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\title{Generalized Linear Multi-Frequency Imaging in VLBI}
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\author{Sergey Likhachev, Vladimir Ladygin, Igor Guirin}
\affil{Astro Space Center of P.N. Lebedev Physical Institute of
Russian Academy of Sciences}

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\contact{Sergey Likhachev}
\email{slikhach@asc.rssi.ru}

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\authormark{Likhachev, Ladygin, \& Guirin}

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\keywords{ VLBI, multi: frequency, imaging, ASL }

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In VLBI, generalized Linear Multi-Frequency Imaging (MFI) consists of 
multi-frequency synthesis (MFS) and multi-frequency analysis (MFA) 
of the VLBI data obtained from observations on various frequencies. 
A set of linear deconvolution MFI algorithms is described.  The algorithms 
make it possible to obtain high quality images interpolated on any given 
frequency inside any given bandwidth, and to derive reliable estimates of spectral 
indexes for radio sources with continuum spectrum.
\end{abstract}

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\section{Statement of the problem}

Let us consider a linear model for intensity $I_{kpq}=I(x_{p},y_{q},\nu
_{k}) $ of the radio source in a point $(x_{p},y_{q})$ on the observational
frequency $\nu _{k}$:

\[
\begin{array}{l}
I_{kpq}\approx \left( {I_{0}}\right) _{pq}+\left( {I_{1}}\right) _{pq}\beta
_{k}\,+\ldots +\left( {I_{N-1}}\right) _{pq}\cdot (\beta _{k})^{N-1}, \\ 
\quad \\ 
\beta _{k}=\frac{\nu _{k}}{\nu _{0}}-1,\quad k=1,\;2,\;\ldots \;,\;K,
\end{array}
\]

\noindent where $\nu _0 $ is reference frequency corresponding to the
intensity $\left( {I_0 } \right)_{pq} $.

If the intensity $I_{kpq} $ in the point $(x_p ,y_q )$ can be approximated
by power law as

\[
I_{kpq}=\left( {I_{0}}\right) _{pq}\cdot \left( {\frac{\nu _{k}}{\nu _{0}}}%
\right) ^{\alpha _{pq}}, 
\]

then we can present it as

\[
I_{kpq}=\left( {I_{0}}\right) _{pq}e^{\xi _{k}\alpha _{pq}}\approx \left( {
I_{0}}\right) _{pq}\cdot \left( {1+\xi _{k}\alpha _{pq}}\right) 
\]

\noindent where $\xi _k = \ln \;\left( {1 + \beta _k } \right) \approx \beta
_k $,

and thus the spectral indexes $\alpha _{pq}=\alpha \left( {%
x_{p},y_{q}}\right) $ can be obtained as

\[
\left( {I_{1}}\right) _{pq}=\alpha _{pq}\cdot \left( {I_{0}}\right) _{pq} 
\]

Let us consider a target function

\[
\rho =\sum\limits_{k=1}^{K}{\sum\limits_{n=0}^{M-1}{\sum\limits_{m=0}^{M-1}{%
w_{knm}\cdot \left\vert {V_{knm}-\hat{V}_{knm}}\right\vert ^{2}}}},
\]

\noindent where, $w_{knm}=w(u_{n},v_{m},\nu _{k})\geq 0$ are weights, ${
V_{knm}}$, ${\hat{V}_{knm}}$ is a measured and a model visibility function
respectively,
\[
{\hat{V}_{knm}=A}_{k}\cdot \sum\limits_{p,q=o}^{M-1}\left[
\sum\limits_{l=o}^{N-1}\left( \widehat{I_{l}}\right) _{pq}\cdot \beta
_{k}^{l}\right] \cdot \exp \left\{ -2\pi i\cdot \left(
u_{n}x_{p}+v_{m}y_{q}\right) \right\} ,
\]

\noindent where, ${A}_{k}$ is a gain coefficient\bigskip\ \bigskip for k-th
antenna, 
\[
\left( \widehat{I_{l}}\right) _{pq}=\Delta ^{2}\varphi _{pq}\cdot \left(
I_{l}\right) _{pq}\left( 1-x_{p}^{2}-y_{q}^{2}\right) ^{-0.5}, 
\]

$\varphi _{pq}$ is a normalized beam, $\Delta $ is a grid step.

