wavelets and scale invariance
Ruzzie Green, "Mother, daughter and doll" (circa 1950)
The concept of self-similarity is linked to the idea that a quantity is equivalent to any of its rescaled versions, up to an amplitude normalization. This equivalence may be deterministic (as it is the case for deterministic fractals, such as the devil's staircase or the von Koch flake), or it may be understood in a probablistic sense (stochastic fractals, such as, e.g., the fractional Brownian motion). In both situations, a wavelet-based multiresolution analysis reveals in a natural way the existence of scale invariance properties and it allows for an efficient estimation of the exponents controlling the scaling laws.