# An Improved Robust Fractal Image Compression Based on M-Estimator
## 1 Introduction  
The idea of fractal image compression (FIC) was originally introduced by [PERSON] et al. in 1987 [1], and the first automatic algorithm for FIC was developed by [PERSON] in 1992 [2]. Due to the high compression ratio and simple decompression method, the FIC algorithm attracts many researchers' interests [3; 4; 5; 6]. However, there are two drawbacks of fractal encoding: its complex computation and its retrieved image quality. Many fractal-coding algorithms have been developed to improve on these drawbacks [7; 8; 9; 10; 11; 12]. Firstly, based on the basic concepts and methods of fractal image compression algorithms [13; 14; 15; 16; 17; 18; 19], a particle swarm optimization (PSO) method by utilizing the visual information of the edge property is proposed, and this method speeds up the encoder 125 times faster with only 0.89 dB decay of image quality in comparison to the full search method [20; 21; 22; 23; 24]. Then, several fractal image compression algorithms based on modified gray-level transform with fitting plane, characteristics of fractal and partitioned iterated function system, and wavelet transform with diamond search are proposed, which have better results in terms of compression speed and quality of results. Thereafter, high-efficiency video coding provided significant performance improvement in the compression ratio [25; 26; 27; 28; 29]. At the same time, various optimization algorithms [30; 31; 32; 33], robustness [34; 35; 36; 37], and feature extraction research [38; 39; 40; 41] provide new ideas for the field of image compression. However, little attention has been given to the robustness of such fractal-based methods until [PERSON] et al. [42] proposed a fractal-based method to enhance and restore a noisy image. The method is based on the Additive White Gaussian Noise (AWGN), whose amplitude distribution follows a Gaussian distribution, while the power spectral density is uniformly distributed. Indeed, one of the original motivations for this study was the observation that a noisy image is somewhat denoised when it is fractally coded. [PERSON] et al. [43]presented an enhanced fractal predictive denoising algorithm for denoising the AWGN images by using a quadratic gray-level function. A new similarity measure for fractal encoding has been introduced by [PERSON] et al. [44], the so-called, Huber Fractal Image Compression (HFIC) schema. It brings the linear Huber regression technique from the robust statistics into the encoding procedure of the fractal image compression. [PERSON] [45] proposed a similar technique of linear robust regression with least absolute derivation (LAD), least trimmed squares (LTS), and Wilcoxon to FIC. The linear regression techniques are the attempts toward the design of robust fractal image compression [46; 47; 48]. However, there is a primary drawback to the current schema: the complexity of computations [49; 50; 51; 52]. It needs almost ten thousand seconds for a full search HFIC algorithm to encode a standard gray-level Lena image. In our analysis, there are many calculations performed that are not required. For example, the fractal graphics compression technology based on a genetic algorithm [53; 54] considers and utilizes the self-transformation characteristics of the image, but its disadvantage is that it requires an extensive search and is time-consuming. The algorithm proposed in this paper can reduce the compression time by effectively eliminating many unnecessary calculations. This approach is shown to significantly improve the time of compression procedures without affecting the quality of the reconstructed image. The proposed algorithm is a robust M-estimator-based fractal image compression algorithm (RMFIC). It is a full-search fractal encoding algorithm using the quadtree partitioning method and it uses the principle of linear robust statistics. The goal of the work is to improve the efficiency and the retrieved image quality of existing robust fractal encoding methods, such as HFIC and LAD-FIC.