# Post-Filtering of Noisy Images Compressed by HEIF
## 4 Post-Filtering Opportunities and Results  
### Post-Filtering Opportunities  
Thus, we have to remove noise that has a probability density function (pdf) close to Gaussian (with possibly heavier tails), is almost white, and has an approximately known variance (MSE of residual noise and distortions that can be either slightly smaller or larger than the variance of the noise \(\sigma^{2}\) in the original image). Although there has been a huge number of image denoising techniques designed thus far (see survey papers [64; 65; 66] and references therein), our intention here was not to find the best filter for our application. Instead, our aim was to approximately estimate the potential of post-filtering and identify the main tendencies. Because of this, we decided to consider the famous block matching 3-dimensional (BM3D) filter [52; 67], which is based on finding similar patches and using Haar and discrete cosine transforms for 3D denoising of the formed sets of similar blocks.  
\begin{table}
\begin{tabular}{l c c c c c c c c c} \hline \hline
**Image** & \(\sigma\) & **QF** & **CR** & **Mean** & **Variance** & **Kurtosis** & **Skewness** & **Stat** & **P** & **Gaussian or not** \\ \hline Frisco & 5 & 34 & 31.58 & \(-\)0.007 & 11.08 & 3.36 & \(-\)0.0029 & 20524 & 0 & No \\ Frisco & 5 & 47 & 5.23 & 0.005 & 22.59 & 0.078 & 0.0028 & 62.7 & 2.38 \(\times\) 10\({}^{-14}\) & No \\ Frisco & 5 & 60 & 2.97 & \(-\)0.005 & 25.14 & 0.010 & 0.0046 & 1.994 & 0.3689 & Yes \\ Frisco & 10 & 24 & 61.14 & \(-\)0.070 & 24.58 & 4.95 & 0.0287 & 28482 & 0 & No \\ Frisco & 10 & 37 & 5.96 & \(-\)0.011 & 85.40 & 0.129 & 0.0025 & 161.7 & 7.61 \(\times\) 10\({}^{-36}\) & No \\ Frisco & 10 & 50 & 3.22 & \(-\)0.001 & 98.96 & 0.018 & 0.0045 & 4.54 & 0.1031 & Yes \\ Diego & 5 & 48 & 3.42 & \(-\)0.043 & 33.69 & 0.032 & 0.0024 & 11.03 & 0.0040 & No \\ Diego & 5 & 56 & 2.61 & \(-\)0.006 & 28.64 & 0.015 & 0.0063 & 4.07 & 0.1302 & Yes \\ Diego & 5 & 64 & 2.11 & \(-\)0.010 & 26.56 & 0.008 & 0.0065 & 2.56 & 0.2785 & Yes \\ [PERSON] & 10 & 40 & 4.20 & \(-\)0.041 & 107.83 & 0.069 & 0.0054 & 50.74 & 9.55 \(\times\) 10\({}^{-12}\) & No \\ [PERSON] & 10 & 58 & 2.25 & \(-\)0.007 & 101.91 & 0.010 & 0.0041 & 1.873 & 0.3919 & Yes \\ [PERSON] & 10 & 56 & 2.37 & \(-\)0.012 & 102.47 & 0.007 & 0.0026 & 0.779 & 0.6772 & Yes \\ \hline \hline \end{tabular}
\end{table}
Table 2: Statistics of the introduced distortions.  
Figure 3: Visualized \(\Delta_{ij}\), \(i=1,..,I_{lm}\), \(j=1,..,I_{lm}\) for the test images Frisco (**a**) and Diego (**b**) corrupted by AWGN with \(\sigma^{2}\) = 100 compressed by HEIF with QF = 50.  
This filter has the following positive features that are useful for the considered application. First, the BM3D filter has a preset parameter \(\beta\) used for the threshold T setting as \(\beta\sigma\). By changing \(\beta\), it is possible to slightly vary the filter properties--a larger \(\beta\) leads to better noise suppression at the expense of worse edge/detail/texture preservation. In our case, this property can be useful to understand how the optimal or quasi-optimal threshold can be adjusted depending on the original \(\sigma^{2}\), and possibly some other information on the image at hand to be compressed. Second, a filter is quite insensitive if the noise pdf is not exactly Gaussian or not exactly white. This property is also important for the considered application. Third, the BM3D filter is one of the best filters in the sense of AWGN suppression, and its performance is comparable to that of neural network-based filters [68] (neural network-based filters mostly demonstrate better performance for unrealistically large values of \(\sigma\)).