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Changeset 200 in ETALON for reconstruction

Timestamp:

Mar 15, 2015, 5:26:16 PM (9 years ago)

Author:

delerue

Message:

Last set of corrections on the paper

Location:

reconstruction/long_paper3

Files:

: 5 edited

phase_reconstruction_paper.aux (modified) (5 diffs)
phase_reconstruction_paper.log (modified) (20 diffs)
phase_reconstruction_paper.pdf (modified) (previous)
phase_reconstruction_paper.synctex.gz (modified) (previous)
phase_reconstruction_paper.tex (modified) (11 diffs)

Legend:

: Unmodified
: Added
: Removed

reconstruction/long_paper3/phase_reconstruction_paper.aux

-                      r199
+                      r200
 \citation{E203prstab}
 \@LN@col{1}
 \@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces Example $\chi ^2$ defect.\relax }}{2}}
+\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces Example of profiles giving very different $\chi ^2$ despite being relatively similar.\relax }}{2}}
 \providecommand*\caption@xref[2]{\@setref\relax\@undefined{#1}}
 \newlabel{Offsine}{{1}{2}}
 …
 \citation{pchip}
 \citation{VBthesis}
-\citation{VBthesis,DESYthesis}
 \@LN@col{1}
 \@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces Detector position for linear sampling with $10^o$ (top) and $5^o$ (bottom) MDD.\relax }}{3}}
 …
 \@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces Comparison of different sampling with number of MDD. Ls -- is linear sampling with $1^o,5^o,10^0$ MDD and Triple sine sapmling; mx mean that in reconstruction was maximum number of detectors (blue and red on figure \ref  {lin12})\relax }}{3}}
 \newlabel{biglin}{{4}{3}}
+\@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces Effect of the sampling frequencies on the $\chi ^2$. \relax }}{3}}
+\newlabel{sampling_chi2}{{5}{3}}
+\citation{VBthesis,DESYthesis}
 \citation{LaiS}
 \@LN@col{1}
-\@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces Comparison of different LF extrapolation: example of spectrum (top), profile (middle) and histogram with mean $\chi ^2$ for each method (bottom). Gauss and Taylor methods are described in the text. "Real LF spectrum" means that the real LF spectrum is used. For this simulation we use the Hilbert method of phase recovery and $A\omega ^B$ high frequency extrapolation. \relax }}{4}}
-\newlabel{lf}{{5}{4}}
 \@LN@col{2}
+\@writefile{toc}{\contentsline {section}{Study of the reconstruction performance}{4}}
+\@writefile{lof}{\contentsline {figure}{\numberline {6}{\ignorespaces Comparison of different LF extrapolation: example of spectrum (top), profile (middle) and histogram with mean $\chi ^2$ for each method (bottom). Gauss and Taylor methods are described in the text. "Real LF spectrum" means that the real LF spectrum is used. For this simulation we use the Hilbert method of phase recovery and $A\omega ^B$ high frequency extrapolation. \relax }}{4}}
+\newlabel{lf}{{6}{4}}
+\@LN@col{1}
+\@writefile{lof}{\contentsline {figure}{\numberline {7}{\ignorespaces Comparison of different HF extrapolation~: example of spectrum (top) and profile (upper middle), histogram with mean $\chi ^2$ for comparison for Gaussian profiles (lower middle) and Lorenzians (bottom). For these simulation we use the Hilbert reconstruction method of phase recovery and Gaussian LF extrapolations.\relax }}{5}}
+\newlabel{hf}{{7}{5}}
+\@writefile{toc}{\contentsline {section}{Study of the reconstruction performance}{5}}
+\@LN@col{2}
+\@writefile{lof}{\contentsline {figure}{\numberline {8}{\ignorespaces Examples of well reconstructed profile. The original profile is in blue and the profiles reconstructed with the Hilbert transform and the full Kramers-Kronig procedures are in red and black respectively.\relax }}{5}}
+\newlabel{good_profiles}{{8}{5}}
 \citation{Pelliccia:2014vba}
 \@LN@col{1}
