(Letter to the Editor)
KAZIMIERZ M. BORKOWSKI
Toruń Radio Astronomy Observatory, Nicolaus Copernicus University, Toruń, Poland
(Received 21 December, 1984)
|Abstract. A theorem, which provides a relationship between the one-dimensional Fourier transforn of a line section across two-dimensional (spatial) spectrum and that across the corresponding (brightness) distribution function, is proved. The theorem is then shown to be relevant in some problems in radio astronomy and possibly in other fields connected with image reconstructions from one-dimensional scans through objects or their spectra.|
Let B(x,y) and V(u,v) be a pair of the two-dimensional Fourier transforms. In what follows we wish to prove the following result.
The one-dimensional Fourier transform, taken at the frequency
ν, of the section across B(x,y) along
the line which passes at the distance ρ
off the origin of the spatial coordinates,
is equal to the one-dimensional inverse Fourier transform, taken at
the distance ρ, of the section along a perpendicular line across V(u,v), placed at the frequency
ν off the origin of the frequency coordinates,
+ z cosθ) e–j2πνz
+ w cosθ,νcosθ
+ w sinθ) ej2πνz dw. (2)
is equal to the one-dimensional inverse Fourier transform, taken at the distance ρ, of the section along a perpendicular line across V(u,v), placed at the frequency ν off the origin of the frequency coordinates,
The Fourier transform of (1) at the frequency ρ is
(Jacobian of rotation of coordinates is unity). Since this expression can be recognized as the expected slice of the spatial spectrum V(u,v), i.e. the Fourier transform of (2), the proof is completed.
Putting ν = 0 in (1) reduces it to a line integral, or the Radon transform, while (2) hecomes now the Fourier transform of a central section through the spectrum. Thus, our theorem can be regarded as a generalization of the well-known projection theorem, which provides a relationship between the Fourier transform of a function and the Fourier transform of its line integrals. The Radon transform plays important role in a few radioastronomical applications (e.g., Bracewell and Riddle, 1967; Perley, 1979), and, to a greater extent, in other areas of science and medicine concerned with image reconstructions from projections through an object (for a review see, e.g., Rowland, 1979; or Durrani and Bisset, 1984).
Interferometers used in radio astronomy sample the source brightness distribution in the spatial frequency domain along tracks which, in practice, rarely reach the center of the uv plane. The tracks are normally ellipses, or sections thereof, so that a linear approximation to the uv tracks leads to a set of sections along lines that pass different distances (frequencies) off the origin of the uv plane. Our theorem provides a relationship between such slices of spectra and original distributions.
Some methods used to reconstruct the distribution of point sources in the field of view of an interferometer are satisfactorily based on consideration of only the phase of the spatial spectrum, or rather its rate of change at maximum amplitude. Notably, Speed (1976; see also Peckham, 1971) used so-called strip transforms, the fringe frequency spectra, to find directions on the sky plane of the responsible sources. Source positions were then determined as the points in which the directions, corrcsponding to different sections in the spatial frequency domain perpendicular to these directions, crossed. Though independently but basically the same method was developed for use in spectral line very long baseline interferometry (VLBI) work by Guiffrida (1977) and Walker (1981) and is known as the multiple-point fringe rate mapping of cosmic masers. It will be noted that these methods make use of the measured (differential) fringe frequencies at the maximum amplitude while neglecting the phase of the fringe spectrum which, if a spectral line is not blended, contains also necessary information on the placement of a point source along the determined direction. This can easily be demonstrated by assuming a single point or Gaussian source and using our theorem (see also Borkowski, 1984).
Finally, we note that our theorem is not limited to any particular class of source distributions, such as point sources. One is thus tempted to speculate about a generalized back projection applicable to noncentral sections through the spatial spectra as met, e.g., in aperture synthesis. Such a technique not only might prove to be faster than present day VLBI mapping techniques based on interpolation of data and two-dimensional Fourier transformation, but it would also free a user of introducing explicitly the unrealistic assumption of zeroes in all empty spaces of the uv plane.
Borkowski, K.M.: 1984, Postępy Astron. XXXII, 205 (in Polish).
Bracewell, R.N. and Riddle, A.C.: 1967, Astrophys. J. 150, 427.
Durrani, T.S. and Bisset, D.: 1984, Geophys. 49, 1180.
Guiffrida, T.S.: 1977, Ph.D. Thesis, M.I.T., Cambridge, Massachusetts.
Peckham, R.J.: 1971, Ph.D. Thesis, University of Manchester, Manchester.
Perley, R.A.: 1979, Astron. J. 84, 1443.
Rowland, S.W.: 1979, Topics Applied Phys. 32, 9.
Speed, B.: 1976, Monthly Notices Roy. Astron. Soc. 177, 137.
Walker, R.C.: 1981, Astron. J. 86, 1323.