# An Improved Robust Fractal Image Compression Based on M-Estimator
## 5 Experimental Results  
In order to evaluate the performance of this proposed algorithm (RMFIC), a computer simulation is performed. The standard gray-level image of [PERSON] and Boats with \(512\times 512\) pixels size was used. The coding block size in the RMFIC is set to be \(32\times 32\), \(16\times 16\), or \(8\times 8\) using the quadtree partitioning method, and the tolerance value \(T=8\). For the proposed algorithm based on [PERSON]'s M-estimator (RMFIC-H), we adopt the object function of Equation (10) as the matching error function. The initial values of the estimated parameters are obtained by Least Squares. The tuning constant parameter \(k=1.345\sigma\), and in the study case, \(\sigma\) is defined to be fixed value 5. For the proposed algorithm based on [PERSON]'s bisquare M-estimator (RMFIC-TB), we adopt MSE (mean-square-error) in Equation (25), as the matching error function. The initial values of the estimated parameters are obtained by Least Squares regression. The tuning constant parameter \(t=4.6851\sigma\), and \(\sigma\) is defined by Equation (15). We use a blank image as the initial image for decoding and the iteration number is 20, which is sufficient to reconstruct the image. The simulation is done under the Microsoft Visual C++ 2008 on Windows XP with SP3, Intel Core2 Duo 2.53 GHz CPU, and 2G RAM platform. \[\text{MSE}=\frac{1}{N}\sum_{i=1}^{N}\left(f_{i}-f_{i}\right)^{2} \tag{25}\]  
where \(f_{i}\) is the grey value of the pixel \(i\) in the original image and \(\hat{f_{i}}\) is the gray level of the pixel \(i\) in the retrieved image. \(N\) denotes the total number of pixels. The distortion between the original image \(f\) and the retrieved image \(\hat{f}\) is measured by the peak signal-to-noise ratio (PSNR). The definition of PSNR is  
\[\text{PSNR}(f,\hat{f})=10\log_{10}\left(\frac{255}{MSE(f,\hat{f})}\right) \tag{26}\]  
To show the quality of the reconstruction of the image, we corrupt the Lena image and the Boats image with salt and pepper noise with noise level as high as 10%. Figure 1 illustrates the retrieved images from the corrupted Lena image by full search FIC with normal least square regression (FIC-LS) and proposed RMFIC using [PERSON]'s and [PERSON]'s robust statistics (RMFIC-H and RMFIC-TB). Figure 1a,b show the original Lena image and corrupted Lena image with a 10% noise level, respectively. Figure 1c-e illustrate the retrieved image from full search FIC-LS, RMFIC-H, and RMFIC-TB methods, respectively, with corresponding qualities of 23.31 db, 28.10 db and 26.46 db by PSNR for Lena corrupted image. Figure 2 is the test case on Boats image which is corrupted by salt and pepper noise with a 10% noise level. Figure 2a,b show the original Boats image, and corrupted Boats image. Figure 2c shows the retrieved image from full search FIC with normal least squares regression with 22.68 db by PSNR. Figure 2d,e illustrate the retrieved image from proposed methods (RMFIC) with corresponding qualities of, 25.57 db and 24.56 db by PSNR for Boats corrupted image. From the experimental results, we can form the conclusion that the RMFIC algorithm using both the Huber case and [PERSON]'s bisquare case show the ability of robustness for FIC. Namely, it is much more insensitive to outliers, compared to LS-based full search FIC. Note that there are several tiny blocks not reconstructed correctly from the retrieved image from RMFIC-TB by vision. The existence of these tiny blocks is due to the definition of [PERSON]'s bisquare M-estimator. From the weight Equation (19), we can see that if \(|e|>k\), the weight value is zero. That means if the outlier is \"detected\", the bisquare estimator can get rid of this value from the data set. However, in the same situation, [PERSON]'s estimator has set the weight value to 1. Thus, the [PERSON]'s bisquare M-estimator sacrifices some data. When encoding the small blocks by fractal, there is a case where pixels are taken as outliers and the weight is taken as zero. That is why these tiny incorrect retrieved blocks exist.