# Post-Filtering of Noisy Images Compressed by HEIF
## 3 Distortion Statistical Analysis  
In lossy image compression based on orthogonal transforms, distortions are due to the operation of transform coefficient quantization. This holds for both wavelet- and DCT-based coders. It has been found by several researchers for DCT-based [59], BPG [60], and JPEG [61] coders that:  
1. The MSE of the introduced distortions increases (in most cases) monotonically if the PCC is changed to increase the CR [59; 60];
2. Introduced distortions are \"non-stationary\", in the sense that they are more intensive for the so-called active image areas (textures, neighborhoods of sharp edges, and high-contrast small-sized objects) [59];
3. Due to this, the integral MSE is usually larger for complex structure images with a larger percentage of pixels that belong to locally active areas [60; 61]; meanwhile, it is less likely to have a non-Gaussian (heavy-tailed) distribution of introduced errors for complex-structure images than for simple-structure ones;
4. There is a tendency to have heavier tails of introduced errors for a larger CR [59];
5. In the lossy compression of noisy images with possible post-filtering, it is worth considering not only the statistics of the introduced errors, but also the statistics of the residual noise and distortions [60];
6. From the post-filtering viewpoint, it is desirable to not only analyze the statistics of the residual noise and distortions, but also their spatial correlation, since the removal of spatially correlated noise is a more complex task than AWGN suppression, and the efficiency of spatially-correlated noise removal is usually less than AWGN suppression for the same noise variance [62]. The statistics and spatial correlation characteristics of distortions may also depend on the noise statistics and spatial correlation if one is dealing with spatially-correlated noise, but this question was out of the scope of our study. Furthermore, we did not consider the case of signal-dependent noise, assuming that either the noise was close to pure additive or that an image contaminated by signal-dependent noise had been converted before compression by a proper variance stabilizing transform to be contaminated by additive noise [41; 56]. Thus, our goal here was to study the statistical characteristics of residual noise/distortions for images corrupted by AWGN and compressed by HEIF. We considered the noise/distortion parameter arrays \(\Delta_{ij}=I_{ij}^{true}-I_{ij}^{c}\), \(i=1,..,I_{Im}\), \(j=1,..,J_{Im}\) for two test images for three values of QF and two values of the noise variance, namely, 25 and 100. QF values were chosen to (1) correspond to the OOP for the image Fricso, or to quality degradation for the image Diego (the smallest considered QF); (2) relate to the case when the MSE\({}_{\text{tc}}\) was still almost the same as \(\sigma^{2}\) (the largest considered QF); and (3) correspond to intermediate situations (intermediate values of QF). To study the statistics of \(\Delta_{ij}\), \(i=1,..,I_{Im}\), \(j=1,..,J_{Im}\) for each case, we calculated the mean, variance, skewness, and kurtosis (note that joint analysis of these parameters allows for the rough analysis of data Gaussianity). In addition, [PERSON] statistical K-squared test was applied [63], which aims to gauge the compatibility of given data with the null hypothesis that the data are a realization of independent, identically distributed Gaussian random variables. This test exploits two parameters. The parameter Stat is determined as \(\text{s}^{2}+\text{k}^{2}\), where s is the z-score returned by the skewtest (test whether the skew is different from the normal distribution), and k is the z-score returned by the kurtosisest (test whether a dataset has normal kurtosis). The parameter P is a 2-sided Chi-squared probability for the hypothesis test. According to the recommendations in [63], we assumed that alpha = 0.05 was a statistically significant level.