 \@writefile{lof}{\contentsline {figure}{\numberline {6}{\ignorespaces Comparison of different HF interpolation:example of spectrum and profile, histo with mean $\chi ^2$ for comparison for Gaussians and Lorenzians.\relax }}{5}}
 \newlabel{hf}{{6}{5}}
+\@writefile{lof}{\contentsline {figure}{\numberline {9}{\ignorespaces Example of poorly reconstructed profile. The original profile is in blue and the profiles reconstructed with the Hilbert transform and the full Kramers-Kronig procedures are in red and black respectively.\relax }}{6}}
+\newlabel{bad_profiles}{{9}{6}}
 \@LN@col{2}
+\@writefile{lof}{\contentsline {figure}{\numberline {7}{\ignorespaces Example of well reconstructed profile. The original profile is in blue and the profiles reconstructed with the Hilbert transform and the full Kramers-Kronig procedures are in red and black respectively.\relax }}{5}}
+\newlabel{good_profiles}{{7}{5}}
+\@writefile{toc}{\contentsline {section}{Discussion}{5}}
+\@writefile{lof}{\contentsline {figure}{\numberline {10}{\ignorespaces {$\Delta _{FWHM}$ (top) and $\chi ^2$ (bottom) distribution of 1000 simulations reconstructed using the Hilbert transform method (black line) and Kramers-Kronig reconstruction method (red line). XXX If the top figure is delta FWHM, then the title should say so XXX }\relax }}{6}}
+\newlabel{profiles_stats_hilbert}{{10}{6}}
+\@writefile{lof}{\contentsline {figure}{\numberline {11}{\ignorespaces $\Delta _{FWXM}$ for 1000 profiles with both methods.\relax }}{6}}
+\newlabel{fwxm}{{11}{6}}
+\@LN@col{1}
+\@writefile{lof}{\contentsline {figure}{\numberline {12}{\ignorespaces Original and reconstructed profile and their difference for two different profiles.\relax }}{7}}
+\newlabel{mod}{{12}{7}}
+\@writefile{lof}{\contentsline {figure}{\numberline {13}{\ignorespaces Effect of scaling the constraints on the parameters $\sigma _i$ and $\mu _i$ on the $\chi ^2$.\relax }}{7}}
+\newlabel{sigma_chi2}{{13}{7}}
+\@LN@col{2}
+\@writefile{lof}{\contentsline {figure}{\numberline {14}{\ignorespaces Distribution of the $\chi ^2$ in the case of a lorenzian distribution.\relax }}{7}}
+\newlabel{lorenz}{{14}{7}}
+\@writefile{lof}{\contentsline {figure}{\numberline {15}{\ignorespaces Mean $\chi ^2$ as function of noise amplitude.\relax }}{7}}
+\newlabel{noise}{{15}{7}}
+\@writefile{toc}{\contentsline {section}{Discussion}{7}}
 \bibcite{OTR_LURE}{1}
 \bibcite{ODR_Cianchi}{2}
 …
 \bibcite{E203prstab}{5}
 \bibcite{Pelliccia:2014vba}{6}
-\@LN@col{1}
-\@writefile{lof}{\contentsline {figure}{\numberline {8}{\ignorespaces Example of poorly reconstructed profile. The original profile is in blue and the profiles reconstructed with the Hilbert transform and the full Kramers-Kronig procedures are in red and black respectively.\relax }}{6}}
-\newlabel{bad_profiles}{{8}{6}}
-\@LN@col{2}
-\@writefile{lof}{\contentsline {figure}{\numberline {9}{\ignorespaces {$\Delta _{FWHM}$ (top) and $\chi ^2$ (bottom) distribution of our 1000 simulations reconstructed using the Hilbert transform method and Kramers-Kronig reconstruction method.}\relax }}{6}}
-\newlabel{profiles_stats_hilbert}{{9}{6}}
-\@writefile{lof}{\contentsline {figure}{\numberline {10}{\ignorespaces $\Delta _{FWXM}$ for 1000 profiles with both methods.\relax }}{6}}
-\newlabel{fwxm}{{10}{6}}
 \bibcite{VBthesis}{7}
 \bibcite{pchip}{8}
 …
 \bibcite{DESYthesis}{10}
 \@LN@col{1}
-\@writefile{lof}{\contentsline {figure}{\numberline {13}{\ignorespaces Effect of scaling the constraints on the parameters $\sigma _i$ and $\mu _i$ on the $\chi ^2$.\relax }}{7}}
-\newlabel{sigma_chi2}{{13}{7}}
-\@writefile{lof}{\contentsline {figure}{\numberline {14}{\ignorespaces Distribution of the $\chi ^2$ in the case of a lorenzian distribution.\relax }}{7}}
-\newlabel{lorenz}{{14}{7}}
-\@writefile{lof}{\contentsline {figure}{\numberline {15}{\ignorespaces Mean $\chi ^2$ as function of noise amplitude.\relax }}{8}}
-\newlabel{noise}{{15}{8}}
 \@LN@col{2}

