The recent definition of fractional Brownian motions on surfaces has raised the statistical issue of estimating the Hurst index characterizing these models. To deal with this open issue, we propose a method which is based on a spectral representation of surfaces built upon their Laplace-Beltrami operator. This method includes a first step where the surface supporting the motion is recovered using a mean curvature flow, and a second one where the Hurst index is estimated by linear regression on the motion spectrum. The method is evaluated on synthetic surfaces. The interest of the method is further illustrated on some fetal cortical surfaces extracted from magnetic resonance images as a means to quantify the brain complexity during the gestational age.

Gyrification index (GI) is an appropriate measure to quantify the complexity of the cerebral cortex. There is, however, no universal agreement on the notion of surface complexity and there are various methods in literature that evaluate different aspects of cortical folding. In this paper, we give two intuitive interpretations on folding quantification based on the magnitude and variation of the mean curvature of the cortical surface. We then present a local spectral analysis of the mean curvature to introduce two local gyrification indices that satisfy our interpretations. For this purpose, the graph windowed Fourier transform is extended to the framework of surfaces discretized with triangular meshes. An adaptive window function is also proposed to deal with the intersubject cortical size variability. The intrinsic nature of the method allows us to compute the degree of folding at different spatial scales. Our experiments show that while more classical surface area-based GIs may fail at differentiating deep folds from very convoluted ones, our spectral GIs overcome this issue. The method is applied to the cortical surfaces of 124 healthy adult subjects of OASIS database and average gyrification maps are computed and compared with other GI definitions. In order to illustrate the capacity of our method to capture and quantify important aspects of gyrification, we study the relationship between brain volume and cortical complexity, and design a scaling analysis with a power law model. Results indicate an allometric relation and confirm the well-known observations that larger brains are more folded. We also perform the scaling analysis at the vertex level to investigate how the degree of folding varies locally with the brain volume. Results reveal that in our healthy adult brain database, cortical regions which are the least folded on average show an increased folding complexity when brain size increases.