Introduction

Various methods are being used to compensate for the resolution loss that is present in image data acquired by the Hubble Space Telescope. The common goal of these methods is to invert the excess point spreading caused by spherical aberration in the primary mirror so as to restore the images to their original design resolution. Inverse problems of this type are notoriously ill-posed. Not only can noise amplification cause serious degradation as improved resolution is sought for such problems but, also, the details of the implementation of any restoration method can have a significant impact on the quality of the restoration, with seemingly insignificant design choices having the potential to cause large changes in the result.

There are a variety of noise sources present in HST image data acquired with a CCD camera. The photo-conversion process by which object light is converted into photoelectrons introduces object-dependent noise characterized statistically as a Poisson random process. Nonideal effects introduce extraneous electrons that are indistinguishable from object-dependent photoelectrons. Examples of this excess ``noise'' include object-independent photoelectrons, bias or ``fat zero'' electrons, and thermo-electrons. We use the term background counts for the cumulative effect of these extraneous electrons. Background counts are Poisson distributed. Read-out noise further contributes to the degradation of images acquired with the charge coupled device camera aboard the HST. This noise is characterized as a Gaussian random process.

We use the following mathematical model due to Snyder, Hammoud, and White (1993) for describing CCD image data acquired by the HST:

where are the data acquired by reading out pixel j of the CCD-camera array, is the number of object-dependent photoelectrons, is the number of background electrons, is readout noise, and is the number of pixels in the CCD camera array. The statistical description of these quantities is as follows. The random variables , , and are statistically independent of each other and of , , and for . Object-dependent counts form a Poisson process with mean-value function , where

where accounts for nonuniform flat-field response, detector efficiency, and the spatial extent of the detector array,

is the HST point spread function (PSF), is the object's intensity function. The flat-field response function is assumed to be known, for example, through a flat-field calibration measurement. The PSF is also assumed to be known, for example, from a theoretical model of HST optics or through observations of an unresolved star. Background counts form a Poisson process with mean-value function ; this mean-value function is assumed to be known, for example, through a dark-field calibration measurement or from HST images taken near but not including the object of interest. Read-out noise forms a sequence of independent, identically distributed, Gaussian random variables with mean and standard deviation ; the constants and are assumed to be known.

One approach for compensating for blurring while recognizing the statistical properties of noise encountered in charge coupled device cameras is based on the method of maximum-likelihood estimation, as discussed by Snyder, Hammoud, and White (1993). This leads to the following iteration for producing a maximum-likelihood estimate of the object's intensity function:

where

and where

is the conditional-mean estimate of in terms of the data and the intensity estimate to be updated. Evaluation of the conditional mean (6) yields

where is the Poisson-Gaussian mixture probability density given by

This iteration is used as follows:

STEP 0.

Initialization: select

STEP 1.

Compute using (3)

STEP 2.

If done, display else and perform STEP 1