reconstruction/long_paper3/phase_reconstruction_paper.log

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reconstruction/long_paper3/phase_reconstruction_paper.tex

-                      r199
+                      r200
 $$\chi^2=\sum_i\omega_i^2(O_i-E_i)^2/N,$$
 where $O_i$ is the observed value , $E_i$ is the expected value, $\omega_i=1/\sqrt{O_i+E_i}$ is the weight of the point, N is the number of points.\par
+However two very similar profiles but with a slight offset, will give a worse $\chi^2$ than a profile with oscilations (see figure \ref{Offsine}).
+However two very similar profiles but with a slight offset, will give a worse $\chi^2$ than a profile with oscillations (see figure \ref{Offsine}). This can be partly mitigated (in the case of horizontal offset) by offsetting one profile with respect to the other until the $\chi^2$ is minimized.
 \begin{figure}[!htb]
  \centering
   \includegraphics*[width=70mm]{a.eps}
   \caption{Example $\chi^2$ defect.}% VH change name of picture and unite with other
+  \caption{Example of profiles giving very different $\chi^2$ despite being relatively similar.}% VH change name of picture and unite with other
    \label{Offsine}
 \end{figure}
 …
 $$
 where $\mbox{rset} = \{ 0.1 ; 0.2 ; 0.5 ; 0.8 ; 0.9\}$, $FWXM_{\mbox{orig}}$ and $FWXM_{\mbox{reco}}$ are the FWXM of the original and reconstructed profiles respectively.
+To ensure that the choice of the parameters $\sigma_i$ and $\mu_i$ for the simulations does not biais significantly the results, their value has been varied and this is shown on figure~\ref{sigma_chi2} XXX can you add FWHM ? XXX.
+\begin{figure}[!htb]
+ \centering
+  \includegraphics*[width=70mm]{newfigures/chi_sigma.eps}\\
+  \includegraphics*[width=70mm]{newfigures/chi_mu.eps}%add new figure
+  \caption{Effect of scaling the constraints on the parameters $\sigma_i$ and $\mu_i$ on the $\chi^2$.}
+   \label{sigma_chi2}
+\end{figure}
 …
+%This section has been moved from elsewhere
+The choice of 33 frequencies for the sampling of the spectrum was made to match the current layout used on E-203. However it is important to check if there is an optimum value. Using the same simulations we used the same spectrum but sampled  with 5 to 120  points. The effect of changing the sampling frequencies on the  $\chi^2$ is shown on figure~\ref{sampling_chi2}. This study uses linear sampling with 1000 profiles for each point and the Hilbert reconstruction method.
+XXX Why you did not do the figure for FWHM ? XXX
+\begin{figure}[!htb]
+ \centering
+  \includegraphics*[width=70mm]{newfigures/Chi_Ndet.eps}
+  \caption{Effect of the sampling frequencies on the $\chi^2$. }
+   \label{sampling_chi2}
+\end{figure}
+%%%% Extrapolation / inerpolations
 After applying the sampling procedure the data need to be interpolated and extrapolated to have a larger number of points in the spectrum. Interpolation is done using Piecewise Cubic Hermite Interpolating Polynomial (PCHIP)~\cite{pchip}, as suggested in \cite{VBthesis}.
 The interpolation function must satisfy the following criteria: it must conserve the slope at the two endpoints (to have a continuous derivative) and respects monotonicity. Cubic Hermite interpolation has been chosen as it matches these requirements. XXX Something is not clear here: which function was used for interpolation? PCHIP or Cubic hermite ? XXX
 …
 $$\rho_{LF}=|F(\omega)|=\sqrt{A+B\omega^2+C\omega^4}$$
 Conditions for A, B and C constants are the same. Comparison of different LF extrapolation can be found in figure~\ref{lf}.
+Conditions for A, B and C constants are the same. Comparison of different LF extrapolation can be found on figure~\ref{lf}.
 \begin{figure}[!htb]
 …
