Image impulse noise removal is a crucial preprocessing step for obtaining high-quality images in various digital image processing applications，including aerospace，biomedical engineering，industrial inspection，and robot vision. Impulse noise contamination can occur during acquisition，transmission，and recording due to different factors such as detector and circuit systems，channel transmission errors，non-ideal media between the scene and the imaging system，and memory location errors in the hardware components ^［1］. Impulse noise affects only some of the pixels of an image while leaving others unaffected. Two major impulse noise models，namely the Salt-and-Pepper Noise （SAPN） model，where the noise pixels assume only the highest and lowest values of the digital number （DN） in the dynamic range of the image，and the Random-Valued Impulse Noise （RVIN） model，where the noise pixels can assume value in the dynamic range of the image with equal probability，have been described in the literature ^［2］.

The median filter （MF） ^［3］ has gained wide application due to its simple structure，fast execution time，and ability to preserve edge information. However，a limitation of the MF is its poor noise removal performance since it replaces the DN value of each pixel with the median found within a specific window. Therefore，various variants of the MF have been developed，such as the weighted median filter （WMF） ^［4］，center-weighted median filter （CWMF） ^［5］，adaptive median filter （AMF） ^［6］，adaptive switching median filter （ASMF） ^［7-9］，progressive switching median filter （PSMF） ^［10］，adaptive dual-threshold median filter （ADTMF） ^［11］，directional difference-based switching median filter （DD-SMF） ^［12］，decision-based filters ^［13］，spatially varying median filter （SVMF） ^［14］，and three-state median filter （TSMF） ^［15］. These methods improve the MF's ability to remove high-density impulse noise by detecting noisy pixels within the window. Furthermore，some enhanced versions，such as the effectively improved boundary discriminative noise detection filter （IBDNDF） ^［16］，the switching adaptive median and fixed weighted mean filter ^［17］，the unsymmetric trimmed adaptive median filter （UTAMF） ^［18］，and the median-type noise detectors and the detail-preserving regularization method ^［19］，also exhibit better performance and can successfully recover meaningful image details even from images damaged by 90% SAPN.

Of course，researchers have also contributed excellent image-denoising strategies from other theoretical aspects. Chen^［20］ proposed a weighted coupled sparse representation model with classified regularization to remove impulse noise in images by dividing image pixels into clear，slightly damaged，and severely damaged pixels. The low-rank Hankel matrix based on the annihilating filter is a compelling image restoration method. Jin^［21］ extended this idea and proposed a sparse and low-rank decomposition of a Hankel-structured matrix for impulse noise removal. In Ref. ［22］，the authors presented an adaptive detail-preserving filter based on a cloud model that can remove up to 95% of SAPN while preserving good image details. Fuzzy theory is often used to solve uncertainty problems and has profound implications for image denoising. For example，the adaptive network-based fuzzy inference system filter and the regression-based neuro-fuzzy network trained by the ABC algorithm can eliminate impulse noise ^［23-24］. Besides，Ahmed^［1］ proposed an iterative adaptive fuzzy filter using alpha-trimmed mean that can retrieve meaningful details even at up to 97% SAPN. Singh^［25］ achieved excellent results on standard test images with 20% to 99% SAPN damage using their proposed adaptive type-2 fuzzy filter.

In recent years，deep learning has been widely applied to image denoising，particularly for Gaussian noise，Gaussian additive white noise，and Gaussian-impulse mixture noise ^［26-30］. However，only a few deep-learning models have been used for impulse noise removal. Notable examples include the denoising framework based on deep convolutional neural network （CNN） for suppressing impulse noise in color images ^［31］，the efficient CNN with particle swarm optimization （PSO） model for high-density impulse noise removal ^［32］，the CNN-based image restoration method for impulse noise removal ^［33］，and the switching filtering technique using deep learning for impulse noise removal ^［34］. They accomplish image denoising through two main processes： 1） utilizing neural networks to detect impulse noise in the image and 2） inputting the detection results into the denoising module to reconstruct the denoised image.

