We describe the CCD image data collected by the HST by using the model discussed in detail by Snyder, Hammoud, and White (1993) in which the data collected at pixel of the CCD detector, , is described as
where is the number of object-dependent photo-electrons, is the number of background-dependent photo-electrons, is CCD read-out noise, is a two-dimensional index, and is the support region for the detector array. We assume that the random variables , , and are statistically independent of each other and of , , and , for , and that the object-dependent photoelectrons are Poisson distributed with the mean function
where is the object's intensity function, is the flat-field response function, is the PSF, and is a collection of parameters that are determined by the optical path-length error (Goodman 1968) due to the spherical aberration, focusing error, or other aberrations. Additionally, we assume that the background-dependent photoelectrons are Poisson distributed with a known mean function , and that the CCD readout noise are identically-distributed Gaussian random-variables with known mean and variance .
Because of the parameterization by , the PSF cannot take an arbitrary functional form, and the trivial solutions previously discussed are avoided. In the monochromatic, space invariant situation, the parameterized PSF is obtained by spatially sampling the continuous function
where is the telescope's pupil function, is the wavelength of the observed light, is the optical path-length error due to the optical aberrations, and is a two-dimensional spatial variable that indexes over the effective telescope pupil. The path-length error is commonly specified through either a point-by-point description
or a polynomial expansion
where the polynomials are typically chosen as the Zernike polynomials, orthogonal over an annular region (Burrows 1990). For a system such as the HST, the discrete PSF can be created by averaging the continuous PSF over detector regions. Mathematically, this procedure is described as
where is a jointly discrete-continuous function that describes the averaging and sampling performed by the CCD detector array.