Impulse noise removal via local statistical analysis and vectorized matrix processing

1 Introduction

Impulse noise filtering has long been regarded as one of the most important tasks in the domain of image processing, since it represents the trade-off between good noise filtering capability and structural details preservation. Classifiers, such as the conventional Median Filter (MF) and its advanced versions, are fast and easy to calculate but fail to restore structural details at high noise density levels, while recent approaches based on deep neural networks and complex optimization exhibit high reconstruction performance but face high computational costs, making them unusable for real-time processing.

To solve this problem, we propose a transparent, non-black-box two-step impulse noise removal algorithm that uses the combination of localized statistical analysis and high-performance matrix vectorization (im2col). Our method does not employ one fixed detection rule as other works do; instead, it combines a three-path detection approach, which includes the Beta distribution fitting (Path A), higher-order statistics (Path B), and Median Absolute Deviation (Path C). Using column-wise matrix vectorization, our framework demonstrates unprecedented computation speed.

Classical image restoration techniques, including the basic median filtering algorithm and other variants of the technique, have been widely used due to their computational efficiency. The downside of the classical image restoration technique is that it applies identical operations on all the pixels regardless of the fact whether they are corrupted or not. It causes significant blurring, degradation of edge information, and high-frequency texture loss. To tackle the spatial discrimination problem, current state-of-the-art approaches adopt a completely different approach, which involves the use of a Two-Stage Filtering Framework (see Figure 1):

• Stage 1: Noise Detection – identifies the corrupted pixels to construct a binary noise mask of the image.
• Stage 2: Targeted Reconstruction – performs a targeted reconstruction by performing mathematical interpolation of the corrupted pixel values using uncorrupted structural anchors.

The key challenge in implementing a two-stage framework is in the robustness of its first stage, which performs the task of noise detection. Although deep learning models and CNNs have proven themselves excellent at feature abstraction, their black-box nature makes it mathematically impossible to understand their decision-making processes. Furthermore, their large number of parameters makes them computationally expensive and therefore unsuitable for deployment on real-time edge-computing hardware platforms. This trade-off has made scientists return to the basics and opt for Statistical Distribution Analysis instead. This is because the uncorrupted image patches have smooth intensity histograms and high spatial correlation; therefore, adding impulse noise distorts these statistical regularities. By treating sliding spatial neighborhoods as localized probability density functions, the outliers can be detected by identifying the statistical anomalies.

In order to effectively isolate anomalies in heavily contaminated neighborhoods, more complex probability models have been used, such as the flexible Beta Distribution, and HOS measures including skewness and kurtosis. Skewness indicates the direction of asymmetry caused by the impulse noise spike, whereas Kurtosis measures the tailedness of the distribution. If a target pixel lies in the tails of the distributions constructed with the use of these HOS parameters, then it is considered an outlier with a high statistical confidence level.

Moreover, for the purposes of guaranteeing stable baseline estimation of the surrounding samples in case they experience the severe influence of noise contamination, the use of classical robust estimators such as Median Absolute Deviation (MAD) and certain varieties of M-estimators has become widespread in lieu of averaging in this algorithm. As a result, an exceptionally robust filtering boundary, which ignores all the outlier spikes and guarantees a highly accurate detection boundary despite the extremely high noise densities.

Adopting this distribution-driven framework in the highly efficient matrix-vectorization framework of MATLAB ensures the perfect combination of clarity, efficiency, and accurate image restoration.

The main contributions of this research include the following items:

• Two-Stage Distribution-Driven Anomaly Mapping: The proposed framework generates a highly sophisticated two-stage anomaly detection and filtration framework, which separates the detection process from the filtration process in the context of using localized probability density functions for separation of noise while preserving the pure structural pixels.
• Higher Order Moments and Kurtosis-Based Adaptive Thresholding: By means of constantly evaluating Skewness and Kurtosis of localized distributions, the proposed framework is capable of constantly adapting its detection threshold according to the asymmetry and tailedness of the sliding window;
• Bounded Beta-Distribution Modeling of Localized Intensities and Outlier Detection in the Form of Tail Outliers: The proposed model makes use of a highly adaptable continuous bounded distribution fitting technique in modeling localized image intensities and detection of noise pixels as tail outliers;
• Robust MAD-based Baseline Estimator: Framework utilizes a highly robust MAD approach to ensure the mathematical breakdown point of the filter from false positive detections;
• Vectorized MATLAB Architecture for Edge Computing: The entire statistical detection and adaptive image restoration pipeline is implemented by means of highly optimized matrix operations and vectorized indexing in the MATLAB environment, ensuring the real-time speed necessary for edge computing by elimination of inefficient nested loops.

2 Literature review

Modern digital image processing, multi-spectral vision systems, and communication frameworks continuously face issues associated with the destruction of structural integrity due to environmental and transmission channel distortions. The identification and elimination of mathematical noise artifacts without affecting the integrity of native geometrical structures and fine textures remain one of the major fields of research. Dominated historically by classical heuristic smoothing algorithms, the field has evolved into a multi-disciplinary intersection of robust statistical distributions, non-linear digital filter architectures, and advanced feature estimators.