Although various variants of median filters，fuzzy filters，and image denoising and restoration models based on deep learning have been developed by previous researchers，only some have explored image noise removal strategies from the fundamental level of digital camera imaging principles. Digital cameras are complex products that integrate optics，mechanical structures，and electronics. During imaging，the image sensor （CCD or CMOS） converts light signals into discrete electronic signals，which are further converted into digital signals. The number of pixels on the image sensor is represented as a matrix of rows and columns，where the numerical signal in each cell corresponds to the DN value in the digital image. Due to design errors in optical devices，the resolution of the image sensor，the wave-particle duality of light，and the complexity of the observed area，neighboring pixels exhibit subtle differences，which can be understood as pixel gradients between adjacent pixels. Therefore，by exploring the camera imaging model，we propose an adaptive iterative median filter based on pixel gradients，summarized as follows.

1） The modulation transfer function （MTF），as a performance indicator of the camera，can calculate the maximum pixel gradient of the captured image and establish a pixel gradient set. Subsequently，the gradient weight root-mean-square error （GWRMSE） set of the original image and the corresponding pixel gradient filtered image is computed. The Gaussian distribution of the GWRMSE set is calculated using a Gaussian function. Finally，the pixel gradient corresponding to the Gaussian distribution's maximum value is considered the optimal pixel gradient for removing the impulse noise in the image.

2） Our proposed filter consists of two types of median filters. The first type includes two orthogonal judgment-based median filters with window sizes 3×3，combined with the optimal pixel gradient for real-time removal of the image's discrete，low-density RVIN and SAPN. The second type is an adaptive window size median filter combined with the optimal pixel gradient and SAPN attribute set for removing high-density SAPN.

3） The adaptive window size and number of iterations are determined based on the density and complexity of the impulse noise in the image，and the proposed filter is used to achieve noise removal.

Digital images are represented using a two-dimensional array，where the rows and columns determine the position of each pixel in the image. The DN value of a pixel represents the grayscale value with statistical spatial characteristics. During the imaging process，the camera discretizes the shape of objects into points，lines，and surfaces. Uniformly distributed objects in the scene should have the same DN value for all pixels in the image. However，due to limitations imposed by the camera's optical system and image sensor materials and design techniques，each pixel only represents the statistical properties of the corresponding region. In other words，the DN values of pixels representing the same object can vary，as shown in Fig. 1. Therefore，we propose using pixel gradients to describe the fluctuation range in non-uniform DN values.

The MTF^［35］ provides an intuitive description of the transfer process of light information through optical media and devices. It comprehensively and objectively characterizes the sharpness of images. The corresponding formula is as follows：

where I(ωx,ωy) and O(ωx,ωy) are the spatial spectrum of the image and object，respectively. H(ωx,ωy) is the Fourier transform of the point spread function，called the optical transfer function. ωx and ωy are 2D spatial frequencies expressed in terms of circular frequencies.

The light distribution of an optical image can be regarded as a combination of countless spatial frequency sinusoidal wave distributions. Specific frequency sinusoidal grating stripes define the modulation depth of the sinusoidal distribution，and testing requires projection brightness with a sinusoidal variation as the target. However，this poses a significant challenge for production. To facilitate laboratory MTF testing，a square wave target corresponding to the limiting spatial frequencies is commonly used^［36］，as shown in Fig. 2（a）.