 \begin{itemize}
 \item $\rho_{HF}(\omega_f)= 0 $ for $ \omega_f >  \omega_{fmax}$
 \item $\rho_{HF}(\omega_f)= $\rho_{real}(\omega_f) $ for $ \omega_f >  \omega_{fmax} $ where $\rho_{real}$ is the real spectrum.
+\item $\rho_{HF}(\omega_f)= \rho_{real}(\omega_f) $ for $ \omega_f >  \omega_{fmax} $ where $\rho_{real}$ is the real spectrum.
 \end{itemize}
 These extrapolation methods are compared in figure~\ref{hf}.
+These extrapolation methods are compared on figure~\ref{hf}.
 …
     \includegraphics*[width=65mm]{new203/HFGauss.eps}
         \includegraphics*[width=65mm]{new203/HFLorenz.eps}
     \caption{Comparison of different HF interpolation:example of spectrum and profile, histo with mean $\chi^2$ for comparison for Gaussians and Lorenzians.}
+    \caption{Comparison of different HF extrapolation~: example of spectrum  (top) and profile (upper middle), histogram with mean $\chi^2$ for comparison for Gaussian profiles (lower middle) and Lorenzians (bottom). For these simulation we use the Hilbert reconstruction method of phase recovery and Gaussian LF extrapolations.}
    \label{hf}
 \end{figure}
+xxx For this simulation we use Hilbert method of phase recovery and Gauss LF extrapolations.}
+\textbf{ At this procedure of spectrum recovery is finished and than we applied our reconstruction techniques to reconstruct the original profile.}
+As can be expected in some cases the reconstruction went very well and in some other cases it was not as convincing. An example of a well reconstructed profile is shown on figure~\ref{good_profiles} and examples of poorly reconstructed profile is shown on figure~\ref{bad_profiles}.
+XXX If you show Lorenzian profiles for HF extrapolation why don't you also show it for LF extrapolation? XXX
+\section{Study of the reconstruction performance}
+After applying extrapolation and interpolation the spectrum recovery is complete and we can apply different reconstruction techniques to reconstruct the original profile. For each reconstruction method some profiles are very well reconstructed whereas some other are not so well reconstructed. Examples of well reconstructed profiles are shown on figure~\ref{good_profiles} and examples of poorly reconstructed profile are shown on figure~\ref{bad_profiles}.
 \begin{figure}[!htb]
 …
   \includegraphics*[width=65mm]{new203/pic/658.eps}\\
    \includegraphics*[width=65mm]{new203/pic/914.eps}
   \caption{Example of well reconstructed profile. The original profile is in blue and the profiles reconstructed with the Hilbert transform and the full Kramers-Kronig procedures are in red and black respectively.}
+  \caption{Examples of well reconstructed profile. The original profile is in blue and the profiles reconstructed with the Hilbert transform and the full Kramers-Kronig procedures are in red and black respectively.}
    \label{good_profiles}
 \end{figure}
 …
+\section{Study of the reconstruction performance}
+{ As we deal with the module of the Fourier transform, we lose the position and direction information of the profile, therefore we first minimize $\chi^2$ to find the correct position and direction.} % VH change and add text
+{The  $\Delta_{FWXM}$ and  $\chi^2$ distribution of the 1000 simulations which we made and then reconstructed using the Hilbert transform method and Kramers-Kornig reconstruction are shown in figure~\ref{profiles_stats_hilbert}. There is a good agreement in FWHM between the two methods indicating that they are both good at finding the bunch length. However we see that the Hilbert method gives lower $\chi^2$ indicating that this method is better at reconstruction the bunch profile.}% VH changed text
