Z-Box merging: Ultra-fast computation of Fractal Dimension and Lacunarity

The applicability of fractal analysis to medical sciences has been well-known for almost thirty years now. However, the sheer volume of data produced by most medical imaging apparatuses, and the extreme inefficiency of most methods of fractal analysis, has presented a roadblock in the width of their application. To remediate that, very fast methods of fractal analysis are required. In a previous work of ours, the Box Merging method was introduced, which implements Box Counting by counting non-empty "boxes" directly from the coordinates of each element of a set. Its chief drawback is that it needs to sort a large array several times, slowing it down. This paper proposes another method that only requires one sorting. Afterwards, it offers a way to also calculate the lacunarity with negligible changes in algorithm structure and running time. To our knowledge, this marks the first time that the lacunarity can be computed in an acceptable time-frame. Finally, after offering some example applications and future improvements, the paper concludes.