When a mid-wave infrared imaging system captures an image of a square wave target，the system outputs the peak intensity Imax corresponding to the transparent slits and the valley intensity Imin corresponding to the target surface patterns，as shown in Fig. 2（b）. In this way，the Contrast Transfer Function （CTF） can be represented as the ratio between the amplitude of signal fluctuations and the average signal level，as follows：

Since the square wave signal can be expanded into a Fourier series of its fundamental frequency and its multiples of AC signals，the CTF can also be expressed as the sum of the MTF of the fundamental frequency and its multiples of the square wave signal as the following：

When the frequency of the square wave signal exceeds 1/3 the cut-off frequency，the higher order terms MTF can be ignored. CTF is redefined as：

Once the optical calibration of the imaging system is completed in the laboratory，the optimal MTF value of the camera is determined. This allows for the calculation of the maximum gradient gmax between adjacent pixels，as mentioned below：

In practical scenarios，due to losses during light propagation，the relative brightness values of the highlights in the image are lower than those of the scene. Conversely，due to the effects of stray light，scattering，and diffraction，the relative brightness values of the shadows in the image are higher than those of the scene. However，for an ideal optical system without aberrations，stray light，scattering，reflection，or absorption，the described situation would not occur. Nonetheless，they all satisfy the following equation：

where Imaxdr denotes the maximum value of the dynamic range of the image acquired by the camera in the corresponding spectral range. Similarly，Imindr represents the minimum value of the dynamic range of the image captured by the camera in the corresponding spectral range.

By utilizing the optimal MTF value of the camera，we can calculate the maximum gradient gmax between adjacent pixels. This allows for the representation of the pixel gradient set of the camera's captured image G，as mentioned below：

where gk is the k-th pixel gradient of the set G.

In addition to the factors influencing the brightness values of the image mentioned earlier，the optical system is also affected by the presence of objects and background in the imaging area. Each pixel captures the DN value corresponding to the imaged area. Specifically，when the pixel's imaged area corresponds to an object's surface，the DN value represents the light information from the object，and the optical system solely influences the gradient between adjacent pixels. However，when the pixel's imaged area falls on the edge of an object，the DN value represents both the light information from the object and the background. The optical system and the background brightness values influence the gradient between adjacent pixels. Therefore，it is necessary to determine an optimal pixel gradient for a specific optical imaging system.

Firstly，based on the proposed filter，we calculated the root-mean-square error rk between the pixel gradient gk corresponding to the filtered image Ik and the original image Io，as shown below：

where M×N is the image size.

To ensure a more precise assessment of the optimal pixel gradient based on the computed rk(Ik) for different pixel gradients gk，we have conducted further calculations of rk(Ik) within a unit pixel gradient，as follows：

where hk(rk,gk) denotes the GWRMSE of rk(Ik) with respect to the pixel gradient gk.

In this manner，we can represent the pixel gradients set G corresponding to the GWRMSE set H as follows：

Subsequently，we have employed a Gaussian function to establish a Gaussian distribution model A for the GWRMSE set H，which can be expressed as：

where fk is the Gaussian function corresponding to the GWRMSE hk. μ denotes the mean value of the set H. σ denotes the standard deviation of the set H. n denotes the full number of elements of the setH.

Finally，we designate the maximum element in set A to correspond to the hk element in set H，which in turn corresponds to the gk element in the set G，as the optimal pixel gradient gopt for image filtering，as follows：

Based on the calculated optimal pixel gradient，we can determine the optimal fluctuation range among neighboring pixels in the original image，including the surface and edge of the feature. In other words，when the pixel gradient between the central pixel and its neighboring pixels is less than the optimal pixel gradient，we consider the central pixel to be not impulse noise； otherwise，it is classified as impulse noise.

The proposed filter utilizes two types of MFs. The first type is primarily employed for real-time removal of dispersed，low-density RVIN and SAPN，employing two orthogonal-based MFs，F1 and F2，with window sizes of 3×3，as depicted in Fig. 3. F1 considers the pixels above，below，left，and right of the center pixel as neighbors to form the orthogonal setup. Similarly，F2uses the pixels in the top-left，bottom-left，top-right，and bottom-right positions as neighbors to establish the orthogonal configuration. By establishing these two orthogonal relationships，we can determine if the center pixel satisfies the range defined by the optimal pixel gradient，allowing for rapid identification of whether the center pixel corresponds to impulse noise.

