New Approaches to Fractal Dimension Estimation With Application to Gray-Scale Images

Abstract

Two new approaches for calculating box-counting fractal dimension (FD) estimates for gray-scale images are considered to overcome some of the limitations of the standard box-counting method, which requires setting a threshold in a pre-processing step. They include weighted gray-level box-counting (W-GBC) FD estimator and the probabilistic gray-level box-counting estimator in the image probability space (i. e., probability being proportional to pixel values) of an image (P-GBC-img). They are contrasted against the standard box-counting FD algorithm (BBC) and the probabilistic gray-level box-counting estimator in the intensity probability space (i. e., probability being proportional to the numerosity of a given range of pixel values) (P-GBC-int). A set of nine synthetic images and a set of 686 real gray-level images of tear film interferometry from normal and dry eye subjects were used for the evaluation of the considered estimators. Strong correlation (Pearson's ρ) was found between BBC and W-GBC (ρ = 0.998, p <; 0.001) and between BBC and P-GBC-img (ρ = 0.993, p <; 0.001) but not between BBC and P-GBC-int (ρ = 0.365, p <; 0.001). A good agreement, for both synthetic and real images, between BBC and the other estimators was achieved only for W-GBC, which additionally showed the highest discriminating power among the considered FD estimators (AUC = 0.697 vs the second best BBC with AUC = 0.638). Also, W-GBC is shown to fulfill the conditions for the recursive downsampling and, in consequence, can be implemented in a computationally efficient manner, particularly for large images. Finally, the W-GBC FD estimator achieves superior performance to that of BBC estimator.

Keywords

• Box-counting
• fractal dimension estimation
• generalized mean
• image entropy