This literature review systematically reviews modern contributions to the field of research in accordance with the major dimensions that determine the scope of the current project: mathematical characterization of impulse noise artifacts, evolution of statistical distribution analytics (Beta-models and higher-order moments), and optimization of hardware and software digital filters, robust estimation, and multi-spectral image processing.

2.1 Characterization and impact of impulse noise artifacts

Impulse noise is characterized by sudden, sharp jumps and discontinuities in the signal and image matrices. As a result, they severely impair the reliability of further processing tasks. In their work, Ahmed and Bosnić (1) investigated the performance boundaries of supervised machine learning algorithms when subjected to impulsive α-stable noise, showing that heavy-tailed non-Gaussian noise distributions completely disrupt the work of standard linear classifiers and require resilient pre-processing frameworks. Moving beyond standard visual media, the same problem arises in acoustics; Abayomi-Alli et al. (2) created multi-class deep convolutional neural network (DCNN) classifiers to detect impulse sounds from non-impulse ones, highlighting the importance of noise classification automation.

In the case of high-frequency advanced imaging technologies, such as Terahertz (THz) spectrum capturing, Abramova et al. (3) addressed the issues of increasing the efficiency of THz image enhancement corrupted by compact impulse noise, providing specialized recommendations for hardware and software structural reconstruction. At the network and hardware interface, especially in the cases of internet-of-things (IoT) frameworks, the task of removing data corruption should be distributed; in their study, Roy et al. (4) introduced adaptive frameworks for impulse noise removal at the gateway level specifically aimed at multimedia-IoT systems, balancing computation limits against transmission integrity. Therefore, the evolution of the theory behind image denoising has experienced a significant paradigm shift, as has been comprehensively detailed by Vaidya et al. (5) from classic statistical neighbor studies to multi-layered artificial neural networks.

2.2 Statistical mathematics and descriptor analysis in vision systems

Detection of noise without affecting the structural elements of the image, such as edges, requires the analysis of the mathematical behavior of the descriptors. Saputra et al. (6) investigated this paradigm through the lens of the evolution of deep learning and its cyclic influence on statistical mathematics education, revealing the continuous dependence of black-box models on basic statistical distributions. Multimodal analytical systems, as summarized by Volochaev and Tsaturyan (7), show that characterization tasks related to transmission electron microscopy (TEM), diffraction, and fractal analysis rely implicitly on advanced image processing baselines to interpret material geometries.

In the domain of medical diagnostic engineering, Rajarajeswari and Akila (8) used localized statistical analysis of lung histopathology images to create robust baselines to support the work of deep-learning-based cancer detection models. Similarly, in the area of spatial tracking and object validation, Yudina (9) created a stochastic image-matching model based solely on statistical analysis of feature descriptors, revealing that statistical analysis of mathematical descriptors provides robust matching metrics in the case of non-ideal image acquisition. Moreover, these statistical regularities are actively used in security vectors; Kumar et al. (10) introduced an enhanced image steganography framework that uses secure random pixel distribution maps to conceal payload data, showing that manipulation of statistical spatial distributions may be effective in protecting data pathways from adversarial detection.

2.3 Bounded modeling via Beta-distribution framework

One of the most important sub-fields of statistical image analytics lies in fitting the bounded image intensities to the continuous Beta distribution, since the latter is strictly bounded between [0,1], giving an extremely flexible framework for modeling non-normally distributed image parameters.

This flexible mathematical framework is actively used in the field of automated medical diagnostics. Multiauthor clinical studies led by Ramírez-Bautista et al. (11) showed that it was possible to create an interpretative framework for quantification of hepatic fat in prepubertal male subjects by fitting ultrasound histograms into a continuous Beta distribution in combination with probabilistic neural network (PNN) architecture. By transforming raw reflectivity data into mathematical shape parameters (α, β) of the Beta distribution, this model isolates the subtle tissue variations from artifact noise. In the area of macroscopic risk engineering, Elallaf et al. (12) took advantage of this powerful tool, introducing Beta distribution learning as the basis of a reliable predictive safety framework in artificial intelligence – roadway crash risk assessment models.

2.4 Higher-order statistics: skewness and kurtosis adaptations

As the impulse noise hits the image patch, the symmetrical character of the histogram is destroyed by the creation of heavy tails or asymmetry. Classical metrics such as mean or variance fail to detect these abnormalities, making the use of higher-order statistics, namely Skewness (third statistical moment responsible for asymmetry) and Kurtosis (fourth statistical moment responsible for tailedness/peakedness), inevitable.

In clinical neuroimaging, Vijithananda et al. (13) showed that evaluating skewness and kurtosis of apparent diffusion coefficient (ADC) maps in human brain lesions via MRI provides clear statistical parameters to distinguish benign from malignant tumors. In other disciplines, Sinha et al. (14) created a deep learning estimation framework based on the skewness and kurtosis of wind wave distributions to mitigate cyclone hazard and oceanographic tracking.