The decision formula for the center pixel，neighboring pixels，and optimal pixel gradient associated with F1 is expressed as follows：

where Δg111，Δg121，Δg131，Δg141，Δg211，Δg221，Δg231，and Δg241 are the differences between the gradient of the central pixel pi,j and the adjacent pixels pi-1,j，pi+1,j，pi,j-1 or pi,j+1 minus the optimal pixel gradient gopt.

The decision formula for the center pixel，neighboring pixels，and optimal pixel gradient associated with F2 is expressed as follows：

where Δg112，Δg122，Δg132，Δg142，Δg212，Δg222，Δg232，and Δg242 are the differences between the gradient of the central pixel pi,j and the adjacent pixels pi-1,j-1，pi+1,j-1，pi+1,j-1 or pi+1,j+1 minus the optimal pixel gradient gopt.

When Δg111≥0，Δg121≥0，Δg131≥0，and Δg141≥0，or Δg211≥0，Δg221≥0，Δg231≥0，and Δg241≥0，or Δg112≥0，Δg122≥0，Δg132≥0，and Δg142≥0，or Δg212≥0，Δg222≥0，Δg232≥0，and Δg242≥0，the central pixelpi,j is classified as impulse noise. Subsequently，we need to establish a 3×3 window centered at pi,j，where all the pixels within the window are utilized to restore the central pixel pi,j quickly. The corresponding set of pixel intensities Wi,j，as follows：

Finally，we use the median value vi,j of the set Wi,j instead of the center noise pixel pi,j，as follows：

where median denotes the median model for computing the set Wi,j.

The second type of MF is primarily designed to remove high-density SAPN. It is an adaptive median filter F3，which employs a variable window size，as illustrated in Fig. 4. The adaptive window size of F3 is established around the central pixel pi,j as a (2T+1)×(2T+1) neighborhood，where the selection of the window size T depends on the SAPN density. The set of pixel intensities within the window is represented as Wi,jT，and it is expressed as follows：

Then，the median value vi,jT of Wi,jT can be computed，and the SAPN within Wi,jT can be effectively removed to form a new set WneT，as described in the following：

where S denotes the salt noise set S=Imaxdr,Imaxdr∈N. P denotes the pepper noise set P=Imindr,Imindr∈N.

Subsequently，we can establish a weight set C to determine whether the center pixel pi,j is a noisy pixel based on the median value vi,jT and the optimal pixel gradient gopt，as shown below：

where c denotes the element in the weight set C. When vi,jT-gopt<Imindr，vi,jT-gopt=Imindr. When vi,jT+gopt>Imaxdr，vi,jT+gopt=Imaxdr.

Finally，we decide whether the center pixel pi,j is a noisy pixel. If pi,j∈C or WneT=∅，then pi,j is considered non-noisy. Otherwise，pi,j is a noisy pixel. We utilize the median value vneT from the set WneT to restorepi,j，as illustrated below：

By proposing the pixel gradient-based adaptive median filter，we can effectively remove dispersed，low-density RVIN and SAPN in remote sensing images in real time and eliminate high-density SAPN in the image. The entire noise removal process is outlined in Algorithm 1. In general，for impulse noise images captured by the same camera，the pixel gradients for noise removal remain consistent due to the fixed MTF. When applying Algorithm 1 for the first time on an impulse noise image，the proposed filter does not damage the regions unaffected by impulse noise interference. Similarly，successfully removed areas of impulse noise are not compromised during the iterative filtering process until all noise within the image is eliminated. It is important to emphasize that the number of iterations （Iters） and the adaptive window size （T） of Algorithm 1 are determined by the SAPN density （D），as indicated in Table 1.

In this section，we use mid-wave infrared （MWIR） images，long-wave infrared （LWIR） images captured by real cameras，and open-access available datasets to validate the effectiveness of the proposed filters and compare them with state-of-the-art impulse noise filters. All experiments are conducted on an Intel Core i7-10700F 2.9 GHz processor with 16 GB RAM and NVIDIA GeForce GTX 1 660 Ti，running MATLAB 2019a.

