Version 1.9 Build 1367

News	FAQ
Search	Home

Next: The Imaging Model Up: Recommendations for the AIPS++ Imaging Model Previous: A linear algebra model for Imaging

The inverse problem

Since the matrix A is nearly always singular, it has a null space. Any vector I^z in the null space will vanish: AI^z = 0. The sky brightness corresponding to I^z is therefore invisible to the Telescope and must be determined by other means such as the non-linear part of a deconvolution algorithm. To see that a linear method will not recover any vectors in the null-space consider a matrix C operating on D. Since D = AI, we see immediately that CD = CAI must also suffer from the same null-space as A. The two most prominent non-linear algorithms are the Maximum Entropy MEM (see e.g. Narayan and Nityananda, 1986) and CLEAN (Högbom, 1974). The MEM image can be defined as that solution to AI = D which has maximum entropy - I^T(lnI - lnI^M - 1) where I^M is a default image. The requirement that the entropy be maximal generates vectors in the null-space. CLEAN can be viewed purely as a way to solve AI = D which implicitly assigns values to the null space vectors I^z. Each identification of a clean component generates vectors in the null space.

Next: The Imaging Model Up: Recommendations for the AIPS++ Imaging Model Previous: A linear algebra model for Imaging