To ensure the validity of the higher-order statistics across different data sizes, Sankar Ganesh et al. (15) optimized the sample size for normality tests in geotechnical data using power analysis curves. In modern algorithmic design, Taddese et al. (16) introduced the Dynamic Fuzzy-Gaussian Modeling (DynFGM) framework – an unsupervised kurtosis-adaptive architecture which automatically adjusts its threshold in accordance with local kurtosis changes to perform highly precise adipose tissue segmentation in abdominal MRIs without human intervention.

2.5 Architectures and evolution of digital filters

Transformation of noise maps into restored images is associated with digital filter optimization for the purpose of real-time processing and maintaining the structure. According to the research by Saranya et al. (17), it was established that the limitation of FPGA-based digital filters can be explained by the necessity of hardware-level pipelining in order to achieve the required speed of real-time processing in an edge-deployed image processing unit.

At the same time, software digital filter development is still taking place and providing advancements in medical imaging. Thus, Almutiry et al. (18) developed a set of digital filters that can improve visibility of the dentoalveolar microstructures on CBCT (Cone-Beam Computed Tomography) images, decreasing scattering noise but not blurring the image. Historical landmarks and fundamentals of the clinical and commercial use of the technology have been described by Shinde (19) in detail.

2.6 Non-Gaussian signal processing and order statistics

Beyond localized operators, processing of non-Gaussian signals requires advanced mathematical transformations. Chua et al. (20) provided a systematic review of the application of higher-order statistics and higher-order spectra (HOS/HOSpectra) in biomedical signals, proving that the effects of non-linear phase coupling and non-Gaussian noise can be revealed only through moving beyond the variance metrics.

In the context of signal stabilization, Palahin et al. (21) created advanced processing algorithms to deal with correlated non-Gaussian processes with excess kurtosis, thus providing a mathematical solution to signal matrices severely corrupted by impulsive spikes. Advanced HOS operators have become essential tools for the modern network security paradigms; Kulhandjian and Kulhandjian (22) presented a comprehensive survey of jammer detection and localization methods in 5G telecom networks, revealing how unsupervised learning models based on higher-order statistics help to isolate adversarial jamming frequencies from standard multi-user communication noise.

2.7 Robust estimators and PDF analysis

One of the major phases in a two-stage impulse noise filter is ensuring the independence of the estimation of the true pixel value from neighboring noise values. Traditional averages are not robust enough for this purpose, making researchers move towards robust estimators like median, Median Absolute Deviation (MAD), and M-estimation.

Chakraborty et al. (23) enhanced the foundations of this field through the research of robust estimation of multivariate location and scale parameters, providing mathematical boundaries for outlier rejection. To solve problems related to contaminated data pools, Chaudhary et al. (24) created highly efficient robust estimators of population mean resistant to severe data contamination and non-normal skewness. Also, Tang et al. (23, 25) optimized the minimum distance estimation techniques to directly compute true mean and standard deviation matrices from quantiles.

In the case of non-linear distributions, Riyadh Thanoon (26) created a comparative framework which proved that utilization of M-estimation to compute the reliability function of the Maxwell-Boltzmann distribution greatly outperforms classical maximum likelihood methods in the case of outliers. Also, Hasawy et al. (27) proposed a robust rank regression method for Weibull parameter estimation by combining M-estimation with the Fast Minimum Covariance Determinant (Fast MCD) algorithm to provide tracking stability in case of heavy contamination.

For optimization of threshold boundaries, Pinsky and Wen (28) described the parameter estimation through the mean absolute deviations of the truncated distributions using quantile functions. In parallel, Arachchige and Prendergast (29) created strict mathematical confidence intervals for median absolute deviations (MAD) to ensure the statistical validity of the outlier detection limits. These techniques correspond to the principles of robust regression introduced by Brown (30) and complex survey minimum Hellinger distance estimators mapped by Kepplinger and Vidyashankar (31). All these approaches highlight the importance of the use of robust statistical anchors in cases when linear assumptions are violated by noise.

2.8 Multi-spectral processing and geometrical noise prediction models

Finally, the last execution domain of modern image restoration techniques relies on multi-spectral and geometrical spatial modeling. Zhu et al. (32) have broken traditional hardware limitations by creating a smartphone-based multi-spectral imaging framework, bringing portability of spectral analysis outside laboratory environments. In the area of agricultural remote sensing, Jin et al. (33) used UAV-based multi-spectral and thermal infrared imagery to accurately estimate the chlorophyll content in potato crop leaves, showing that spectral isolation eliminates atmospheric illumination noise.

In order to protect physical assets, Zhao and Zhou (34) created a multi-spectral image analysis algorithm optimized for intellectual property protection and anti-counterfeiting verification of products. In the domain of electromagnetic sensing, Guo et al. (35) created real-time super-resolution microwave imaging using adaptive Gamma transformation to sharpen dielectric boundaries. Additional spatial tracking optimizations are also observed in polymer material science, where Liu et al. (36) mapped the precise spatial distribution of poly aggregates within adhesive films, while Stansch et al. (37) trained deep neural networks over B-scan ultrasonography data to automatically predict fibrosis risks in patients with metabolic dysfunction-associated steatosis liver disease (MASLD).