+The  $\Delta_{FWXM}$ and  $\chi^2$ distribution of the 1000 simulations which we made and then reconstructed using the Hilbert transform method and Kramers-Kornig reconstruction are shown in on figure~\ref{profiles_stats_hilbert}. There is a good agreement in FWHM between the two methods indicating that they are both good at finding the bunch length. However we see that the Hilbert method gives lower $\chi^2$ indicating that this method is better at reconstruction the bunch profile.
 \begin{figure}[!htb]
 …
   \includegraphics*[width=70mm]{new203/pic/2.eps} \\
   \includegraphics*[width=70mm]{new203/pic/3.eps}
   \caption{{$\Delta_{FWHM}$  (top)  and $\chi^2$ (bottom) distribution of our 1000 simulations reconstructed using the Hilbert transform method and Kramers-Kronig reconstruction method.}}% VH change name of picture and unite with other
+  \caption{{$\Delta_{FWHM}$  (top)  and $\chi^2$ (bottom) distribution of 1000 simulations reconstructed using the Hilbert transform  method (black line) and Kramers-Kronig reconstruction method (red line).   XXX If the top figure is delta FWHM, then the title should say so XXX }}% VH change name of picture and unite with other
    \label{profiles_stats_hilbert}
 \end{figure}
 % VH add this block
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+{Figure~\ref{fwxm} shows that at different height of profiles the quality of reconstruction varies. There is a better agreement in the tails than at the top of the profile. On figure \ref{mod} You can see modulus of difference original and reconstructed profiles. Maximum of this related to the top of profile.}
+Figure~\ref{fwxm} shows the different values the FWXH for different values of X. This shows that at   different height of the profiles the quality of reconstruction varies: There is a better agreement in the tails (X=10\%) than at the top of the profile (X=90\%).
+ Figure~\ref{mod} shows the modulus of the difference between the original and reconstructed profiles. One can see oscillations in the difference between the original and reconstructed profile.
+     XXX Figure \ref{mod}  needs to be fixed:
+     Please use subplot to split the figure in 2 parts:
+     The upper 66\% of the figure should show the original and reconstructed profile.
+     The lower 33\% should show the difference (your dashed line). At the moment we have no indication of the scale to be applied to the dashed line. It would be important to have one.
+     XXX
 \begin{figure}[!htb]
  \centering
 …
   \includegraphics*[width=65mm]{new203/pic/6.eps}\\
     \includegraphics*[width=65mm]{new203/pic/7.eps}
     \caption{Explanation to figure (\ref{fwxm})}
+    \caption{Original and reconstructed profile and their difference for two different profiles.}
    \label{mod}
 \end{figure}
+The choice of 33 frequencies for the sampling of the spectrum was made to match the current layout used on E-203. However it is important to check if there is an optimum value. Using the same simulations we used different sampling ranging from 5 to 120 spectrum points. The effect of changing the sampling frequencies on the  $\chi^2$ is shown on figure~\ref{sampling_chi2}.\textbf{Fow this study we use linear sampling with differnt number of points in it, 1000 profiles for each point and Hilbert reconstruction algorithm.}
+\begin{figure}[!htb]
+ \centering
+  \includegraphics*[width=70mm]{newfigures/Chi_Ndet.eps}
+  \caption{Effect of the sampling frequencies on the $\chi^2$. }
+   \label{sampling_chi2}
+\end{figure}
+% Effect of noise.
+% Another important point to check is the effect of the noise on the
+We have also checked how the choice of the constraints on $\sigma_i$ affects the accuracy of the reconstruction. This effect on $\chi^2$ is shown on figure~\ref{sigma_chi2} XXX can you add FWHM ? XXX.
