Multifractal analysis has become an invaluable tool for estimating local regularities in experimental data. However, while multifractal analysis is thoroughly outlined for fields defined everywhere on compact supports, it is less well suited for the analysis of processes that exist only on restricted and partial supports, possibly with heterogeneous densities. Disentangling the multifractal measures of external geometry (inhomogeneous support) from intrinsic multifractality (mark) at arbitrary spatial scales is a major challenge and requires new conceptual and methodological procedures. Moreover, the question of how cross-dependencies in local regularities between the components of the signal fluctuate is rarely addressed in the context of such inhomogeneous complex systems. These two issues, which are crucial for a relevant application of multifractal analysis to geospatial data, are addressed in this presentation by introducing a local point process-oriented univariate and bivariate multifractal analysis. The relevance of the proposed construction is illustrated using synthetic data created by combining homogeneous and inhomogeneous point processes with multifractal textures. Finally, we present real-world applications using examples from urban geography.
