More changes

main.tex

@@ -125,40 +125,52 @@ another object is needed:
 Note, that for Gaussian, Poissonian and Bernoulli likelihoods this term doesn't
 need to be calculated and implemented because NIFTy computes it automatically.
 
+That's it. The rest of this paper explains what these formulae mean and how to
+compute them. From now one, our discussion will become increasingly specific.
+
 \section*{Becoming more specific}
-If the likelihood is Gaussian ($\mathcal H(d|s) \propto (d-R(s))^\dagger N^{-1}
-(d-R(s))$) or Poissonian ($\mathcal H(d|s) \propto - \log (R(s))^\dagger
-d+\sum_i R(s)_i$ ), NIFTy needs:
+If the likelihood is Gaussian
+\begin{align*}
+\mathcal H(d|s) \propto (d-R(s))^\dagger N^{-1}(d-R(s))
+\end{align*}
+or Poissonian 
+\begin{align*}
+\mathcal H(d|s) \propto - \log (R(s))^\dagger d+\sum_i R(s)_i,
+\end{align*}
+NIFTy needs:
 \begin{itemize}
-\item $R$.
-\item $R'^\dagger := (\frac{\partial}{\partial s} R(s))^\dagger$.
-\item $R' = \frac{\partial}{\partial s} R(s)$. (only for 2nd order minimization)
+\item $R(s)$.
+\item $R'^\dagger := \left(\left.\frac{d R}{d s}\right|_{s=\text{postition}}\right)^\dagger$.
+\item Only for 2nd order minimization: $R' = \left.\frac{d R}{d s}\right|_{s=\text{postition}}$.
 \end{itemize}
 
 
 \section*{Even more specific}
 Since NIFTy is implemented in python and is based on numpy let us be as specific
-as possible and talk about numpy arrays. In the end, $s$ and $d$ will be a numpy
-array defined in a python script.
+as possible and talk about numpy arrays. In the end, $s$ and $d$ will be numpy
+arrays.
 
 The array \texttt{d} is created by a script which reads in the actual data which
 drops out of an instrument. This array will remain constant throughout the IFT
 algorithm since the data is given and never will be changed.
 
-The array \texttt{s} is obviously not constant; only its shape won't change in the
-course of the algorithm. NIFTy will store here the reconstruction of the
-physical field which the user wants to infer. \texttt{s} would be a
+The array \texttt{s} is obviously not constant; only its shape won't change in
+the course of the algorithm. In \texttt{s}, NIFTy will store the reconstruction
+of the physical field which the user wants to infer. \texttt{s} would be a
 one-dimensional array for a time-series, two-dimensional for a multi-frequency
 time-series or for a single-frequency image of the sky, three-dimensional for a
 multi-frequency image of sky, etc. To cut a long story short: be aware of the
 shape of \texttt{s} and \texttt{d}!
 
-Now, the response appears. When we talk about \enquote{the response} we mean a
-function \texttt{R(s)} which takes an array of shape \texttt{s.shape} and returns an
-array of shape \texttt{d.shape}. This function shall be made such that it
-simulates the measurement device. Assume one knows the signal \texttt{s}, what
-would be the data \texttt{d} in a noiseless measurement? Make sure that the
-following code runs:
+\subsection*{The response}
+When we talk about \enquote{the response} we mean a function \texttt{R(s)} which
+takes an array of shape \texttt{s.shape} and returns an array of shape
+\texttt{d.shape}. This function shall be made such that it simulates the
+measurement device. Assume one knows the signal \texttt{s}, what would be the
+data \texttt{d} in a noiseless measurement? Make sure that you understand what a
+response is to 100\%. If you have questions about it, do not hesitate to ask!
+This is one of the crucial parts of the whole process. The following code shall
+be running.
 
 \begin{lstlisting}
 import numpy as np
@@ -171,10 +183,11 @@ response_out = R(np.ones(shp))
 if response_out.shape == d.shape:
     print('Yay!')
 else:
-    raise ValueError('Output of response has not the correct shape.')
+    raise ValueError('Output of response doesn't have the correct shape.')
 \end{lstlisting}
 
-Next, $R'$ and $R'^\dagger$ needs to be implemented. Since $R$ is a
+\subsection*{Derivative of response}
+Next, $R'$ and $R'^\dagger$ need to be implemented. Since $R$ is a
 function:
 \begin{align*}
   R: \mathbb R^{s.shape} \to \mathbb R^{d.shape},