Abstract :
To filter an image by an aperture function A(r), or in Fourier space by its transform ??(f), requires that A(r) and ??(f) are physically realizable and mathematically exist as transform pairs. This constrains both functions to being square-integrable to guarantee their uniform convergence both in spatial and frequency domains. This paper presents a general method for deriving filter forms which maximize image SNR and also satisfy the existence constraints. We show that asymptotic limiting forms of the constrained filters reduce to the "ideal observer" result, but demonstrate that the "ideal" filter form has a divergent square integral in some simple cases important in medical imaging. We apply these properly constrained filter forms to radiographic image and CT image models, and derive the range of arguable signal-to-noise advantage, which "more" optimum filters may exhibit with respect to the simple "matched" filter limit.