To validate the optimal pixel gradient gopt computation model，we utilized 100 640×512，16-bit MWIR raw images and 100 320×256，16-bit LWIR raw images acquired by a thermal infrared array scanning camera，and the noise distribution models corresponding to the partial MWIR and LWIR raw images are shown in Fig. 5. The optical system of the thermal infrared array scanning camera is shown in Fig. 6，with corresponding parameters listed in Table 2. After laboratory optical calibration of the MWIR camera，the average MTF was measured to be 0.281 0 （the mean MTF value for the eight directions and centers of the image sensor）. We are considering the maximum dynamic range of the MWIR camera as Imaxdr=11000 and the minimum dynamic range as Imindr=8000； the maximum pixel gradient achievable by this camera is gmax=1073. Consequently，the pixel gradient set G=1≤gk≤1073is established. Subsequently，utilizing the proposed method，we compute the GWRMSE set H and the corresponding Gaussian distribution set A. By analyzing the distribution curves of the GWRMSE function hk(rk,gk) and the Gaussian function fk(hk) with respect to the pixel gradient gk，we determined that the optimal pixel gradient gopt for a single MWIR raw image is 142，as depicted in Fig. 7.

To validate the optimal pixel gradient gopt computation model，we utilized 100 640×512，16-bit MWIR raw images and 100 320×256，16-bit LWIR raw images acquired by a thermal infrared array scanning camera，as shown in Fig. 5. The optical system of the thermal infrared array scanning camera is shown in Fig. 6，with corresponding parameters listed in Table 2. After laboratory optical calibration of the MWIR camera，the average MTF was measured to be 0.2810 （the mean MTF value for the eight directions and centers of the image sensor）. We are considering the maximum dynamic range of the MWIR camera as Imaxdr=11000 and the minimum dynamic range as Imindr=8000； the maximum pixel gradient achievable by this camera is gmax=1073. Consequently，the pixel gradient set G=1≤gk≤1073 is established.

Subsequently，utilizing the proposed method，we compute the GWRMSE set H and the corresponding Gaussian distribution set A. By analyzing the distribution curves of the GWRMSE function hk(rk,gk) and the Gaussian function fk(hk) with respect to the pixel gradient gk，we determined that the optimal pixel gradient gopt for a single MWIR raw image is 142，as depicted in Fig. 7.

Since the MTF of an optical imaging system remains constant，we assert that the optimal pixel gradient for the MWIR camera is also fixed. We analyzed the optimal pixel gradients of 100 MWIR raw images to substantiate this claim. As depicted in Fig. 8（a），all tested images exhibited an optimal pixel gradient of 142. This further validates that our proposed method can consistently compute the optimal pixel gradient for impulse noise removal in the raw images based on the MTF calibrated of the optical imaging system.

Upon obtaining the optimal pixel gradient for impulse noise removal in the raw images，we applied our proposed filter to remove low-density impulse noise from 100 raw images. The impulse noise density D∈[0,1]，the adaptive window size T=0，and the partial filtering results are illustrated in Fig. 9. Through comparison，and it is evident that the proposed pixel gradient-based filter effectively eliminates impulse noise while preserving and restoring the fine details of the raw images. To provide a quantitative assessment，we conducted statistical analysis on the inconsistent pixels between the original and denoised images. For the 100 test images，the maximum proportion of inconsistent pixels was 0.396 7%，the minimum proportion was 0.221 1%，and the average proportion was 0.244 9%，as shown in Fig. 8（b）. This indicates that the proposed filter preserves over 99.5% of the pixels without compromising their integrity during impulse noise removal. The denoised image corresponding to the maximum proportion is depicted in Fig. 9（b），exhibiting complex high-intensity features such as buildings with significant variations among adjacent pixels. The denoised image corresponding to the minimum proportion is illustrated in Fig. 9（d），characterized by homogeneous low-intensity features like farmlands and mountains with minimal fluctuation between neighboring pixels. Throughout the entire noise removal process，the average computation time for all the images was 0.0080s，with a maximum of 0.0170s and a minimum of 0s，as displayed in Fig. 8（c）. Therefore，our proposed filter enables real-time impulse noise removal in MWIR images，yielding excellent denoising results.