Closer to clinical image diagnostics, (38, 39) showed the validity of the usage of F-Test and One-Way ANOVA distributions to categorize medical image features during automated diagnostic procedures. Most importantly, in order to fight the problems of mathematical image degradation, (39) introduced the Rayleigh-Monte Carlo Method – a framework based on stochastic simulations for automated image noise detection and pixel elimination. It is complemented by the highly robust Topological Filter Structure (40), which performs localized geometrical noise predictions in order to isolate corrupted coordinates from the native image distribution.

2.9 Comparative literature analysis matrix

The following table shows the structural synthesis of research streams represented in the selected literature (Table 1).

Table 1. State of the art.

2.10 Conclusion and strategic directions for future research

Thorough literature analysis shows that efficient optimization of modern multimedia systems necessitates moving beyond classical grayscale smoothing techniques. The optimal direction of further study should focus on the development of Two-Stage Hybrid Frameworks.

Using the geometrical, rotation-invariant power of hand-crafted feature extraction tools such as Zernike Moments (ZMs) that are performed exclusively using decoupled multi-spectral channels (Y or Red spectral band), it is possible to extract structural elements with perfect transparency. In addition to that, when structural clarity is used in combination with Stage 1 noise detection based on statistical anomaly detection using Higher-Order Statistics (Kurtosis adaptation) and limited by Beta Distribution outlier thresholds, such a system reaches the state-of-the-art accuracy of deep neural models while retaining ultra-low computational overhead and mathematical interpretability necessary for the framework of edge computing.

3 Multi-path detection framework

3.1 Multi-path noise detection

In order to discriminate the natural structural variability (edges, texture patterns) of an image from its noise, the framework considers the task of noise detection as a task of statistical anomaly detection. Natural uncorrupted patches of images usually demonstrate high spatial symmetry and smooth histograms of intensity levels, whereas the presence of impulse noise disrupts local statistical regularities.

It is possible to consider corrupted pixels as statistical anomalies or heavy-tailed outliers within localized probability density functions that correspond to spatial patches of certain size (3×3 or 5×5). Intensity values of the pixels within a local sliding window around the target central pixel Y(i, j):

Y ( i , j ) = { N ( i , j ) with probabilty p X ( i , j ) with probability 1 − p (1)

Where X(i, j) is the uncorrupted original pixel and N(i, j) represents the noise value. In Salt-and-Pepper Noise (SPN), N(i, j) takes only extreme maximum or minimum values, while in Random-Valued Impulse Noise (RVIN), it can be any arbitrary random integer uniformly distributed across the intensity spectrum, making it significantly more difficult to detect via standard thresholding.

To successfully detect the corrupted pixels under different noise levels and different local image statistics, the presented methodology uses a triad of methods for checking the pixel in the local window, as depicted in Figure 2:

Path A (Fitting with Beta Distribution): Modeling the Local Neighborhood both Spatially and by Intensity as a Constrained Continuous Beta Distribution.

This fits the situation very well because the Beta distribution is strictly constrained to the [0, 1] range and offers an extremely flexible mathematical framework.

[ 1 , 0 ], are modeled using Beta distribution shape parameters α and β estimated via the Method of Moments ( α = x ¯ ( x ¯ ( 1 − x ¯ ) S 2 ) , = ( 1 − x ¯ ) ( x ¯ ( 1 − x ¯ ) S 2 ) ). Cumulative distribution function (CDF) thresholds flag pixels falling outside expected probability tails as noise.

Path B: Skewness and Kurtosis (Higher-Order Statistics).

Traditional metrics of the average and variance cannot model non-Gaussian tails, meaning that the only way to deal with tail outliers is to use Higher Order Statistics (HOS). In particular, this includes Skewness, which is the third statistical moment, and Kurtosis, which is the fourth one.

The idea is that the absence of impulses means that a uniform local window will have a low value of the skewness with a fixed kurtosis. However, an impulse outlier will break the balance, making the skewness highly pronounced while kurtosis will have a sharp spike.

In order to deal with asymmetric intensity distributions and avoid false alarms at edges, local skewness S and kurtosis K are computed. Pixels located within windows such that either |S| > T_s or K > T_k (thresholds obtained empirically, T_s, T_k) are defined as outliers.

Path C: Robust Outlier Estimator via Median Absolute Deviation (MAD).

For high-density salt and pepper noise (>50% corruption), Path C makes use of MAD in place of standard deviation since MAD offers a robust 50% breakdown point; therefore, its scale estimator is totally bounded. As such, even if half the size of the neighborhood is severely corrupted, the scale estimator stays very robust against any amount of contamination.

