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ex-6: went on writing on the convolution

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Giù Marcer 2020-03-17 23:32:45 +01:00 committed by rnhmjoj
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notes/sections/6.md
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@ -4,7 +4,7 @@
The diffraction of a plane wave thorough a round slit must be simulated by The diffraction of a plane wave thorough a round slit must be simulated by
generating $N =$ 50'000 points according to the intensity distribution generating $N =$ 50'000 points according to the intensity distribution
$I(\theta)$ on a screen at a great distance $L$ from the slit iself: $I(\theta)$ on a screen at a great distance $L$ from the slit itself:
$$ $$
I(\theta) = \frac{E^2}{2} \left( \frac{2 \pi a^2 \cos{\theta}}{L} I(\theta) = \frac{E^2}{2} \left( \frac{2 \pi a^2 \cos{\theta}}{L}
@ -21,7 +21,7 @@ where:
- $L$ default set $L = \SI{1}{m}$. - $L$ default set $L = \SI{1}{m}$.
\begin{figure} \begin{figure}
\hypertarget{fig:fenditure}{% \hypertarget{fig:slit}{%
\centering \centering
\begin{tikzpicture} \begin{tikzpicture}
\definecolor{cyclamen}{RGB}{146, 24, 43} \definecolor{cyclamen}{RGB}{146, 24, 43}
@ -46,7 +46,7 @@ where:
\node [cyclamen] at (5.5,-0.4) {$\theta$}; \node [cyclamen] at (5.5,-0.4) {$\theta$};
\node [rotate=-90] at (10.2,0) {screen}; \node [rotate=-90] at (10.2,0) {screen};
\end{tikzpicture} \end{tikzpicture}
\caption{Fraunhofer diffraction.}\label{fig:fenditure} \caption{Fraunhofer diffraction.}
} }
\end{figure} \end{figure}
@ -85,30 +85,47 @@ omitted:
&\thus \theta = \text{acos} (1 -x) &\thus \theta = \text{acos} (1 -x)
\end{align*} \end{align*}
The sample was stored and plotted in a histogram with a customizable number $n$ The sample was binned and stored in a histogram with a customizable number $n$
of bins default set $n = 150$. In \textcolor{red}{fig} an example is shown. of bins default set $n = 150$. In @fig:original an example is shown.
\textcolor{red}{missing plot.} ![Example of sorted points according to
$I(\theta)$.](images/6_original.pdf){#fig:original}
## Gaussian noise convolution ## Gaussian noise convolution
The sample must then be smeared with a gaussian noise with the aim to recover The sample must then be smeared with a Gaussian noise with the aim to recover
the original sample afterwards, implementing a deconvolution routine. the original sample afterwards, implementing a deconvolution routine.
For this purpose, a 'kernel' histogram with a odd number $m$ of bins and the For this purpose, a 'kernel' histogram with a odd number $m$ of bins and the
same bin width of the previous one, but a smaller number of them ($m < n$), was same bin width of the previous one, but a smaller number of them ($m < n$), was
filled with $m$ points according to a gaussian distribution with mean $\mu$, filled with $m$ points according to a Gaussian distribution with mean $\mu$,
corresponding to the central bin, and variance $\sigma$. corresponding to the central bin, and variance $\sigma$.
Then, the original histogram was convolved with the kernel in order to obtain Then, the original histogram was convolved with the kernel in order to obtain
the smeared signal. The procedure is summed up in \textcolor{red}{fig}. the smeared signal. The result is shown in @fig:convolved.
\textcolor{red}{missing plots.} ![Same sample of @fig:original convolved with the
kernel.](images/6_convolved.pdf){#fig:convolved}
The third histogram was obtained by keeping the same edges of the original The convolution was implemented as follow. Consider the definition of
signal and a number of bins n +m -1 (?). convolution of two functions $f(x)$ and $g(x)$:
The convolution was obtained by permorming the dot product between the invere
kernel and the clean signal for each relative position of the two histograms. $$
For a better understaing, see \textcolor{red}{fig}. f*g (x) = \int \limits_{- \infty}^{+ \infty} dy f(y) g(x - y)
$$
Since a histogram is made of discrete values, a discrete convolution of the
signal $s$ and the kernel $k$ must be computed. Hence, the procedure boils
down to a dot product between $s$ and the reverse histogram of $k$ for each
relative position of the two histograms. Namely, if $c_i$ is the $i^{\text{th}}$
bin of the convoluted histogram:
$$
c_i = \sum_j k_j s_{i - j}
$$
where $j$ runs over the bins of the kernel.
