An Approximated Box Height for Differential-Box-Counting Method to Estimate Fractal Dimensions of Gray-Scale Images

Chinmaya Panigrahy,Dionisio Rodríguez-Esparragón,Nihar Mahato,Ayan Seal,Consuelo Gonzalo-Martín,Angel Garcia-Pedrero

doi:10.3390/e19100534

Abstract

The Fractal Dimension (FD) of an image defines the roughness using a real number which is highly associated with the human perception of surface roughness. It has been applied successfully for many computer vision applications such as texture analysis, segmentation and classification. Several techniques can be found in literature to estimate FD. One such technique is Differential Box Counting (DBC). Its performance is influenced by many parameters. In particular, the box height is directly related to the gray-level variations over image grid, which badly affects the performance of DBC. In this work, a new method for estimating box height is proposed without changing the other parameters of DBC. The proposed box height has been determined empirically and depends only on the image size. All the experiments have been performed on simulated Fractal Brownian Motion (FBM) Database and Brodatz Database. It has been proved experimentally that the proposed box height allow to improve the performance of DBC, Shifting DBC, Improved DBC and Improved Triangle DBC, which are closer to actual FD values of the simulated FBM images.

Highlights

IntroductionGeometry, which was introduced by Mandelbrot [1] in 1983
A natural scene could be treated as complex objects that could be represented using FractalGeometry, which was introduced by Mandelbrot [1] in 1983
In [13], Sarkar et al proposed a method known as Differential Box Counting (DBC) to compute Fractal Dimension (FD) using box counting (BC) technique, which became popular in research community because it is simple, easy to interpret and implement

Summary

Introduction

Geometry, which was introduced by Mandelbrot [1] in 1983. Fractal Geometry is used to define self-similar elements independent of scale, known as fractal set. In [13], Sarkar et al proposed a method known as DBC to compute FD using box counting (BC) technique, which became popular in research community because it is simple, easy to interpret and implement. It suffers from various drawbacks like over-counting of boxes along z-direction (quantization-computation), over-counting of boxes along xy-direction, lack of a proper box height, inappropriate use of grid sizes.

Methods

Results

Conclusion