The BID algorithm converges slowly, so its convergence might be
accelerated by multiplying the residual by a constant before adding it
into , thereby boosting the correction factor.
The reason this method fails requires examination of the content of
the residual image.
The residual image contains (a) the aspects of the image that have not yet been accounted for as instances of the PSF, and (b) any structure in the observed image that arises from any process other than a convolution of the original image with the PSF. Any such extraneous information is noise. Sources of such noise include data errors occurring after the optical image is formed, errors in the estimate of the PSF, variations in the real PSF across the image field, Poisson-distributed photon noise in imagery of faint sources, or image effects caused by signals (cosmic rays in astronomy, scattered X-rays in medical imaging) impinging on the detector that are not subject to the blurring effects of the optical system. The portion of the residual image attributable to the PSF decreases in each iteration, but the noise content passes through essentially unchanged. Multiplying the residual image by some factor to boost the speed of convergence also strengthens the noise being added back into the estimate of the object. This noise quickly dominates the signal and makes the procedure unstable.
The instability can be fought by introducing nonlinear constraints
intended to suppress corrections due to noise. Typical constraints
include clamping the values in so that they lie within a known
reasonable dynamic range, or limiting the amount of adjustment that
can occur at any pixel (i.e., clamping
). These limits may
be imposed by setting thresholds or by introducing a damping, or
relaxation function.
Such constraints have two problems. First, they actually just delay the inevitable; noise problems are still serious. Second, the nonlinearities introduced in the constraints may cause artifacts in the estimate of the true image, destroying the quantitative information that is the point of many scientific studies. The mathematical analysis of the nonlinear methods is much more difficult; depending on the nonlinearity employed, a closed form for the estimate may not exist since different pixels may be treated in different ways.