## 1 Introduction  
Noise is a major source of digital image contamination. Image noise may arise from image data quantization, transmission errors, electronic interference from the imaging hardware, as well as other sources [1]. The noise in digital images affects the real image signal, resulting in an image quality decline. This influence will affect the performance of digital image application areas such as segmentation, retrieval, edge extraction, etc. To obtain reliable results, efforts have been made in many applications to achieve effective noise suppression. Image de-noising, in the field of image processing, is a commonly used method to make the image visually prominent and obtain enough useful information [2, 3]. The major function of image de-noising is to recover the original image from a noisy measurement and retain the image structure information as much as possible [4]. Denoising is a well-known method used from time to time but produces an ill-posed problem in image processing. To date, many de-noising digital image schemes have been continuously developed, from pixel level filtering methods, such as Gaussian filtering, bilateral filtering, and total variation regularization, to patch level filtering methods, such as non-local means [3, 5, 6], block-matching 3D filtering (BM3D) [4], and low-rank regularization, etc. [3, 7]. [PERSON] et al. introduced bilateral filtering as a nonlinear filter that combines domain and range filtering [8]. This filter replaces each pixel with the weighted average of its neighbors. The weight assigned to each neighbor decreases with both the distance in the image plane and the distance on the intensity axis [6]. This filter produces a smooth image while preserving the edges. It demonstrated great effectiveness for a variety of computer vision and computer graphics problems [6]. The non-local means filtering proposed by [PERSON] et al. [4] is a spatial image de-noising algorithm based on bilateral filtering. This de-noising method replaces each pixel in the noisy image using the weighted pixel average with the related surrounding neighborhoods. The weighting function is determined by the similarity between neighborhoods [9]. It was applied to image de-noising in various specific fields because of its superior performance over other methods, such as the NLM filter and bilateral filtering [9; 10]. To estimate white Gaussian noise in images, a work surveyed six methods and found that the noise estimation using standard deviation measurement in residual images was most dependable [11]. The residual image was obtained by subtracting the original and a low-pass-filter-smoothed image. [PERSON] et al. measured the noise level directly from the residual image of a bit plane. They claimed that the noise level in the original image might be overestimated due to the nature of the quantization involved [12]. A residual image (RE) can be defined as \(\text{RE}=\text{OR}-\text{SM}\) using the subtraction value between an original (OR) image and its smoothed (SM) version. The subtraction can effectively remove the signal part and leave the noise with edge parts [12; 13]. Many RE-based state-of-the-art de-noise algorithms have been proposed in recent decades. [PERSON] et al. developed an RE correlation regularization de-noising scheme that minimizes the correlation between neighboring RE patches [14]. They claimed that correlation-based regularization can help to produce better results than K-SVD (a dictionary learning algorithm), both qualitatively and quantitatively. However, a computational burden drawback exists in the algorithm. [PERSON] et al. proposed a residual-based method that combined the bilateral filter and structure adaptive kernel filter for Gaussian noise de-noising [15]. The noise was suppressed efficiently and this combined method showed acceptable de-noising performance for heavy Gaussian noise. However, the iterations are a time-consuming task [1]. This method uses a novel criterion to determine wavelet coefficient thresholds by minimizing the computed RE image kurtosis.