Space-bandwidth product and accuracy of the optical Mellin transform.

The optical Mellin transform has recently been success­ fully realized and used to construct useful scale invariant optical pattern recognition systems, in Doppler signal pro­ cessing, and in the correlation of nonvertical imagery, among other applications. The Mellin transform of f(x) can be synthesized optically by applying a coordinate transformation ξ = lnx to the input coordinate x and an optical Fourier transform to ƒ[exp(ξ)]. We now examine the space-band­ width product requirements N' in the coordinate transformed space and derive expressions for their relationship to the input space-bandwidth product N and the accuracy of the optical Mellin transform. We denote the minimum resolution element size in x and ξ space by ΔX and Δξ, respectively. We consider an input function defined over 0 ≤ x < xma×. The corresponding function in ξ space is defined over — ∞ ≤ ξ ≤ lnxmax for full accuracy recording. By virtue of the required input coordi­ nate transformation, the portion of the input function lying at or near x = 0 maps into ξ = — ∞ in coordinate transformed space. To avoid the difficulty associated with this region, we restrict the domain of the input function to δ ≤ x ≤ xma×. If we denote the number of resolution elements in the interval 0 < x < δ by M, we find the spatial extent of the input function in terms of its space-bandwidth N to satisfy