Astro Space Center, Profsojuznaya 84/32, GSP-7, 117997, Moscow, Russia

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 project Astro Space Locator (ASL) for Windows
developed at the Astro Space Center.

** Definition 1.1.** A space of complex functions dual
to the brightness distribution and risen by an
operator , is called a space of * visibility functions* or
* spatial coherency*.

(1) |

** Definition 1.2.** Let us define * an object* as a domain of the Universe that
is a subject 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 object as a result of creation
* by unknown spatial brightness distribution*

(the object) of

i.e.,

In other words, we have an original object located somewhere in the space (Universe) and we can observe only some projection of this object on this space. For example, one of the projections of the object can be its electromagnetic emission of the object in a given spectral band and in a given moment of time.

Let us consider a metrics

(2) |

There exist a few possible approximating functions:

- Orthogonal approximation:

If is a Fourier basis, the orthogonal approximation is known in VLBI as a CLEAN algorithm. - Bi-orthogonal approximation:

If is a 2-D complex function (a model), the bi-orthogonal approximation is known in VLBI as a self-calibration algorithm. - Non-parametrical approximation:

(known in VLBI as a MEM algorithm). - Mathematical programming approximation:

(known in VLBI as a so-called NNLS algorithm).

Let us consider an expression

(3) |

Let us consider a discrepancy

(4) |

(5) |

Let us represent * a complex function*
as a time series in the neighborhood of a point
. Then

(6) |

Let us introduce the following notations:

- let us call

as*a fringe rate*; - let us call

as*a fringe delay*.

Both values are complex ones and can be represented as

(7) |

** Example.** If
(no amplitude calibration) then

(8) |

and obtain a well-known expression for phase calibration (see Schwab 1981).

A value describes derivatives of the second order that is necessary to take into account for Space VLBI imaging.

** Definition 5.1.** If for any three radio telescopes

- there exists its closing, i.e.,

- any two baselines

- and

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.''

The software project, Astro Space Locator (ASL) for Windows 9x/NT/2000 (code name ASL_Spider 1.0) is developed by the Laboratory for Mathematical Methods 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.

A generalized self-calibration (GSC) algorithm was developed. The solution was obtained as a non-linear optimization in the Hilbert space . 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.

Schwab, F. R. 1981, VLA Scientific Memorandum, No. 136, NRAO

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