Commit 60817431 authored by Philipp Arras's avatar Philipp Arras
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Some changes

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\usepackage{csquotes} \usepackage{csquotes}
\usepackage{hyperref} \usepackage{hyperref}
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\usepackage{amsmath, amssymb}
\graphicspath{{img/}} \graphicspath{{img/}}
\usepackage[backend=biber, sorting=none]{biblatex} \usepackage[backend=biber, sorting=none]{biblatex}
\addbibresource{bib.bib} \addbibresource{bib.bib}
\title{IFT Collaboration Guide} \title{IFT Collaboration Guide}
Philipp Arras
\begin{document} \begin{document}
\maketitle \VerbatimFootnotes
\section*{The need for a guide} \section*{The need for a guide}
\section*{Mathematical description of what NIFTy needs} \section*{Mathematical description of what NIFTy needs}
For first order minimization: For first order minimization:
\begin{itemize} \begin{itemize}
...@@ -46,6 +79,100 @@ d+\sum_i R(s)_i$ ), NIFTy needs: ...@@ -46,6 +79,100 @@ d+\sum_i R(s)_i$ ), NIFTy needs:
\end{itemize} \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.
The array \verb|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 \verb|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. \verb|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 \verb|s| and \verb|d|!
Now, the response appears. When we talk about \enquote{the response} we mean a
function \verb|R(s)| which takes an array of shape \verb|s.shape| and returns an
array of shape \verb|d.shape|. This function shall be made such that it
simulates the measurement device. Assume one knows the signal \verb|s|, what
would be the data \verb|d| in a noiseless measurement? Make sure that the
following code runs:
import numpy as np
# Load the data and store it in a numpy array called 'd'.
# Store the shape of s in 'shp'.
# Store the function which implements the response in 'R'.
response_out = R(np.ones(shp))
if response_out.shape == d.shape:
raise ValueError('Output of response has not the correct shape.')
Next, $R'$ and $R'^\dagger$ needs to be implemented. Since $R$ is a
R: \mathbb R^{s.shape} \to \mathbb R^{d.shape},
its gradient at the position \verb|s=position| is a linear map of the following shape:
\footnote{There are various ways to think about derivatives and gradients of
multi-dimensional functions. A different view on gradients would be that at a
given point $s=s_0$ the gradient is a matrix with \verb|s.size| columns and
\verb|d.size| rows. Obviously, it is not feasible to store such a matrix on a
computer due to its size. There we think of this matrix in terms of a linear
map which maps an array of shape \verb|s.shape| to an array of shape
\verb|d.shape|. This linear map shall be implemented on the computer in terms
of a function. Think of this map as a linear approximation to \verb|R| based
at \verb|s_0|.}
\left. \frac{dR}{ds}\right|_{s=position} = R': \mathbb R^{s.shape} \to \mathbb R^{d.shape}
What needs to be implemented is a function \verb|R_prime(position, s0)| which
takes the arguments \verb|position| (which is an array of shape \verb|s.shape|
and determines the position at which we want to calculate the derivative) and
the array \verb|s0| which shall be the derivative taken of.
\verb|R_prime| is nonlinear in \verb|position| in general and linear in
\verb|s0|. The output of \verb|R_prime| is of shape \verb|d.shape|.
Finally, we need to implement the function \verb|R_prime_adjoint(postition,
d0)|. It implements the adjoint action of the derivative: $R'^\dagger$. It takes
a \verb|position| of shape \verb|s.shape| as well. After the position is set
we've got a linear function again which shall be the adjoint to
\verb|R_prime(position, _)|. Thus, \verb|d0| has the same shape as the data
\verb|d| and the output of \verb|R_prime| has the shape \verb|s.shape|.
In order to test your implementation make sure that the following code runs
# Test adjointness
position = np.random.random(s.shape)
s0 = np.random.random(s.shape)
d0 = np.random.random(d.shape)
res0 = d0.vdot(R_prime(position, s0))
res1 = s0.vdot(R_prime_adjoint(position, d0))
np.testing.assert_allclose(res0, res1)
# Test derivative
dir = np.random.random(s.shape)
dir /= dir.var()
eps = 1e-8
finite_diff = (R(position + eps*dir)-R(position)) / eps
derivative = R_prime(position, dir)
np.testing.assert_allclose(finite_diff, derivative)
\section*{The same in NIFTy language}
\section*{Example $\gamma$-ray imaging} \section*{Example $\gamma$-ray imaging}
The information a $\gamma$-ray astronomer would provide to the algorithm (in the The information a $\gamma$-ray astronomer would provide to the algorithm (in the
simplest case): simplest case):
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