Similarly，after laboratory optical calibration of the LWIR camera，the average MTF was measured to be 0.2704. Considering the maximum dynamic range of the LWIR camera as Imaxdr=12000 and the minimum dynamic range as Imindr=9000，the maximum pixel gradient achievable by this camera is gmax=1033. Consequently，the pixel gradient set G=1≤gk≤1033is established for the LWIR camera. Therefore，the optimal pixel gradient for the 100 LWIR raw images is 137，as illustrated in Fig. 10（a）.

The impulse noise density D∈[0,1] of the LWIR raw images，so the adaptive window size of the proposed filter to be T=0. Through quantitative analysis and visual inspection of the denoised images from the 100 LWIR raw images，the maximum proportion of inconsistent pixels was found to be 0.319 4%，the minimum proportion was 0.215 4%，and the average proportion was 0.228 4%，as shown in Fig. 10（b）. This indicates that over 99.5% of the LWIR image pixels remain intact，largely preserving the original pixel while effectively removing the impulse noise in the images，as illustrated in Fig. 11. Specifically，because there is a stripe noise ^［37］ in Fig. 11 （a），Fig. 11（b） represents the denoised image corresponding to the maximum proportion. At the same time，Fig. 11（d） depicts the denoised image corresponding to the minimum proportion. Throughout the entire noise removal process，the average computation time for all the images was 0.001 6 s，with a maximum of 0.016 0 s and a minimum of 0s，as displayed in Fig. 10（c）. Therefore，the proposed optimal pixel gradient model and filter demonstrate real-time impulse noise removal in LWIR camera-captured images.

In this section，we compare the proposed method with seven filtering methods，including Median Filter （MF） ^［3］，Blind Denoising Convolutional Neural Network （BdCNN） ^［38］，High-order Nonconvex Total Variation and Overlapping Group Sparsity （HONTV-OGS） fusion method ^［39］，𝓁0 Overlapping Group Sparse Total Variation （𝓁0-OGSTV） method ^［40］，Distribution Transformation Network （DTN） ^［41］，Deep Convolutional Neural Network （DCNN） ^［33］，and Rank-deficient Hankel Matrix-based Low-rank Approximation （ALOHA） method ^［21］，using the MWIR and LWIR raw images. The denoised results of the MWIR and LWIR raw images are presented in Fig. 12 and 13，respectively. It is evident that all seven advanced filtering methods and our proposed method effectively remove discrete and low-density RVIN from the images while restoring rich image details. Additionally，we computed the average proportion of inconsistent pixels before and after denoising for 100 MWIR and 100 LWIR original images，as shown in Table 3. It is clear that our proposed pixel gradient-based method successfully removes RVIN in the original images while preserving over 99.5% of the original pixels. In contrast，the other methods have caused varying degrees of damage to the original pixels in the images.

This section compares the proposed filter with seven advanced impulse noise filtering methods through simulation experiments to evaluate its denoising performance on high-density SAPN. The quality of the denoised images is quantitatively evaluated using the Peak Signal-to-Noise Ratio （PSNR） and Structural Similarity Index （SSIM）. PSNR measures the noise density relative to the original image ^［42］. SSIM，which approximates human visual perception，assesses the similarity between the processed and original images ^［42］. Generally，higher values of PSNR and SSIM indicate that the quality of the processed image approaches that of the original image. The definitions of PSNR and SSIM are as follows：

where MSE is the mean square error of the original image Io and the processed image Ip.

where μIo，μIp，σIo，σIp，and σIoIp are the local means，standard deviations，and cross-covariance for images Ioand Ip，respectively. C1and C2are two constants，C1=0.01(Imaxdr-Imindr)2，C2=0.03(Imaxdr-Imindr)2.

