Ultra-fast computation of fractal dimension for RGB images

The fractal dimension (FD) is a quantitative parameter widely used to analyze digital images in many application fields such as image segmentation, feature extraction, object recognition, texture analysis, and image compression and denoising, among many others. A variety of algorithms have been previously proposed for estimating the FD, however most of them are limited to binary or gray-scale images only. In recent years, several authors have proposed algorithms for computing the FD of color images. Nevertheless, almost all these methods are computationally inefficient when analyzing large images. Nowadays, color images can be very large in size, and there is a growing trend toward even larger datasets. This implies that the time required to calculate the FD of such datasets can become extremely long. In this paper we present a very efficient GPU algorithm, implemented in CUDA, for computing the FD of RGB color images. Our solution is an extension to RGB of the differential box-counting (DBC) algorithm for gray-scale images. Our implementation simplifies the box-counting computation to very simple operations which are easily combined across iterations. We evaluated our algorithm on two distinct hardware/software platforms using a set of images of increasing size. The performance of our method was compared against two recent FD algorithms for RGB images: a fast box-merging GPU algorithm, and the most advanced approach based on extending the DBC method. The results showed that our GPU algorithm performed very well and achieved speedups of up to 7.9x and 6172.6x regarding these algorithms, respectively. In addition, our algorithm achieved average error rates similar to those obtained by the two reference algorithms when estimating the FD for synthetic images with known FD values, and even outperformed them when processing large images. These results suggest that our GPU algorithm offers a highly reliable and ultra-fast solution for estimating the FD of color images.