Optimum Image Reconstruction From Chop Measurements

Gerald Moriarty-Schieven (JAC), Doug Johnstone (Univ. of Toronto), Christine Wilson (McMaster U.), Jean Giannakopoulou-Creighton (IPAC), & Eric Gregersen (McMaster U.)

Technological improvements in submm instrumentation over the last few years have enabled astronomers to map large (hundreds of square arcminutes) areas of star-forming regions (Motte, Andre and Neri (1998, A&A, 336, 150); Johnstone and Bally (1999, ApJ, 510, L49); Wilson et al. (1999, ApJ, 513, L139)) with reasonable resolution (8-14" beams) and high sensitivity (rms noise ~0.01 Jy/beam). While the capability of submm instruments has improved dramatically, e.g. SCUBA, removing the interference produced at submm wavelengths by the rapidly varying atmosphere of the Earth still requires complex chopping procedures during the measurement process at the telescope. Instead of taking individual flux measurements at each position on the sky, difference measurements are taken between two locations separated by a chop distance and direction. By taking these difference measurements quickly, usually at a rate of several Hertz, the foreground atmosphere is effectively frozen in time and the signal provides a direct measure of the difference in flux between the two locations within the molecular cloud. Producing a flux map of the molecular cloud from a set of difference measurements requires deconvolving the chop beam, careful consideration of the sources of noise during the observations, and an understanding of the propagation of these errors through the reconstruction technique.

Many techniques for reconstructing an image of the sky from such chop maps have been proposed including the Emerson Fourier deconvolution method in common use at the JCMT, and maximum entropy techniques. Cosmologists, anticipating significant advances in satellite observations of the Cosmic Microwave Background have also considered matrix inversion solutions to reconstruct images from chop maps (Wright et al. (1996, ApJ, 458, L53). However, only a limited number of comparative tests have been performed to analyze the strengths and weaknesses of each individual technique (see for example Richer (1992, MNRAS, 254, 165) for a discussion on maximum entropy techniques, and Jenness et al. (1998, Proc. SPIE 3357, 548); and Jenness et al. (2000, ASP conf. series, in press) who talks about the merits of the Emerson fourier deconvolution method compared to the classical Emerson-Klein-Haslam technique). We have recently analyzed several techniques for making maps: in particular the Emerson Fourier deconvolution, and matrix inversion techniques.

The matrix inversion technique reconstructs the image as follows. The set of difference (chop) measurements can be represented by a matrix D = CS + N, where D is the set of chop measurements, C is the chop configuration, S is the sky brightness, and N is the noise. Cosmologists, in preparing for data sets with similar sized matrices, have spent considerable effort in obtaining methods for determining S, and have found that an iterative scheme works extremely well and converges quickly (Wright et al. 1996). For a map of size 256x256, approximately 100 iterations are needed. The technique is very computer intensive compared to the Emerson Fourier method, requiring several minutes on a Pentium III Linux computer compared to a fraction of a minute for the latter method. However, the matrix inversion technique has the advantage in that it uses all available information (e.g. the noise properties of each bolometer, etc.).

Using an artificial data set with known noise properties (see Figure 1), we analyzed the two principal techniques for constructing images of the sky from the chop data. The best image reconstructions were produced using the matrix inversion technique (Figure 3), especially when the noise was variable across the image and/or there is structure near the edge of the map. The Emerson Fourier deconvolution technique (Figure 2) is an efficient algorithm but suffers edge effects and diffusion when the noise is non-uniform (Figure 4).

This work has been submitted to the Astrophysical Journal (Johnstone et al. 2000a).