For a better understanding, see @fig:dot_conv. Thus, the third histogram was
obtained with $n + m - 1$ bins, a number greater than the initial one.
\begin{figure} \begin{figure}
\hypertarget{fig:dot_conv}{% \hypertarget{fig:dot_conv}{%
@ -116,43 +133,43 @@ For a better understaing, see \textcolor{red}{fig}.
\begin{tikzpicture} \begin{tikzpicture}
\definecolor{cyclamen}{RGB}{146, 24, 43} \definecolor{cyclamen}{RGB}{146, 24, 43}
% original histogram % original histogram
\draw [thick, cyclamen, fill=cyclamen!15!white] (0.0,0) rectangle (0.5,2.5); \draw [thick, cyclamen, fill=cyclamen!05!white] (0.0,0) rectangle (0.5,2.5);
\draw [thick, cyclamen, fill=cyclamen!15!white] (0.5,0) rectangle (1.0,2.8); \draw [thick, cyclamen, fill=cyclamen!05!white] (0.5,0) rectangle (1.0,2.8);
\draw [thick, cyclamen, fill=cyclamen!15!white] (1.0,0) rectangle (1.5,2.3); \draw [thick, cyclamen, fill=cyclamen!25!white] (1.0,0) rectangle (1.5,2.3);
\draw [thick, cyclamen, fill=cyclamen!15!white] (1.5,0) rectangle (2.0,1.8); \draw [thick, cyclamen, fill=cyclamen!25!white] (1.5,0) rectangle (2.0,1.8);
\draw [thick, cyclamen, fill=cyclamen!15!white] (2.0,0) rectangle (2.5,1.4); \draw [thick, cyclamen, fill=cyclamen!25!white] (2.0,0) rectangle (2.5,1.4);
\draw [thick, cyclamen, fill=cyclamen!15!white] (2.5,0) rectangle (3.0,1.0); \draw [thick, cyclamen, fill=cyclamen!25!white] (2.5,0) rectangle (3.0,1.0);
\draw [thick, cyclamen, fill=cyclamen!15!white] (3.0,0) rectangle (3.5,1.0); \draw [thick, cyclamen, fill=cyclamen!25!white] (3.0,0) rectangle (3.5,1.0);
\draw [thick, cyclamen, fill=cyclamen!15!white] (3.5,0) rectangle (4.0,0.6); \draw [thick, cyclamen, fill=cyclamen!05!white] (3.5,0) rectangle (4.0,0.6);
\draw [thick, cyclamen, fill=cyclamen!15!white] (4.0,0) rectangle (4.5,0.4); \draw [thick, cyclamen, fill=cyclamen!05!white] (4.0,0) rectangle (4.5,0.4);
\draw [thick, cyclamen, fill=cyclamen!15!white] (4.5,0) rectangle (5.0,0.2); \draw [thick, cyclamen, fill=cyclamen!05!white] (4.5,0) rectangle (5.0,0.2);
\draw [thick, cyclamen, fill=cyclamen!15!white] (5.0,0) rectangle (5.5,0.2); \draw [thick, cyclamen, fill=cyclamen!05!white] (5.0,0) rectangle (5.5,0.2);
\draw [thick, cyclamen] (6.0,0) -- (6.0,0.2); \draw [thick, cyclamen] (6.0,0) -- (6.0,0.2);
\draw [thick, cyclamen] (6.5,0) -- (6.5,0.2); \draw [thick, cyclamen] (6.5,0) -- (6.5,0.2);
\draw [thick, <->] (0,3.3) -- (0,0) -- (7,0); \draw [thick, <->] (0,3.3) -- (0,0) -- (7,0);
% kernel histogram % kernel histogram
\draw [thick, cyclamen, fill=cyclamen!15!white] (1.0,-1) rectangle (1.5,-1.2); \draw [thick, cyclamen, fill=cyclamen!25!white] (1.0,-1) rectangle (1.5,-1.2);
\draw [thick, cyclamen, fill=cyclamen!15!white] (1.5,-1) rectangle (2.0,-1.4); \draw [thick, cyclamen, fill=cyclamen!25!white] (1.5,-1) rectangle (2.0,-1.6);
\draw [thick, cyclamen, fill=cyclamen!15!white] (2.0,-1) rectangle (2.5,-1.8); \draw [thick, cyclamen, fill=cyclamen!25!white] (2.0,-1) rectangle (2.5,-1.8);
\draw [thick, cyclamen, fill=cyclamen!15!white] (2.5,-1) rectangle (3.0,-1.4); \draw [thick, cyclamen, fill=cyclamen!25!white] (2.5,-1) rectangle (3.0,-1.6);
\draw [thick, cyclamen, fill=cyclamen!15!white] (3.0,-1) rectangle (3.5,-1.2); \draw [thick, cyclamen, fill=cyclamen!25!white] (3.0,-1) rectangle (3.5,-1.2);