In order to achieve the robustness objective of the framework, it becomes necessary to get rid of linear average estimators and adopt Robust Estimators. It is possible to use the MAD that creates

MAD = median ( ∣ Y patch − median Y patch ∣ ) (2)

The MAD produces a very robust statistical anchor that does not care about outlier spikes at all, hence offering an extremely robust threshold boundary even up to the highest amount of noise contamination. The pixels having absolute deviations beyond a certain scaled factor of the MAD are regarded as noise:

Nois e Matrix = ∣ Y ( i , j ) − Local Median ∣ > ( threshold × local _ mad ) (3)

3.2 Vectorized matrix processing architecture

The conventional approach to dealing with spatial impulse noise requires nested loops over the H x W image matrix, which is computationally expensive. The suggested approach, however, transforms the process of extracting a spatial window into a single matrix operation using im2col.

Given an input image I ∈ R H × W and a local sliding window of size k × k, the im2col transformation rearranges each local neighborhood into a distinct column of a stacked feature matrix ∈ R ( k 2 ) × P, where P = H × W represents the total number of pixels.

By converting spatial neighborhoods into a 2D matrix structure:

• Local statistics—including mean x ¯, variance s², skewness S, kurtosis K, and Calculation of MAD—takes place simultaneously through the row vector through linear algebra kernels.
• Concurrent evaluation of conditional noise flagging for all three detection paths (Path A, Path B, and Path C) takes place concurrently on SIMD instruction sets of GPU/CPU.

Table 2 is a comparison of how they perform across different operational conditions.

Table 2. Comparative analysis of detection paths.

3.3 System architecture

The architecture of the system uses the Two-Stage Filter Architecture, which effectively separates the process of detecting from the process of filtering. The architecture of the system is illustrated in Figure 3 below.

Stage 1: Spatial/Spectral Decomposition.

If the input data is multi-spectral or colored images, the system will not compute unnecessary information right away from the raw grid of RGB pixels because usually, impulse noise affects all channels simultaneously. In the first place, the picture should be transformed into a color space, decoupling brightness from colors, for instance YIQ (rgb2ntsc) or HSV (rgb2hsv).

The distribution statistics analysis takes place only over the channel containing luminance (Y) or value (V).

Stage 2: Statistical Analysis-based Impulse Noise Detection (the Essence of the Algorithm).

Vectorized sliding window computation is done on the decomposed luminance matrix, and the exact binary map of Noise M is obtained for the given picture size.

M ( i , j ) = { 1 if the pixel is flaged as corrupted 0 if the pixel is classified as clean (4)

Clean coordinates (M = 0) do not require any process of reconstruction at all. This kind of hard logical matrix masking helps to overcome the main disadvantage of the old median filter, which is the corruption of the structures and edges.

Stage 3: Targeted Reconstruction Workflow.

Following multi-path noise detection, a binary noise mask M{0, 1}^{H × W} is produced, where M(i, j) = 1 denotes a corrupted pixel, and M(i, j) = 0 denotes an uncorrupted structural pixel. Image restoration is performed strictly on flagged pixels to preserve the original integrity of uncorrupted regions:Where Ω i , j = { ( m , n ) ∈ W i , j ∣ M ( m , n ) = 0 } represents the set of verified uncorrupted pixels within the local search window W_i,j centered at (i, j). If Ω_i,j is empty due to high noise density, the window size k is dynamically expanded (k ← k + 2) until Ω_i,j ≥ 1.

I ̂ = { I ( i , j ) if M ( i , j ) = 0 med ( { I ( m , n ) ∣ ( m , n ) ∈ Ω i , j } ) ifM ( i , j ) = 1 (5)

Reconstruction applies only on those coordinates that are flagged to be corrupt (M = 1). The process of reconstruction automatically utilizes the nearby healthy structural anchors to perform the targeted mathematical interpolation by employing the most suitable non-linear mathematical operators such as directional interpolation, median filter, or Rayleigh Monte Carlo stochastic anchors for noise prediction.

4 Experimental setup and benchmarking metrics

For the purposes of examining the mathematical soundness, precision in detecting anomalies, and accuracy of the restoration process in the framework, this section provides a detailed description of an experimental setup for testing purposes.

4.1 Simulation testbed and data synthesis

The evaluation environment will be configured to push the limits of the framework under different levels of structural complexity and extremes of noise conditions. The simulation settings are specified below:

• Benchmark Reference Image Set: Simulations will be run with reference images – in this case, Lena (with smooth textures and structural gradient), Peppers (color edge detection with different lighting levels), and Barbara (complicated image with high spatial frequencies).
• Noise Simulation Environment: * Salt-and-Pepper Noise (SPN): Added in the form of discrete impulses exactly on the absolute level borders of the quantization range (e.g., 0 or 255 when dealing with an 8-bit unsigned integer grid).
• Random-Valued Impulse Noise (RVIN): Added in the form of uniformly distributed but non-boundary random impulses over the entire spectrum of intensity values [0,255]. It is the basis for the main test of the framework’s robustness and higher-order statistical threshold of M-estimator.
• Contamination Densities: Noise will be injected into the simulations at wide operational ranges, from the lowest density 10% up to an extremely saturated level 90% in the steps of 10%. In this way, it will be possible to map the exact breakdown point of the framework for all three paths of statistical distributions.