+\begin{figure}[!htb]
+ \centering
+  \includegraphics*[width=70mm]{newfigures/chi_sigma.eps}\\
+  \includegraphics*[width=70mm]{newfigures/chi_mu.eps}%add new figure
+  \caption{Effect of scaling the constraints on the parameters $\sigma_i$ and $\mu_i$ on the $\chi^2$.}
+   \label{sigma_chi2}
+\end{figure}
+While doing this work we also became aware of the discussion in~\cite{Pelliccia:2014vba} where it is argued that these reconstruction method have more difficulties with lorentzian profiles rather than gaussian profiles. Therefore we also simulated 1000 lorenzian profiles and performed a similar study. This is shown on figure~\ref{lorenz}. Although the $\chi^2$ is slightly worse in that case than in the case of gaussian profiles we still have a good agreement between the original and reconstructed profiles.
+While doing this work we also became aware of the discussion in~\cite{Pelliccia:2014vba} where it is argued that these reconstruction method have more difficulties with lorentzian profiles  than gaussian profiles. Therefore we also simulated 1000 lorenzian profiles and performed a similar study. This is shown on figure~\ref{lorenz}. Although the $\chi^2$ is slightly worse in that case than in the case of gaussian profiles we still have a good agreement between the original and reconstructed profiles.
 \begin{figure}[!htb]
 …
   \includegraphics*[width=70mm]{THPME088f9.eps} \\
 %  \includegraphics*[width=75mm]{THPME088f10.eps}
   \caption{Distribution of the $\chi^2$ in the case of a lorenzian distribution.}
+  \caption{Distribution of the $\chi^2$ in the case of a lorenzian distribution. XXX Can you add delta FWHM?  XXX}
    \label{lorenz}
 \end{figure}
+{So far we have only considered the ideal case where no noise is added to the measured spectrum. However in a real experiment a noise component has to be added to the measured spectrum. This noise was added as follow :
+% Effect of noise.
+So far we have only considered the ideal case where no noise is added to the measured spectrum. However in a real experiment a noise component has to be added to the measured spectrum. This noise was added as follow :
 $$O_i' = O_i \times (1 + n_i) N_{max}$$
+ where $O_i$ is the observed value,  $O_i$ is the observed value with noise, $n_i$ is a random number between 0 and 1, and $N_{max}$ is the
+maximum noise for that simulation (depending on the case this can be 5\%, 10\%, 20\%, 30\%, 40\% or  50\%). This study was done using linear sampling with 33 samples 1000 Number of simulations for each noise value. The figure~\ref{noise} shows how the $\chi^2$ is modified when this noise component is added.
+ }
+XXX I think it should be XXX
+$$O_i' = O_i \times [1 + ( n_i N_{max}) ] $$
+ where $O_i$ is the observed value,  $O_i'$ is the observed value with noise, $n_i$ is a random number between 0 and 1 (all numbers between 0 and 1 been equiprobable XXX Please check if this is true or if your random function has a gaussian distribution XXX), and $N_{max}$ is the
+maximum noise for that simulation (depending on the case this can be 5\%, 10\%, 20\%, 30\%, 40\% or  50\%). This study was done using linear sampling with 33 samples  and 1000 simulated profiles for each noise value. The figure~\ref{noise} shows how the $\chi^2$ is modified when this noise component is added.
 \begin{figure}[!htb]
  \centering
   \includegraphics*[width=70mm]{newfigures/new2/noise.eps}\\
   \caption{Mean $\chi^2$ as function of noise amplitude.}
+  \caption{Mean $\chi^2$ as function of noise amplitude. XXX Can you add delta FWHM?  XXX}
    \label{noise}
 \end{figure}

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