+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.
+
+
+
