Fractal dimension (FD) is a useful metric for the analysis of natural images that exhibit a high degree of complexity, randomness and irregularity in color and texture. Several approaches exist in the literature to measure FD of gray-scale images. The aim of this study is to introduce a FD estimation method for color images with color proximity in Lab space. The proposed method uses a xy-plane partitioning–shifting mechanism, where the divisors of image size are used as grid sizes. The proposed method simulates on synthesized color fractal Brownian motion (FBM) images, publicly available Brodatz database, Google color fractal images and noisy Brodatz database. The random midpoint displacement algorithm for the formation of gray-scale images is extended in this work to synthesize color FBM images. Noisy Brodatz database is obtained by adding salt-and-pepper noise with different noise densities to understand the behavior of FD. The experimental results illustrate that the proposed method is effective and efficient and outperforms the three state-of-the-art methods by observing the values of two proposed metrics, namely average error and average computed FD. A new mathematical expression for estimating FD of a color image is demonstrated, which relies on the number of edge pixels of individual color channel using multiple linear regression.

中文翻译：

---

实验室空间中合成和自然彩色图像的分形维数

分形维数（FD）是用于分析自然图像的有用度量，这些自然图像在颜色和纹理上表现出高度的复杂性，随机性和不规则性。文献中存在几种测量灰度图像FD的方法。这项研究的目的是介绍在实验室空间中具有彩色接近度的彩色图像的FD估计方法。建议的方法使用xy平面分割移位机制，其中图像大小的除数用作网格大小。该方法对合成的彩色分形布朗运动图像，公开可用的Brodatz数据库，Google彩色分形图像和嘈杂的Brodatz数据库进行了仿真。在这项工作中扩展了用于形成灰度图像的随机中点位移算法，以合成彩色FBM图像。通过添加具有不同噪声密度的椒盐噪声来了解FD的行为，可以得到嘈杂的Brodatz数据库。实验结果表明，该方法是有效且高效的，并且通过观察两个建议指标（平均误差和平均计算FD）的值，其性能优于三种最新方法。