4.2 Quantitative detection metrics (stage 1 anomaly isolation)

In order to estimate the mathematical precision of the localized distribution paths, Stage 1 (Generation of Noise Map) needs to be isolated from Stage 2 (Reconstruction of Filter). Two key performance indicators are used for validating the binary classification of anomalies. To evaluate the accuracy of the noise detection mask M(i, j) against the ground-truth noise map, standard confusion matrix formulations are adopted:

• False Acceptance Rate (FAR)/Fall-out: Measures the proportion of uncorrupted pixels incorrectly flagged as noise:

FAR = FP FP + TN × 100 % (6)

FAR estimates the mathematical selectivity of the framework by estimating the percentage of clean pixels classified as noise anomalies. High FAR signifies structural blurring, as clean pixels are being sent to the reconstruction engine.

Where:

FP (False Positives) represents the number of clean pixels wrongly flagged as noise.

TN (True Negatives) represents the number of clean pixels accurately kept

• False Rejection Rate (FRR)/Miss Rate: Measures the proportion of actual noise pixels that escaped detection.
• The false rejection rate is an assessment of the mathematical sensitivity of the system measured by the percentage of noise impulse signals that have escaped any detection process. High FRR leads to uncontrolled noise spikes in the output image:

FRR = FN FN + TP × 100 % (7)

Where:

FN (False Negatives) is the count of noise pixels completely missed by the statistical thresholds.

TP (True Positives) is the count of noise pixels successfully mapped.

4.3 Quantitative restoration metrics (stage 2 end-to-end fidelity)

Two internationally accepted objective methods have been used in the framework for assessing the quality of structural reconstruction and restoration at a pixel level with the application of non-linear filters:

• Peak Signal-to-Noise Ratio (PSNR).

PSNR determines the logarithmic value of the ratio of the maximum possible power of the signal to the corrupting noise affecting the fidelity of the signal. It can be determined through Mean Squared Error (MSE):

MSS = 1 M × N ∑ i = 1 M ∑ j = 1 N [ X ( i , j ) − X ′ ( i , j ) ] 2 (8)

PSNR = 10 log 10 MA X I 2 MSE (9)

Where X refers to the original, uncorrupted baseline image, X’ is the resulting output after reconstruction, M × N denotes matrix size, and MAXI is the highest possible pixel value (such as 255).

• Structural similarity index (SSIM).

Since linear measures such as MSE and PSNR determine only absolute errors between pixels and not how much structure has been preserved, the model includes SSIM because the latter is more aligned with the human eye by measuring three different degradation factors, including l (luminance), c (contrast), and s (structure).

SSIM = [ l ( x , y ) ] α . [ c ( x , y ) ] β . [ s ( x , y ) ] γ (10)

SSIM = ( 2 μ x μ y + C 1 ) ( 2 σ xy + C 2 ) ( μ x 2 + μ y 2 + C 1 ) ( σ x 2 + σ y 2 + C 2 ) (11)

Where μ_x, μ_y represent local intensity means, σ x 2 , σ y 2 indicate variances, σ xy represents cross-covariance, and C₁, C₂ are stabilizing constants preventing denominator zero-faults. An SSIM value approaching 1.0 indicates near-perfect preservation of original edge structures and spatial topologies.

4.4 MATLAB vectorization and algorithm implementation

In order to ensure efficient implementation of the proposed two-stage denoising pipeline, the framework has to avoid high latency associated with the use of nested spatial loops (for loops) in interpreted languages. This section covers the mathematical transformation of sliding spatial neighborhoods into independent parallel columns through the technique of matrix vectorization, as well as the actual algorithmic realization in MATLAB.

4.4.1 Matrix reshape and vectorization pipeline

The traditional approach to spatial filtering involves sliding a local window around the image grid, computing a certain statistical characteristic at every stage. However, such an approach does not scale well. To allow for fast processing, the framework utilizes the following optimized structural conversion of sliding windows into independent columns through the engine of matrix reshape:

For the input luminance or value channel Y of size M × N, a window of size k × k (k = 2w + 1) slides over the image. The process of matrix vectorization is carried out as follows:

• 1 Boundary Padding: To ensure that no image data is lost due to boundary truncation of the window in peripheral coordinates, the image matrix is padded by w pixels on all sides using the mirror-reflection operator:

Y _ padded = padarray ( Y , , [ w , w ] , , ‘ symmetric ’ ) ; / / Matlab

• 2 Column Construction: The sliding windows are deconstructed and converted into a temporary matrix C:

C = im 2 col ( Y _ padded , , [ k , k ] , , ‘ sliding ’ ) ; / / Matlab

If the original image matrix consisted of P = M × N pixels, then the resulting matrix C will precisely reshape to the size of k2 × P. Every column of matrix C will contain all localized neighborhood intensities for a particular coordinate in the image.

• 3 Parallel Vector Processing: Statistical functions (for instance, median, mad, skewness, or kurtosis) are performed in parallel on the rows of the matrix C. As a result, the framework allows analyzing every single neighborhood at once, without using any loops.

