# Scale invariance and $1/f^2$ power spectrum

In the paper

Occlusion Models for Natural Images : A Statistical Study of a Scale-Invariant Dead Leaves Model; Lee, A. B. Mumford, D. B. Huang, J.; International Journal of Computer Vision

I read that full scale invariance implies a power spectrum of $1/f^2$, i.e. that for a two dimensional stationary random process $Pr\{I(x,y)\}=Pr\{I(sx,sy)\}$ implies a $1/f^2$ form of the Fourier transform of the auto-covariance matrix.

I read this in several articles by now, but I am not able to find I proof that I could follow or to prove it myself. I am also confused about the direction of implication. One proof I found showed, that a $1/f^2$ power spectrum implies a scale invariant auto-covariance matrix. However, the authors from the paper above seem to imply the opposite. Additionally, a flat power spectrum (i.e. white noise) would also fulfill the scale invariance property from above but would clearly be not $1/f^2$.

In summary, I would be grateful if someone could

• hint to literature, where this is shown in detail
• provide a proof
• clarify the direction of implication or the conditions under which this is usually shown.