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

Piotr D Szyperski,D Robert Iskander

doi:10.1109/access.2019.2960256

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.

Highlights

There are different approaches to calculating fractal dimension (FD) of an image, including the binary box-counting algorithm (BBC) and its differential extensions, the differential box counting algorithm (DBC) [1], [2], as well as the multitude of other image fractal descriptors [3]–[17]
Following experiments on the simulated interference data, the real data consisting of 686 gray-level interferometric images – originally of size 720 × 576 px from which square central parts of size 512 × 512 px have been extracted as inputs – of human tear film acquired in an in-vivo manner were added to the test set
Plotted values are normalized by adaptive linear rescaling of ordinates to match their minima and maxima with abscissæ and assert positive direction of output variation. These results indicate that the proposed estimators of FD provide agreeable outcomes to the traditional BBC

Summary

Introduction

There are different approaches to calculating fractal dimension (FD) of an image, including the binary box-counting algorithm (BBC) and its differential extensions, the differential box counting algorithm (DBC) [1], [2], as well as the multitude of other image fractal descriptors [3]–[17]. For gray-scale images, the classical BBC FD estimator requires a pre-processing step of transforming the image into binary values, where the result depends on the used threshold [21], [22]. Limits the generality of the BBC method to a pre-processed class of input images. The aim of this work is to develop methods that overcome the limitation of BBC. Following the philosophy of Huang et al [23], methods should have the ability of high discriminative power in application in which interferometry is used to non-invasively assess the kinetics of human tear film [24]–[26]

Objectives

Methods

Results

Conclusion