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Philipp Arras
whatisalikelihood
Commits
53d1c9b2
Commit
53d1c9b2
authored
Aug 04, 2020
by
Philipp Arras
Browse files
Merge branch 'master' into vincent_update
parents
fc53aac7
269a8536
Pipeline
#79997
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in 32 seconds
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53d1c9b2
...
...
@@ 125,7 +125,6 @@ If the likelihood is Gaussian (with noise covariance $N$ and response $R$)
\begin{align*}
\mathcal
H(ds)
\propto
(dR(s))
^
\dagger
N
^{
1
}
(dR(s))
\end{align*}
or Poissonian
\begin{align*}
\mathcal
H(ds)
\propto

\log
(R(s))
^
\dagger
d+
\sum
_
i R(s)
_
i,
...
...
@@ 152,7 +151,7 @@ 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
onedimensional array for a timeseries, twodimensional for a multifrequency
timeseries or for a singlefrequency image of the sky, threedimensional for a
multifrequency image of sky, etc. To cut a long story short: be aware of the
multifrequency image of
the
sky, etc. To cut a long story short: be aware of the
shape of
\texttt
{
s
}
and
\texttt
{
d
}
!
\subsection*
{
The response
}
...
...
@@ 176,7 +175,7 @@ response_out = R(np.ones(shp))
if response
_
out.shape == d.shape:
print('Yay!')
else:
raise ValueError('Output of response does
n'
t have the correct shape.')
raise ValueError('Output of response does
no
t have the correct shape.')
\end{lstlisting}
\subsection*
{
Derivative of response
}
...
...
@@ 201,7 +200,7 @@ following shape: \footnote{There are various ways to think about derivatives and
What needs to be implemented is a function
\texttt
{
R
\_
prime(position, s0)
}
which
takes the arguments
\texttt
{
position
}
(which is an array of shape
\texttt
{
s.shape
}
and determines the position at which we want to calculate the derivative) and
the array
\texttt
{
s0
}
which
shall be
the derivative taken
of
.
the array
\texttt
{
s0
}
of
which the derivative
shall be
taken.
\texttt
{
R
\_
prime
}
is nonlinear in
\texttt
{
position
}
in general and linear in
\texttt
{
s0
}
. The output of
\texttt
{
R
\_
prime
}
is of shape
\texttt
{
d.shape
}
.
...
...
@@ 314,7 +313,7 @@ Why is this already sufficient?
(
R
(
s
))
^
\dagger
d
+
\sum
_
i R
(
s
)
_
i
$
. Implementing
$
R
$
and stating that the data is
Poissonian determines this form.
\item
Since
$
R
$
is a composition of a convolution and a sampling, both of which
is a
linear operation,
$
R
$
itself is a linear operator.
\footnote
{
I.e.
$
R
(
\alpha
are
linear operation
s
,
$
R
$
itself is a linear operator.
\footnote
{
I.e.
$
R
(
\alpha
s
_
1
+
s
_
2
)
=
\alpha
R
(
s
_
1
)
+
R
(
s
_
2
)
$
.
}
Thus,
$
R'
=
R
$
and
$
R'
^
\dagger
=
R
^
\dagger
$
. All in all, we need an implementation for
$
R
$
and
$
R
^
\dagger
$
.
\end{itemize}
...
...
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