The Richardson-Lucy (R-L) algorithm (Richardson 1972, Lucy 1974) is the technique most widely used for restoring HST images. The standard R-L method has a number of characteristics that make it well-suited to HST data:
The R-L iteration can be derived very simply if we start with the imaging equation and the equation for Poisson statistics. The imaging equation tells how the true image is blurred by the PSF:
where is the unblurred object, is the PSF (the fraction of light coming from true location that gets scattered into observed pixel ), and is the noiseless blurry image. If the PSF does not vary with position in the image, then and the sum becomes a convolution. The probability of getting counts in a pixel when the mean expected number of counts is is given by the Poisson distribution:
and so the joint likelihood of getting the observed counts in each pixel given the expected counts is
The maximum likelihood solution occurs where all partial derivatives of with respect to are zero:
The R-L iteration is simply
It is clear from a comparison of Eqs. (4) and (5) that if the R-L iteration converges (as has been proven by Shepp &Vardi 1982), meaning that the correction factor approaches unity as the iterations proceed, then it must indeed converge to the maximum likelihood solution for Poisson statistics in the data.