Many computer and robot vision applications require multi-scale image
analysis. Classically, this has been accomplished through the use of a
linear scale-space, which is constructed by convolution of visual input
with Gaussian kernels of varying size (scale). This has been shown to
be equivalent to the solution of a linear diffusion equation on an
infinite domain, as the Gaussian is the Green's function of such a
system (Koenderink, 1984). Recently, much work has been focused on the
use of a variable conductance function resulting in anisotropic
diffusion described by a nonlinear partial differential equation (PDE).
The use of anisotropic diffusion with a conductance coefficient which is
a decreasing function of the gradient magnitude has been shown to
enhance edges, while decreasing some types of noise (Perona and Malik,
1987). Unfortunately, the solution of the anisotropic diffusion
equation requires the numerical integration of a nonlinear PDE which is
a costly process when carried out on a uniform mesh such as a typical
image. In this paper we show that the complex log transformation,
variants of which are universally used in mammalian retino-cortical
systems, allows the nonlinear diffusion equation to be integrated at
exponentially enhanced rates due to the non-uniform mesh spacing
inherent in the log domain. The enhanced integration rates, coupled
with the intrinsic compression of the complex log transformation, yields
a speed increase of between two and three orders of magnitude,
providing a means of performing rapid image enhancement using
anisotropic diffusion.