Fast computation of fractal dimension for 2D, 3D and 4D data

The box-counting (BC) algorithm is one of the most popular methods for calculating the fractal dimension (FD) of binary data. FD analysis has many important applications in the biomedical field, such as cancer detection from 2D computed axial tomography images, Alzheimer's disease diagnosis from magnetic resonance 3D volumetric data, and consciousness states characterization based on 4D data extracted from electroencephalography (EEG) signals, among many others. Currently, these kinds of applications use data whose size and amount can be very large, with high computation times needed to calculate the BC of the whole datasets. In this study we present a very efficient parallel implementation of the BC algorithm for its execution on Graphics Processing Units (GPU). Our algorithm can process 2D, 3D and 4D data and we tested it on two platforms with different hardware configurations. The results showed speedups of up to 92.38 x (2D), 57.27 x (3D) and 75.73 x (4D) with respect to the corresponding CPU single-thread implementations of the same algorithm. Against an OpenMP multi-thread CPU implementation, our GPU algorithm achieved speedups of up to 16.12 x (2D), 6.86 x (3D) and 7.49 x (4D). We have also compared our algorithm to a previous GPU implementation of the BC algorithm in 3D, achieving a speedup of up to 4.79 x . Finally, as a practical application of our GPU BC algorithm a study comparing the FD of 4D data extracted from the EEGs of a schizophrenia patient and a healthy subject was performed. The computation time for processing 40 4D matrices was reduced from three hours (sequential CPU) to less than three minutes with our GPU algorithm.