The problem of the optimization can be presented as a solution of \ the
following system of linear equations:

\[
\left( {D_0 } \right)_{pq} = 0,\;\ldots ,\;\left( {D_{N - 1} } \right)_{pq}
= 0 
\]

\noindent for a vector of intensity $\left( \widehat{\mathbf{I}}\right)
_{rt}=\left( {\left( \widehat{\mathbf{I}}{_{0}}\right) _{rt},\;\left( 
\widehat{\mathbf{I}}{_{1}}\right) _{rt},\;\ldots ,\;\left( \widehat{\mathbf{I
}}{_{N-1}}\right) _{rt}}\right) ^{T}$, where the $m$-th residual map $\left(
{D_{m}}\right) _{pq}$ can be defined as:
\begin{eqnarray}
\left( {D_{m}}\right) _{pq} &=&\sum\limits_{k=1}^{K}{\beta _{k}^{m}\cdot }
\left\{ {D_{kpq}}-{\sum\limits_{i=0}^{M-1}{\sum\limits_{l=0}^{M-1}{
B_{k,p-i,q-l}\cdot }}}\sum\limits_{n=0}^{N-1}{\left( \widehat{\mathbf{I}}{
_{n}}\right) _{il}\cdot \beta _{k}^{n}}\right\} , \\
m &=&0,\;1,\;\ldots ,\;N-1,
\end{eqnarray}

where, ${D_{kpq}=}\sum\limits_{n,m=o}^{M-1}w_{knm}\cdot V_{knm}\cdot \exp
\left\{ 2\pi i\left( u_{n}x_{p}+v_{m}y_{q}\right) \right\} $ is a k-th
"dirty" map at the point $\left( x_{p},y_{q}\right) $,

$B_{k,p-i,q-l}=\sum\limits_{n,m=o}^{M-1}w_{knm}\exp \left\{ 2\pi i\left[
u_{n}\left( x_{p}-x_{i}\right) +v_{m}\left( y_{q}-y_{l}\right) \right]
\right\} $ \ is a k-th "dirty" beam at the point $\left(
x_{p}-x_{i},y_{q}-y_{l}\right) $.

\section{Solution of the problem}

Let us choose the following initial conditions: ${\left( \widehat{
\mathbf{I}}{_{m}}\right) _{il}^{\left( 0\right) }=0}$ for all $m,i,l$ and
form \bigskip initial arrays $\left( {D_{m}}\right) _{pq}^{\left( 0\right)
},m=0,\;1,\;\ldots ,\;N-1,$ and $\left( \widehat{B}{_{m}}\right)
_{pq}=\sum\limits_{k=1}^{K}A_{k}^{2}\cdot \left( {\beta _{k}}\right)
^{m}\cdot B_{kpq}$, $m=0,...,2N-2$.

Calculation of the next s-th step ($s=1,\;2,\;\ldots )$ begins from the
choice of the point $(x_{p},y_{q})$, of the map maximum

\[
\varepsilon ^{(s-1)}=\max_{x_{r}^{2}+y_{t}^{2}<\;1;\quad 0\;\leq \;m\;<\;N}{
\left\vert {\left( {D_{m}}\right) _{rt}^{(s-1)}}\right\vert }. 
\]

