X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/dmems12.git/blobdiff_plain/099b94e08988d44f4f22b13a2da77f029c6ec29a..cf45cd222f858f35722966b715fb0540cd2c7420:/dmems12.tex

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}