Optimum spatial filter and uncertainty

Abstract

This paper deals with a basic aspect of spatial filters that analyze spatial waveforms such as visual patterns. In this paper, the n -dimensional spatial filters are assumed to be linear, homogeneous, isotropic and low-pass. The uncertainty relations between spatial resolution and spatial frequency resolution of the spatial filters are obtained by means of the variational method. The optimum filters in the sense of minimizing the product of their resolutions are obtained. The following three cases are considered: Case 1 . In this case, the impulse response f (x) and the transfer function Φ(ξ) are allowed to spread over the entire spatial domain E n and spatial angular frequency domain ɛ{su n }, respectively. Case 2 . In this case, f (x) vanishes outside the hypersphere of radius R with the center at the origin of E n . Case 3} . In this case, Φ(ξ) vanishes outside the hypersphere of radius P with the center at the origin of ɛ{su n }.