Is the classical autocorrelation function appropriate for spatial signals defined on fractal supports?: The necessity of the generalized autocorrelation function

When applied to signals defined on fractal sets, the classical autocorrelation function has generally been exploited through its power law properties, the main hypothesis being that the exponent involved in this power law is uniquely defined. In this paper, we show that different power laws can likely be retrieved for the same signal. This non-uniqueness turns out to be associated to the uncertainty in determination of the exponent value. To avoid such degeneracy, we propose to use a generalized form of the autocorrelation function, a version of which we have previously introduced in the context of characterization of fractal sets.