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Astronomical Data Analysis Software and Systems XII ASP Conference Series, Vol. 295, 2003 H. E. Payne, R. I. Jedrzejewski, and R. N. Hook, eds.

Generalized Self-Calibration for Space VLBI Image Reconstruction
Sergey F. Likhachev Astro Space Center, Profsojuznaya 84/32, GSP-7, 117997, Moscow, Russia Abstract. Generalized self-calibration (GSC) algorithm as a solution of a non-linear optimization problem is considered. The algorithm allows to work easily with the first and the second derivatives of a visibility function phases and amplitudes. This approach is important for high orbiting Space VLBI data processing. The implementation of the GSC-algorithm for radio astronomy image restoration is shown. The comparison with other self-calibration algorithms is demonstrated. The GSC-algorithm was implemented in the radio astronomy imaging software pro ject Astro Space Locator (ASL) for Windows developed at the Astro Space Center.

1.

Image reconstruction: Basic concepts

Definition 1.1. A space of complex functions V (u) dual to the brightness distribution I (x) and risen by an operator F , is called a space of visibility functions or spatial coherency.


V (u) = F {I (x)} =
-

I (x)exp {-2i x, u } dx

(1)

Definition 1.2. Let us define an object as a domain of the Universe that is a sub ject of the investigation whose brightness distribution could be represented as a 2-D function with infinite spatial frequency spectra. Definition 1.3. Let us define an image of the ob ject as a result of creation by unknown spatial brightness distribution I
tr ue

(x)

(the ob ject) of an il lumination distribution J i.e., J
tr ue tr ue

(x) ,
tr ue

(x) = (H I

)(x) . ted somewhere in the space of this ob ject on this space. can be its electromagnetic in a given moment of time.

In other words, we have an original ob ject loca (Universe) and we can observe only some pro jection For example, one of the pro jections of the ob ject emission of the ob ject in a given spectral band and 191

c Copyright 2003 Astronomical Society of the Pacific. All rights reserved.


192 2.

Likhachev VLBI Image Restoration as an Approximation Procedure

Let us consider a metrics =
ij

w

ij

^ Vij - Vij

2

min

(2)

^ where Vij is a measurements of the visibility function and Vij is an approximating function. There exist a few possible approximating functions: 1. Orthogonal approximation: ^ Vij =
ij

cij

ij

If ij is a Fourier basis, the orthogonal approximation is known in VLBI as a CLEAN algorithm. 2. Bi-orthogonal approximation: ^ V = g· ·g Ї If ij is a 2-D complex function (a model), the bi-orthogonal approximation is known in VLBI as a self-calibration algorithm. 3. Non-parametrical approximation: lg Ii - · max
i

(known in VLBI as a MEM algorithm). 4. Mathematical programming approximation: D =B·I I0 (known in VLBI as a so-called NNLS algorithm). 3. Generalized Self-Calibration (GSC) as a Problem of Non-linear Optimization

Let us consider an expression V
t

= gt V

tr ue g t Їt

+ t

(3)

where gt = diag [gi ], dim {gt } = [N - 1 в N - 1], Vt = {Vij kl } visibility matrix was measured on the baseline (i, j) for a given moment of time tk and tr frequency l , dim {Vt } = [N - 1 в N - 1], Vtue true visibility function for the baseline (i, j), for a given moment of time tk and frequency l , dim {Vt } = [N - 1 в N - 1], t additive noise.


Generalized Self-Calibration for Space VLBI Let us consider a discrepancy ^g zt = gt Vt Їt - V It is necessary to obtain: arg min = zt
g
i

193

t

(4) (5)

2

tr ^ ^ ^ where Vt is a model of Vtue , Vt is upper triangular matrix with Vij = 0. The solution gt obtained on the basis generalized Newton's algorithm with pseudo-inversion.

4.

Local Approximation of Gains

Let us represent a complex function gi (t , ) as a time series in the neighborhood of a point (t0 ,0 ). Then gi t0 +t, 0 + = gi (t0 ,0 )+ gi (t0 ,0 ) gi (t0 ,0 ) t + + t O t2 + 2 (6)

Let us introduce the following notations: · let us call i (t0 ,0 ) ri = t as a fringe rate; · let us call i (t0 ,0 ) i = as a fringe delay. Both values are complex ones and can be represented as gi t0 +t, 0 + = ai (t0 ,0 ) ai (t0 ,0 ) ai (t0 ,0 )+ t + + t i · [ai (t0 ,0 ) rit + ai (t0 ,0 ) ]}в exp {i · i (t0 ,0 )} + O t2 +
2

(7)

Example. If ai (t0 ,0 ) = const (no amplitude calibration) then gi t0 +t, 0 + = {const · [rit + ]}в exp i · i (t0 ,0 )+ O t2 +
2

+2k 2

+ (8)

and obtain a well-known expression for phase calibration (see Schwab 1981). A value O t2 + 2 describes derivatives of the second order that is necessary to take into account for Space VLBI imaging.


194 5.

Likhachev Some Imaging Problems for High Orbiting Space VLBI

Definition 5.1. If for any three radio telescopes 1. there exists its closing, i.e., 1 + 2 + 3 = 0; 2. any two baselines b13 & b23 > D 3. and
Ear th

,

b23 - b13 / b12 1,

then the VLBI can be called high orbiting space VLBI. In case of a High Orbiting SVLB mission a good (u,v)-coverage does not guarantee high quality images because is an "apogee phase gap." 6. Implementation of GSC in the ASL for Windows
12



rms

The software pro ject, Astro Space Locator (ASL) for Windows 9x/NT/2000 (code name ASL Spider 1.0) is developed by the Laboratory for Mathematical Methods1 of the ASC to provide a free software package for VLBI data processing. We used the Microsoft Windows NT/2000 and MS Visual C++ 6.0 on IBM compatible PCs as the platform from which to make data processing and reconstruction of VLBI images. 7. Outcomes

A generalized self-calibration (GSC) algorithm was developed. The solution was obtained as a non-linear optimization in the Hilbert space L2 . GCS describes not only the first derivatives but also of the second derivatives that is necessary to take into account for Space VLBI imaging. A global fringe fitting procedure is just an initialization (zero iteration) of GSC. GSC allows to obtain more stable and reliable results than traditional self-calibration algorithms. References Schwab, F. R. 1981, VLA Scientific Memorandum, No. 136, NRAO

1

http://platon.asc.rssi.ru/dpd/asl/asl.html