Spatial aliasing and scale invariance

Examples of scale-invariant filtering

When we consider all functions with vanishing gradient, we notice that the gradient vanishes whether it is represented as $(1,-1)/\Delta x$ or as $(1,0,-1)/2\Delta x$ . Likewise for the Laplacian, in one dimension or more. Likewise for the wave equation, as long as there is no viscosity and as long as the time axis and space axes are stretched by the same amount. The notion of ``dip filter'' seems to have no formal definition, but the idea that the spectrum should depend mainly on slope in Fourier space implies a filter that is scale-invariant. I expect the most fruitful applications to be with dip filters.

Resonance or viscosity or damping easily spoils scale-invariance. The resonant frequency of a filter shifts if we stretch the time axis. The difference equations

$$y_t - \alpha y_{t-1} =$$ 0 (5)

$$y_t - \alpha^2 y_{t-2} =$$ 0 (6)

both have the same solution $y_t = y_0 \alpha^{-t}$ . One difference equation has the filter $(1,-\alpha)$ , while the other has the filter $(1,0,-\alpha^2)$ , and $\alpha$ is not equal to $\alpha^2$ . Although these operators differ, when $\alpha \approx 1$ they might provide the same general utility, say as a roughening operator in a fitting goal.

Another aspect to scale-invariance work is the presence of ``parasitic'' solutions, which exist but are not desired. For example, another solution to $y_t - y_{t-2}=0$ is the one that oscillates at the Nyquist frequency.

(Viscosity does not necessarily introduce an inherent length and thereby spoil scale-invariance. The approximate frequency independence of sound absorption per wavelength typical in real rocks is a consequence of physical inhomogeneity at all scales. See for example Kjartansson's constant Q viscosity, described in IEI. Kjartansson teaches that the decaying solutions $t^{-\gamma}$ are scale-invariant. There is no ``decay time'' for the function $t^{-\gamma}$ . Differential equations of finite order and difference equations of finite order cannot produce $t^{-\gamma}$ damping, yet we know that such damping is important in observations. It is easy to manufacture $t^{-\gamma}$ damping in Fourier space; $\exp[(-i\omega)^{\gamma+1}]$ is used. Presumably, difference equations can make reasonable approximations over a reasonable frequency range.)

Spatial aliasing and scale invariance