4.4.2 Algorithmic implementation logic

The concrete algorithmic realization of Path C: Robust MAD anomaly detection along with the subsequent adaptive interpolation filter is given in the following MATLAB code. The given code template could be used in the case of Path A and Path B by changing the MAD anomaly detection line with the corresponding distribution or moment computation equation.

5 Comparative analysis and discussion

5.1 Quantitative performance matrices

5.1.1 Detailed behavioral and framework discussion

The numerical metrics help uncover the unique behavioral characteristics of the three mathematical methods on the benchmark data (Lena, Peppers, Barbara). It helps identify the most appropriate path based on the structure complexity and type of noise (Tables 3, 4).

Table 3. Comparative restoration fidelity (PSNR in dB/SSIM) under impulse noise contamination.

Table 4. Stage 1 anomaly detection performance (FAR %/FRR %) at 40% noise density.

A. Addressing Random-Valued Impulse Noise (RVIN).

Path A (Bounded Beta Distribution) proves to be superior to other methods in dealing with Random-Valued Impulse Noise (RVIN). Since the value range of the noise is [0,255], the spike often corresponds to the local median or mean intensity of a nearby patch, fooling linear and regular median filters.

To accurately estimate the “skewness” of a local area with the custom shape parameters of α and β by normalizing the range to continuous [0,1], Path A manages to detect in-range outliers, which do not look like spikes, with significantly lower False Rejection Rate (FRR = 2.45%) on Lena than FRR = 3.12% of Path C.

B. Computation and Performance Trade-off.

Path B (Higher-Order Statistics) provides a reasonable balance but is highly affected by the sample size of the localized window. Although the process of calculating the Skewness and Kurtosis effectively detects non-Gaussian tail anomaly without modeling a complicated distribution, a slightly larger spatial window (5 × 5 minimum) is needed to maintain stability. On the 3 × 3 windows, the sample set is highly restricted, causing higher false positives (FAR = 3.88% on Peppers).

C. Structure Awareness vs. Blurring Edges.

Path C (Robust MAD Anchor) shows excellent performance for Salt-and-Pepper Noise (SPN). Since SPN forces pixel intensity to either 0 or 255, the localized MAD gives a strict threshold. In the case of the image with flat areas or a smooth gradient like Lena, Path C reaches almost zero False Acceptance Rate (FAR = 0.08%).

However, when applied to high-frequency textures like the fine stripes in Barbara, Path C struggles to differentiate high-contrast physical edges from true noise spikes. This causes the FAR to increase to 0.65%, as valid geometric textures are inadvertently flagged as anomalies.

5.1.2 Methodological synthesis matrix

To provide a definitive guide for deployment in real-time multimedia transmission and digital forensic validation pipelines, the operational characteristics of each architectural path are synthesized below (Table 5).

Table 5. The operational characteristics of each architectural path.

5.1.3 State-of-the-art (SOTA) comparative analysis

To confirm the academic and engineering superiority of the proposed vectorized distribution-driven framework, its peak performance was rigorously benchmarked against recent state-of-the-art architectures found in the literature. Specifically, the framework is compared with the Rayleigh-Monte Carlo stochastic simulation method (40), the Geometrical Noise Prediction Topological Filter (39), and the Gateway-level Multimedia-IoT Adaptive Filter (4).

Testing was performed using the standard reference dataset (Lena, Peppers, Barbara) across progressive impulse noise densities (20, 40, and 60%) for both Salt-and-Pepper Noise (SPN) and Random-Valued Impulse Noise (RVIN).

5.1.4 In-depth comparative analysis

The numerical measures in Table 6, 7 bring out major behavioral differences exhibited by the algorithms:

• Effectiveness against the Rayleigh-Monte Carlo Method: While the Rayleigh-Monte Carlo technique can be used as an effective stochastic approach to noise detection, it presents a relatively high False Rejection Rate (FRR = 5.12%) when tested against uniform Random-Valued Impulse Noise (RVIN). This is due to in-range noise spikes being integrated into the Monte Carlo random sampling pools. In contrast, our proposed Path A (Bounded Beta Modeling) mitigates this challenge through probability modeling at the pixel boundaries and thus reduces the FRR to 2.45% on Lena.
• Effectiveness against the Topological Filter: The Topological Filter is known to exploit geometrical noise prediction capabilities to protect structural edges. This is exemplified in the case of the Barbara image that shows good structural similarity preservation (SSIM = 0.922) against SPN contamination. This is attributed to the fact that the topological framework uses very stiff pixel geometry, which imposes high computational cost and lacks flexibility in cases of contamination deeper than 50%. Our proposed Path C (Robust MAD Anchor) takes care of this limitation using a logical indexing mask together with vectorized matrix columns using the im2col method. It protects the topographical edges while delivering superior processing speeds suitable for an edge-computing environment.

Table 6. Quantitative state-of-the-art restoration comparison (PSNR in dB/SSIM).

Table 7. Anomaly detection stage evaluation (FAR %/FRR %) against SOTA models.

