Abstract

Mathematical properties and potential uses of a new class of integral transforms are described. Its input functions f(x) or f(x,w) must have limited support, as is true of any image. For a 1-D input f(x), the 1-D integral logarithmic transform is defined as F(y) = ∫f(xy)d(logx). For a 2-D input function f(x,w) the alternative 2-D transforms F(y,z) = ∫∫f(xy,wz)d(logx)d(logw) or G(y,z) = ∫∫f_ρ(yr, zθ)d(logr)dθ are defined. In the latter, function f_ρ (r,0) is the 1:1 polar remapping of input function f(x,w). Mathematically, it is found that transforms F(y) and F(y,z) are invariant to linear change of scale (magnification) of input coordinates x or (x,w). This is without incurring an objectionable magnification-dependent translation in the output, as is incurred by simple 1:1 log-polar mapping. Of further interest, transform G(y,z) is invariant to both magnification and rotation of the input function f(x,w). Also, the transform F(y) due to a nonlinear power-law change of scale f(x) is itself a power-law change of scale on the original F(y). Finally, the log-transforms are conveniently inverted: for example, in the 1-D case of a given F(y), the input f(x) can be computed as f(x) = (x/x_o)F'(x/x_o), where x_O is the upper support coordinate for f(x). An optical implementation of transform F(y,z) is described.

© 1991 Optical Society of America