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The point spread function and the small-angle approximation to the radiative transfer equation

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Application of the small-angle approximation to the RTE (see Small-angle approximation ...) results in a closed-form solution for the point spread function (PSF) (Wells WH 1969, see also Small-angle approximation to the RTE with application to ocean waters). The PSF of a scattering medium determines the loss of resolution in an image taken through that medium. Therefore, it is a valuable tool that is widely used for predicting image quality and detection limits (McLean JW and Voss 1991, Voss KJ and Chapin 1990, Kuga Y and Ishimaru 1985, Mertens LE and Replogle 1977, see also Point spread function and imaging).

To calculate the PSF, the scattering phase function (SPF) must be known (see Small-angle approximation to the RTE with application to ocean waters). Wells, who first formulated the solution (Wells WH 1969), used an algebraic form for the SPF (for example, McLean JW and Voss 1991, their Equation 6) that had a plausible shape (similar to that of ocean water) and could be integrated to obtain an exact, analytical PSF. The purpose was merely to show a simple example of the theory without resorting to numerical modeling.

Unfortunately, data were published soon after showing good agreement between the measured spatial frequency decay function (the Fourier transform of the SPF) and the decay function generated from the Wells algebraic SPF (Hodara H 1973). Consequently, the Wells' algebraic function was used to validate the SAA even though it had become clear that the function was not a good fit to data (McLean JW and Voss 1991). Subsequently, a more realistic, modified version of the algebraic SPF has been used (McLean JW and Freeman 1996). Another commonly used analytic SPF is the Henyey-Greenstein function (HG, for example Jonasz M and Fournier 2007, pp. 259-261 - where the HG function in its various forms and fitting to data are discussed). However, that function also has been shown to be inconsistent with data for ocean waters (Kattawar GW 1975). Clearly, more accurate results will be obtained by using measured scattering functions and numerically calculating the PSF

CITATION: Swanson N. L. 2008. The point spread function and the small-angle approximation to the radiative transfer equation (www.tpdsci.com/Tpc/RTESAAPSF.php). In: Top. Part. Disp. Sci. (www.tpdsci.com).

HISTORY: Published: 28-Oct-2008 Modified: 17-Jun-2009 Peer-reviewed: 31-Oct-2008

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