## 1 Introduction
[PERSON] et al. estimated noise in chest CT image data with varying image quality using RE [16]. The RE should possess the statistical properties of contaminating noise. However, it is very likely that the residual patch will contain remnants of the clean image patch [14]. As a result, at high noise levels, the RE usually contains structures from the clean image patch; thus, it does not contain contaminating noise. [PERSON] et al. applied a statistical test on the RE and found that the RE did not contain only pure noise, as there were structures present [13]. They de-noised the RE with an adaptive Wiener filter first and then parts of the cleaned RE, where one of the hypothesis tests was rejected, were added back into the de-noised image. The idea is to first smooth the image's flat regions, and then work on the details. Their iterative scheme produced gains in both PSNR and SSIM. Adaptive de-noising involves newly proposed algorithms that perform local image computation in contrast to global de-noising [17]. These schemes improve the image quality and edge conservation more effectively than the global methods [18; 19; 20; 21; 22; 23; 24]. The adaptive image de-noising filter has become a promising research area [20]. A nonlinear module, named the Moran statistic, showed a correlation with the noise levels of medical images [12]. The Moran's Z measurements presented high consistency with the variation in smoothing and sharpening in images [25]. This calculation proved that the spatial correlation corresponds well with the variation in image spatial properties [26; 27]. Both [PERSON] et al. and [PERSON] et al. recently proposed using the Moran statistic calculation as an index for image de-noising [25; 26; 27; 28]. In their work, the Moran-statistic-based adaptive image filter algorithms produced better image quality than global methods. Fuzzy C-means (FCM) clustering is an unsupervised scheme successfully applied to feature analysis, clustering, and classifier designs [29; 30]. The FCM algorithm classifies the image by grouping similar data points in the feature space into clusters if the images are presented in \"feature spaces\". The FCM algorithm assigns pixels to each category by using fuzzy memberships. Each point in the feature space partially belongs to the clusters using the fuzzy membership function [31]. FCM has been used to decrease contamination from a mixed combination of impulse and Gaussian noise on digital images. The results demonstrate that this noise reduction technique outperforms state-of-the-art filters with respect to the metric PSNR [32]. This work proposes estimating the smoothness and sharpness spatial information in RE images using the Moran statistic. The Moran's Z measurements are then used as feature data input into FCM. Three clusters are pre-assumed for the FCM in this work: they are heavy noise, medium noise, and less noisy areas (i.e., structure areas). The rates for each position partially belong to the clusters determined by an FCM membership function. The membership function uses weights to calculate the weighted sum of each position. Each pixel in a noisy image is a linear combination of the product of three SMs with membership functions in the same position. The average filters with different windows (3 \(\times\) 3, 5 \(\times\) 5) and a Gaussian filter are a priori applied to this noisy image to obtain three SMs. The results show that this scheme works better than the non-adaptive schemes. This scheme's performance is evaluated in terms of the peak signal-to-noise ratio (PSNR), structural similarity index (SSIM), and comparison to the bilateral filter and NLM.