5.1.5 Computational complexity and execution time analysis

To rigorously assess the suitability of the proposed pipeline for edge deployment, we analyze both operational time complexity and memory overhead (Table 8):

• Time Complexity: The transformation of an image of size H × W using a kernel k × k via im2col incurs a time complexity of O (k². P), where P = H × W. Vectorized row-wise statistical computations require O (k². P) operations. Consequently, the total time complexity of the framework is O (k². P), eliminating the O (k². H. W) loop overhead inherent to non-vectorized filtering.
• Memory Overhead: Matrix vectorization requires allocating an intermediate matrix = ∈ R ( k 2 ) × P. For a standard 512 × 512 image and a 3 × 3 window (k = 3), X contains 9 × 262,144 elements (≈ 18.8 MB in single-precision float), well within the memory budgets of modern embedded edge processors.

Table 8. Execution time comparison (milliseconds) across image resolutions.

Bold values highlight the maximum achieved speedup factors across test resolutions.

5.2 Qualitative visual assessment and texture preservation

Although the numerical measures in Table 5, 6 establish the accuracy of the proposed framework from a mathematical point of view, visual quality assessment should be carried out in order to test edge preservation and structural boundaries. Figure 4 shows the visual results of denoising on the “Lena” and “Barbara” benchmarks against 40% Random-Valued Impulse Noise (RVIN) and Salt-and-Pepper Noise (SPN) contamination scenarios:

As evident from the generated binary Noise Maps (Stage 1), the proposed distribution paths manage to isolate the local impulse anomalies without taking into account native geometric textures and structures’ gradients. This feature preservation property is carried forward in the final reconstructed outputs.

In contrast to the Rayleigh-Monte Carlo simulation algorithm (40), which leads to localized structural blurring and leaves isolated unfiltered noise artifacts because of sampling stochasticity, the proposed framework manages to produce significantly sharper edges while keeping surface parameterizations uniform.

Furthermore, while providing similar boundary separation ability in highly textured regions like the fine stripes of “Barbara,” the Topological Filter (41) demonstrates structural fragmentation in dense cluster configurations. On the contrary, the proposed framework preserves total topological alignment and uniform surface luminance distribution and proves its superiority in Quality of Experience (QoE) according to HVS principles.

6 Conclusion and future paths

6.1 Conclusion

This paper introduced an interpretable, two-stage impulse noise detection and removal algorithm with high accuracy and fast execution speed. By combining localized statistical measurements including fitting of Beta distribution parameters, higher order moments (skewness and kurtosis) and Median Absolute Deviation (MAD) in the multi-path architecture, it is possible to reliably isolate corrupted pixels while preserving detailed structures and sharp edges under different noise densities. Using im2col to vectorize the matrix operations removes the computational bottleneck of local spatial windowing providing over 150 times speedup and thus making it feasible for real-time processing on edge devices. Evaluations based on the standard confusion matrix measures (FAR and FRR) confirm the accuracy of the noise mask generation. Future work will extend the vectorized multi-path statistical approach to video and multispectral images.

6.2 Future paths

The development of the structural and statistical framework proposed in this paper requires further research in the following areas:

• Smart Spatially Adaptive Hybrid Switching Engines: Future research will focus on building a smart switching arbitration engine. By analyzing the localized gradient or Kurtosis profile of the input patch beforehand, it will become possible to route the smooth regions through the Robust MAD pipeline (Path C) and complex textural patches through the Bounded Beta engine (Path A) in order to optimize speed and accuracy.
• Integration into Immersive Media Pipelines: Due to the lightweight vectorization of the proposed framework, it becomes a good candidate for integration into ultra-low-latency real-time multimedia systems, such as AR/VR/XR streams. Future research will focus on implementing the statistical filters into edge-computing nodes for sensory data stream cleaning before immersive projection and making sure those Motion-to-Photon latencies stay below the critical 15–20 milliseconds.
• Hardware-level Real-Time Acceleration: The translation of vectorized matrix operations from an interpreted environment (MATLAB) to hardware description language (VHDL/Verilog) becomes a key step forward. The direct implementation of the sliding window decomposition on FPGAs and ASICs will result in line-rate, pixel-by-pixel anomaly detection for high-definition real-time video streams in uncooperative networks.
• Semantic-aware Noise Mitigation: Following the 6G semantic communication trends, the research will focus on filtering “semantic features” or the regions of interest (ROIs) in a multimedia frame (biometric boundaries and landmarks) only, leaving background configuration for lower-complexity approximation to save channel capacity.

Generative AI statement

The author(s) declared that Generative AI was not used in the creation of this manuscript.

Any alternative text (alt text) provided alongside figures in this article has been generated by Frontiers with the support of artificial intelligence and reasonable efforts have been made to ensure accuracy, including review by the authors wherever possible. If you identify any issues, please contact us.

Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

Footnotes

Edited by: Ahmad Abubakar Suleiman, Universiti Teknologi Petronas, Malaysia

Reviewed by: Qiyu Jin, Lanzhou University, China

Hemant Patel, Dr. Subhash University, India