise to the profile : each group of two +pixels has its intensity added to a random number picked in $[-N,N]$ +(NB: it should be noticed that picking a new value for each pixel does +not distort enough the profile). The absolute error on the result is +evaluated by comparing the difference between the reference and +computed phase, out of $2\pi$, expressed in percents. That is : $err = +100\times \frac{|\theta_{ref} - \theta_{comp}|}{2\pi}$. + +Table \ref{tab:algo_prec} gives the maximum and average error for the two algorithms and increasing values of $N$. + +\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 & 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} +\caption{Error (in \%) for cosinus profiles, with noise.} +\label{tab:algo_prec} +\end{center} +\end{table} + +These results show that the two algorithms are very close, with a +slight advantage for LSQ. Furthemore, both behave very well against +noise. Assuming the experimental ratio of 50 (see above), an error of +1 percent on phase correspond to an error of 0.5nm on the lever +deflection, which is very close to the best precision. + +Obviously, it is very hard to predict which level of noise will be +present in real experiments and how it will distort the +profiles. Nevertheless, 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, and relatively +close to most of that come from experiments. Figure \ref{fig:noise60} +shows a sample of worst profile for $N=30$. It is completly distorted, +largely beyond the worst experimental ones. + +\begin{figure}[ht] +\begin{center} + \includegraphics[width=9cm]{intens-noise20-spl} +\end{center} +\caption{Sample of worst profile for N=10} +\label{fig:noise20} +\end{figure} + +\begin{figure}[ht] +\begin{center} + \includegraphics[width=9cm]{intens-noise60-lsq} +\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 $atan$ operation. In both cases, it is proportional +to numbers of pixels $M$. For LSQ, it also depends on $nb_s$ and for +SPL on $N = k\times M$, i.e. the number of interpolated points. + +We assume that $M=20$, $nb_s=1024$, $k=4$, all possible parts are +already in lookup tables and only arithmetic operations (+, -, *, /) +are taken account. Translating the two algorithms in C code, we obtain +about 400 operations for LSQ and 1340 (plus the unknown for $atan$) +for SPL. Even if the result is largely in favor of LSQ, we can notice +that executing the C code (compiled with \tt{-O3}) of SPL on an +2.33GHz Core 2 Duo only takes 6.5µs in average, which is under our +desing goals. The final decision is thus driven by the third criterion.\\ + +The Spartan 6 used in our architecture has hard constraint : it has no +floating point units. Thus, all computations have to be done with +integers. + + + \subsection{VHDL design paradigms} \subsection{VHDL implementation}