Uniform partitions and a dimensions spectrum for lacunar measures

A multifractal analysis has been proposed to investigate complex systems, which exhibit a nonuniform local structure. Multifractal measures are normalized mass distributions inhomogeneous at every scale. To every point P is associated the exponent α of the power law rα describing the asymptotic behaviour as r→0 of the mass of a sphere of radius r and centre P. The measure properties are specified by the dimension spectrum f(α) of the set of points P with the same scaling exponent α. This spectrum is nontrivial even when the mass distribution has no holes (measure with Euclidean support) and the box counting dimension is an integer. In this case it can be evaluated by considering a sequence of uniform grids, whose spacing decreases as {2-n}, known as uniform partitions. We consider here a family of iterated function system fractal measures with support on the unit square with random defects specified by three parameters: a scale , a probability p and an attenuation factor η. Given a partition with a diameter less than any cell is chosen with probability p and its mass is multiplied by η: the corresponding pre-measure is obtained after normalization. In all the examples examined the f(α) spectrum appears to widen monotonically as η decreases. This result suggests the possibility of using the method to detect the presence of defects in sponge-like structures, such as cancellous bone (sponge-like), by analysing their radiographs since the grey intensity is related to the mass density of the structure. A widening of the f(α) spectrum is indeed observed in osteoporotic bone radiographs compared to normal bone radiographs. This may be attributed to the rarefaction of the cancellous bone.