\draw [thick, <->] (1,-2) -- (1,-1) -- (4,-1); \draw [thick, <->] (1,-2) -- (1,-1) -- (4,-1);
% arrows % arrows
\draw [thick, <->] (1.25,-0.2) -- (1.25,-0.8); \draw [thick, cyclamen, <->] (1.25,-0.2) -- (1.25,-0.8);
\draw [thick, <->] (1.75,-0.2) -- (1.75,-0.8); \draw [thick, cyclamen, <->] (1.75,-0.2) -- (1.75,-0.8);
\draw [thick, <->] (2.25,-0.2) -- (2.25,-0.8); \draw [thick, cyclamen, <->] (2.25,-0.2) -- (2.25,-0.8);
\draw [thick, <->] (2.75,-0.2) -- (2.75,-0.8); \draw [thick, cyclamen, <->] (2.75,-0.2) -- (2.75,-0.8);
\draw [thick, <->] (3.25,-0.2) -- (3.25,-0.8); \draw [thick, cyclamen, <->] (3.25,-0.2) -- (3.25,-0.8);
\draw [thick, ->] (2.25,-2.0) -- (2.25,-4.2); \draw [thick, cyclamen, ->] (2.25,-2.0) -- (2.25,-4.2);
% smeared histogram % smeared histogram
\begin{scope}[shift={(0,-1)}] \begin{scope}[shift={(0,-1)}]
\draw [thick, cyclamen, fill=cyclamen!15!white] (-1.0,-4.5) rectangle (-0.5,-4.3); \draw [thick, cyclamen, fill=cyclamen!05!white] (-1.0,-4.5) rectangle (-0.5,-4.3);
\draw [thick, cyclamen, fill=cyclamen!15!white] (-0.5,-4.5) rectangle ( 0.0,-4.2); \draw [thick, cyclamen, fill=cyclamen!05!white] (-0.5,-4.5) rectangle ( 0.0,-4.2);
\draw [thick, cyclamen, fill=cyclamen!15!white] ( 0.0,-4.5) rectangle ( 0.5,-2.0); \draw [thick, cyclamen, fill=cyclamen!05!white] ( 0.0,-4.5) rectangle ( 0.5,-2.0);
\draw [thick, cyclamen, fill=cyclamen!15!white] ( 0.5,-4.5) rectangle ( 1.0,-1.6); \draw [thick, cyclamen, fill=cyclamen!05!white] ( 0.5,-4.5) rectangle ( 1.0,-1.6);
\draw [thick, cyclamen, fill=cyclamen!15!white] ( 1.0,-4.5) rectangle ( 1.5,-2.3); \draw [thick, cyclamen, fill=cyclamen!05!white] ( 1.0,-4.5) rectangle ( 1.5,-2.3);
\draw [thick, cyclamen, fill=cyclamen!15!white] ( 1.5,-4.5) rectangle ( 2.0,-2.9); \draw [thick, cyclamen, fill=cyclamen!05!white] ( 1.5,-4.5) rectangle ( 2.0,-2.9);
\draw [thick, cyclamen, fill=cyclamen!15!white] ( 2.0,-4.5) rectangle ( 2.5,-3.4); \draw [thick, cyclamen, fill=cyclamen!25!white] ( 2.0,-4.5) rectangle ( 2.5,-3.4);
\draw [thick, cyclamen] (3.0,-4.5) -- (3.0,-4.3); \draw [thick, cyclamen] (3.0,-4.5) -- (3.0,-4.3);
\draw [thick, cyclamen] (3.5,-4.5) -- (3.5,-4.3); \draw [thick, cyclamen] (3.5,-4.5) -- (3.5,-4.3);
\draw [thick, cyclamen] (4.0,-4.5) -- (4.0,-4.3); \draw [thick, cyclamen] (4.0,-4.5) -- (4.0,-4.3);
@ -165,9 +182,18 @@ For a better understaing, see \textcolor{red}{fig}.
\draw [thick, cyclamen] (7.5,-4.5) -- (7.5,-4.3); \draw [thick, cyclamen] (7.5,-4.5) -- (7.5,-4.3);
\draw [thick, <->] (-1,-2.5) -- (-1,-4.5) -- (8,-4.5); \draw [thick, <->] (-1,-2.5) -- (-1,-4.5) -- (8,-4.5);
\end{scope} \end{scope}
% nodes
\node [above] at (2.25,-5.5) {$c_i$};
\node [above] at (3.25,0) {$s_i$};
\node [above] at (1.95,0) {$s_{i-3}$};
\node [below] at (1.75,-1) {$k_3$};
\end{tikzpicture} \end{tikzpicture}
\caption{Dot product as a step of the convolution between the original signal \caption{Dot product as a step of the convolution between the original signal
(above) and the kernel (below). The final result is the lower (above) and the kernel (center). The final result is the lower
fledging histogram.}\label{fig:dot_conv} fledging histogram.}\label{fig:dot_conv}
} }
\end{figure} \end{figure}
\textcolor{red}{Missing various $\sigma$ comparison.}
## Unfolding