Firstly，we selected six different types of 512x512，8-bit publicly available grayscale test images for SAPN simulation，as shown in Fig. 14. Since the camera's MTF of the simulated images is unknown，use SPAN to destroy the standard test images. Then，according to the characteristics of SAPN，the optimal pixel gradient gopt=255 for all test images and the salt noise set S=255 and pepper noise set P=0. Finally，the proposed method and seven comparative denoising methods are applied to the standard test images for SAPN removal，and the quality of the resulting images is quantitatively evaluated using PSNR and SSIM. Meanwhile，the computational efficiency of all denoising methods is assessed based on their running time under the same conditions.

Figs. 15，16，and 17 present the restoration results of the test images after being contaminated with SAPN for both the comparative and proposed methods. For the 20% SAPN corrupted test image，all the methods can effectively remove the impulse noise from the image and maintain good image details，except for the MF method restored image，which suffers from severe smoothing. For the test images damaged by 50% SAPN，the MF and BdCNN methods recovery results suffer

from severe smoothing. For the test images damaged by 80% SAPN，the restoration results of the MF，BdCNN，and HONTV-OGS methods exhibit severe distortion. In contrast，the other methods produce excellent denoising results，maintaining the integrity of the image content，including our proposed method. However，our method exhibits poor recovery results compared to the other four methods. Although the object's structure in the image is intact，the gradient of the pixels at the object's edges falls off so much that the replacement pixels are easily affected by the neighboring pixels and do not reach the gray value of the original pixels. Moreover，our proposed method can effectively restore meaningful image details even for images severely damaged by extreme SAPN. As demonstrated in Fig. 18，although the image's texture suffers severe damage and smoothing，the edges of the objects in the image are well restored and preserved，and the visual analysis can still distinguish different features.

Table 4 presents the PSNR and SSIM of all test images after being corrupted by SAPN and restored using all filtering methods. Correspondingly，Table 5 provides the average running time of all filtering methods for denoising all test images. Through comprehensive analysis，our proposed method demonstrates excellent denoising results compared to the MF，BdCNN，and HONTV-OGS methods.

Compared to the 𝓁0-OGSTV，DTN，DCNN，and ALOHA methods，although our proposed method does not exhibit superior image restoration，it requires less denoising time. In summary，our proposed denoising method performs well，showing excellent PSNR and SSIM compared to some faster methods. In terms of methods with superior PSNR and SSIM performance，our proposed method demonstrates a speedier computation time. Besides，our proposed method can remove SAPN from 0% to 99%.

This paper proposes a pixel gradients-based adaptive iterative median filter to remove low-density impulse noise and 0% to 99% SAPN in single-channel images. Firstly，based on the imaging principle of optical cameras，we utilize the camera’s MTF to obtain the maximum pixel gradient of the original image and establish the pixel gradients set of that camera. Next，we calculate the GWRMSE between the original image and the corresponding pixel gradient filtered image. Finally，the optimal pixel gradient is determined as the one corresponding to the maximum value of the Gaussian distribution of the GWRMSE set. By considering the noise density and complexity of the impulse noise，we can determine the adaptive window size of the proposed filter and the required number of iterations to remove the impulse noise in the image. Extensive experimental results demonstrate that the developed filter exhibits robustness in removing 8-bit and 16-bit single-channel impulse noise images，effectively removing low-density RVIN from real thermal infrared camera-acquired images in real-time. Compared to other advanced methods，this approach achieves a preservation rate of 99.5% of the original pixels during denoising. Furthermore，for high-density SAPN in images，our proposed filter obtains competitive results compared to advanced filtering methods，showing better PSNR and SSIM compared to faster filtering methods and faster computation time compared to methods with superior PSNR and SSIM. Additionally，for images severely damaged by extreme SAPN，the proposed method can recover meaningful image details.