The diffusion equation is given by, $\partial h /\partial t = \nabla \cdot (c \cdot \nabla h) --(1)$. But in the Perona - Malik nonlinear diffusion equation the choice of c given by two forms PM1 : $c = e ^{(-(|\nabla h|/\lambda)^2)} --(2)$ and PM2 : $c = 1/(1+ (|\nabla h|^2 / \lambda^2)) -- (3)$ where $\lambda$ is edge threshold (which is some quantile of gradient).
The question I have is in equation 1 the units on the diffusion coefficient c have to be $L^2/T$ for dimensional consistency. However equations (3) and (4) which offer forms of the diffusion coefficient are dimensionless since gradient and hence lambda is dimensionless.
How is the dimenionality of the nonlinear diffusion equation in Perona-Malik 1989 filter satisfied?
I am sure I am missing on something very fundamental here. I would appreciate any indicators.
