An iterative deconvolution algorithm based on the method of error metric minimization has been developed (Jefferies &Christou, 1993). This is a blind deconvolution algorithm in that it restores both the object and point spread function (PSF) from an observed image. A detailed discussion of the algorithm, which is based on the work of Lane (1992), has been previously presented by Jefferies &Christou (1993). Here we only highlight the main points.
In order determine both the object and PSF from a noisy measurement of their convolution, it is necessary to enforce known physical constraints. The basic constraints are that the restored object and PSF at each iteration should: (1) convolve to the observed image, (2) have real, non-negative values, and (3) exist over finite regions of space. In addition, other constraints can also be applied. These include: (1) a band-limit constraint which prevents the trivial solution of the observed image and a delta function, (2) the use of multiple observations, (3) the use of only the high signal-to-noise region of the Fourier domain, and (4) the use of object Fourier spectrum information, e.g., as extracted from speckle imaging. We note that the multiple observations constraint can be handled in two ways, either sequentially or in parallel. The former uses the reconstructed object from one observation as the input for the next observation. The latter, which we utilize, constrains the object to all observations simultaneously.
The algorithm is deterministic unlike maximum likelihood techniques which explicitly model the noise in the observations. However non-systematic noise is handled by the use of multiple observations. Here the noise and the PSF are different for each image and it is only the object that is common to all the images.
Constraining the reconstruction to initial estimates of the object's Fourier components, i.e., the amplitude and phase, also prove to be very powerful especially for application to speckle interferometric data. The Fourier components, obtained from an ensemble average power spectrum for the amplitude and cross-spectra or bispectra for the phases, have significantly improved SNR over individual specklegrams, by a factor of the square root of the number of specklegrams and is typically of the order of 30. When using the Fourier amplitude as a constraint the algorithm now approximates one of ``phase retrieval.'' When using both amplitude and phase estimates it approximates an iterative deconvolution algorithm. We note that use of either of these constraints is not limited to the speckle imaging application but can also be applied to single images if an estimate of the PSF is known. Of course, in this case, the Fourier component SNR will be as noisy as the data and appropriate weighting has to be applied. Typically, we apply the Fourier domain constraints for the first few iterations and then relax them so as to avoid systematic errors in the Fourier quotients due to differences in the object and reference PSFs.