Now it's possible to specify a vector \bigskip $\left( \widehat{
\mathbf{I}}\right) _{pq}$:
\[
\left( \widehat{\mathbf{I}}\right) _{pq}^{(s)}=\left( \widehat{\mathbf{I}}
\right) _{pq}^{(s-1)}+\gamma \mathbf{E}^{-1}\cdot \left( \mathbf{D}\right)
_{pq}^{(s-1)},
\]

and the residual maps $\left( \mathbf{D}\right) _{rt}=\left\{ {\left( {D_{0}}
\right) _{rt},\left( {D_{1}}\right) _{rt},...,\left( {D_{N-1}}\right) _{rt}}
\right\} ^{T}$:
\[
\left( \mathbf{D}\right) _{rt}^{(s)}=\left( \mathbf{D}\right) _{rt}^{(s-1)}-
\widehat{\mathbf{B}}_{r-p,t-q}\cdot \left[ \left( \widehat{\mathbf{I}}
\right) _{pq}^{(s)}-\left( \widehat{\mathbf{I}}\right) _{pq}^{(s-1)}\right]
.
\]


Here $\mathbf{E=}\left( E_{ij}\right) $ is a positive defined
matrix of maximum values of \ weighted "dirty" beams, $E_{ij}=\left( 
\widehat{B}_{i+j}\right) _{0,0},i,j=0,...,N-1$; $\gamma $ is a loop gain.
The process of the iteration can be completed if $\varepsilon ^{\left(
s-1\right) }<\varepsilon $, where $\varepsilon $ is a given accuracy.
Otherwise it is necessary to suppose $s=s+1$ and to calculate the next 
$\ \varepsilon ^{\left( s\right) }.$Conditions of the convergence of the
algorithm above is $0<\gamma <2$, $1\leq N\leq K$.

The developed algorithm is nothing other than the \textit{
multi-frequency linear deconvolution}, itself. this is described in 
more detail this procedure by Likhachev, et al. (2003). Notice that the developed
algorithm allows to synthesize and analyze of high-quality VLBI images
directly from the visibility data measured on a few frequencies, 
without analyses of the images itself. In case of multi-frequency linear
deconvolution, it is possible to synthesize an image of a radio source at any
intermediate frequency \textit{inside} any given frequency band. Thus, 
\textit{spectral interpolation} of the image is feasible. 
This part of the algorithm is carry out the \textit{synthesis}
of the image itself. However, the algorithm also makes it possible to obtain an 
estimate of the \textit{spectral index} for a given radio source, i.e., it
implements the \textit{\ analysis }of the image. It is clear
that multi-frequency imaging (MFI) will provide the highest angular
resolution possible for any VLBI project due to its improved $\left(
u,v\right) $-coverage.

\section{Implementation of the linear deconvolution algorithm}

The algorithm described above was implemented in the software,
\textbf{\textit{Astro Space Locator (ASL) for Windows}} (\htmladdnormallink{http://platon.asc.rssi.ru/dpd/asl/asl.html}\/).
It was developed by the Laboratory for Mathematical Methods of the Astro Space Center
(Likhachev, 2003).

\begin{figure}
  \plottwo{P6-1f1.eps}{P6-1f2.eps}
  %\plottwo{P6-1_1.eps}{P6-1_2.eps}
  \caption{3C84 observations at 11 (interpolated) and 15 GHz}
\end{figure}

Fig.1 shows two deconvolved images of 3C84 as observed on the VLBA
at 11 and 15 GHz respectively. Due to the better (u,v)-coverage, the quality
and angular resolution of the interpolated MFS-image at 11 GHz
is much better than for the same source at 15 GHz.

\section{Acknowledgments}

The authors thank Jon Romney (NRAO) for providing the observational
data used in testing the new MFI algorithms.

\begin{references}

\reference Likhachev, S. F., Ladygin, V. A., \& Guirin, I. A. 2003, Lebedev Phys. Institute
Preprint, 31, 30p.
\reference Likhachev, S. F.,Multi-Frequency Imaging for VLBI, Future Directions in
High Resolution Astronomy, in print.

\end{references}

% Do not place any material after the references section

\end{document}
