Of fractal and Fourier: A measure for local shape complexity for neurological applications

Background: Local shape complexity can be biologically meaningful as a marker of disease, trauma, or change in brain structure over time. Fractal dimensionality (FD) is currently the dominant measure of local shape complexity used in neuroimaging but its limitations are not well understood.

New method: Elliptical Fourier harmonic power requirement (HPR) may provide complementary information to FD. We benchmarked the performance of FD and HPR on a series of simulated shapes, systematically manipulating aspects of local shape complexity, and a series of clinical contours (glioma tumour cores and stroke lesions from the BRATS and ATLAS datasets). HPR was calculated as the point of 99.9% harmonic power. FD was calculated at six resolutions (8 × 8, 16 × 16, 32 × 32, 64 × 64, 128 × 128, and 256 × 256), by using an approach which computationally indexes the complexity of the shape boundary (i.e. the number of cells defining the contour) relative to the total grid size.

Results and comparison with existing methods: PR and FD were moderately positively correlated (r ≈ 0.2 to 0.8 depending on shape properties), and both were sensitive to the frequency and amplitude of local complexity. FD was most biased by rotation, while HPR was more biased by global shape features such as deep invaginations. FD indicated an aggregate measure of complexity across the whole contour, while HPR indicated the point of highest complexity.

Conclusions: The HPR index provides conceptually distinct local complexity information from the current FD standard. Future research will benefit from using these complementary measures.