Noise Addition and Removal Process

• Noise addition and removal process is defined by the integration of statistical noise models with filtering and transform-domain techniques to simulate and reduce unwanted interference.
• Advanced methods such as sparse component analysis, wavelet transforms, and diffusion models are employed to address various noise types in fields like biomedical imaging, remote sensing, and quantum measurements.
• Hybrid approaches, including deep CNNs, variational regularization, and quantum-inspired algorithms, improve performance metrics such as PSNR and SSIM while preserving critical signal features.

Noise addition and removal processes are fundamental procedures in signal and image analysis, affecting fields as diverse as digital communications, biomedical imaging, seismic exploration, remote sensing, and quantum-enabled measurement. Contemporary research covers an array of statistical models, transform-domain methods, sparse reconstruction, advanced filtering, diffusion processes, hardware quantum simulation, and even constructive uses of noise. The following sections provide a technical overview of the models, principles, algorithms, and applications referenced in recent research, emphasizing their mathematical structure, typical use-cases, and performance-critical considerations.

1. Mathematical Noise Models and Statistical Assumptions

Noise in imaging and signal systems is described statistically, with models tailored to the source and nature of corruption:

• Additive White Gaussian Noise (AWGN): The most common, where $y = x + n$ and $n \sim \mathcal{N}(0, \sigma^2)$ are independent identically distributed.
• Impulse Noise: Includes “salt-and-pepper” (pixels randomly set to minimum or maximum values) and random-valued impulse noise. Such contamination is non-Gaussian, heavy-tailed, and spatially sparse (0812.2892, Kazerooni et al., 2011).
• Multiplicative (Speckle) Noise: Modeled as $y = x \cdot \epsilon$, with $\epsilon$ random—typical in coherent imaging like SAR or ultrasound (Vuong et al., 19 Aug 2024, Akbar et al., 29 Oct 2024).
• Signal-Dependent Noise: Found in camera sensors; the model often takes the form $\sigma_y(x) = \sqrt{\sigma_r^2 + \sigma_s^2 x}$, with shot (Poisson) and read (Gaussian) noise components (Pearl et al., 2023).
• Periodic, Poisson, Rayleigh, Exponential, Erlang Noise: Each with distinct probability distributions, representing physical processes such as electrical interference, photon counting, or laser/optical effects (Akbar et al., 29 Oct 2024).

The suitability of removal or restoration techniques depends critically on these statistical properties. For example, transform-based denoising typically targets additive Gaussian contamination, while sparse component analysis exploits the spatial sparsity of impulse noise.

2. Transform-Domain and Sparse Component Analysis Approaches

Many state-of-the-art removal methods leverage transform sparsity:

• Discrete Cosine Transform (DCT) Sparsity: Natural images have compact DCT representations; thus, coefficients beyond the effective rank (esp. in zigzag order as in JPEG) are nearly zero (0812.2892). By modeling the noisy image $X$ as $X = S + E$ and observing $T(X) = T(S) + T(E)$ in the DCT domain, one sets up an underdetermined linear system, $x = H \cdot Z(E)$, where $x$ is the “zero region” of the observed coefficients. Sparse component analysis (SCA) is then used, with algorithms such as basis pursuit, FOCUSS, or smoothed-$\ell_0$ methods, to reconstruct sparse impulse error patterns. The salt-and-pepper variant simplifies the detection and leads to a truncation of the mixing matrix.
• Wavelet-Based Sparse Signal Processing: Impulse noise removal (especially salt-and-pepper) can be recast as a sparse reconstruction problem in the wavelet (e.g., dual-tree complex wavelet) domain (Kazerooni et al., 2011). Using the Iterative Method with Adaptive Thresholding (IMAT), a 2D extension of greedy sparse recovery, the method alternates between enforcing data consistency in known “clean” pixel locations and sparsity via hard-thresholding in the wavelet domain.
• Diffusion Model Adaptations: Advanced methods adapt denoising diffusion probabilistic models (DDPMs) for spatially-varying, realistic noise (Pearl et al., 2023). For example, SVNR assigns each pixel a spatially-dependent noise “time” embedding. The noise removal process is then a series of reverse diffusion steps that begin directly from the noisy image, allowing spatial adaptation and alignment with the underlying sensor noise statistics.

3. Filtering, Variational, and Hybrid Regularization Schemes

Noise filtering encompasses both classical and advanced adaptive strategies:

• Median and Nonlinear Filters: Median filters are optimal for impulse (salt-and-pepper) noise but are suboptimal at higher densities or for mixed noise; adaptive schemes switch dynamically between median and mean filtering based on local corruption density or window statistics (Dash et al., 2015, Satpathy et al., 2022). Cascaded approaches (DMF+MDBUTMF, etc.) split the denoising into preliminary (low density) and secondary (high density) filtering stages, significantly improving restoration over classical methods.
• Variational Regularization: For mixed noise (AWGN + impulse), standard pipelines apply a rank order filter followed by a Gaussian denoiser. However, the presence of heavy-tailed residuals degrades performance (Islam et al., 2018). A variational step with an $\ell_1$-norm data fidelity term (robust to outliers) and local, edge-preserving regularizer significantly “Gaussianizes” the noise and improves subsequent denoising.
• Joint Adaptive Statistical Priors (JASP): The De-JASP variational scheme (Eslahi et al., 2015) integrates both local (curvelet transform–based) and non-local statistical priors (grouped 3D transform–based self-similarity). An alternating minimization (split Bregman) algorithm alternates between quadratic, soft-thresholding, and shrinkage steps, achieving strong results (PSNR, SSIM) across Gaussian–impulse mixtures.
• Minimum Mean Square Error (MMSE) Estimation: Classical MMSE estimators combine local statistics (sample covariance) with assumed noise models, often requiring pseudo-inverse solvers and SVD-based computation to account for local structure and noise (Lee et al., 2019).

4. Algorithmic Innovations in Complex or Physical Domains

Recent research extends beyond standard digital filtering:

• Entropy Quantum Computing (EQC): A hardware-based method observes that in quantum-limited measurements (e.g., LiDAR), noise quanta (photons) obey Poissonian statistics (Huang et al., 12 Feb 2025). The EQC approach reconstructs the noise photon distribution across measurement modes by minimizing a Hamiltonian whose ground state represents maximum spatial (or temporal) signal correlation under the Poissonian constraint. Optimization is carried out physically (e.g., Dirac-3 hardware with quantum Zeno effect), providing a hardware acceleration over NP-hard classical optimization for strong-noise regimes.
• SDE-based Perception-Oriented Despeckling: Multiplicative (speckle) noise is modeled as a Geometric Brownian Motion SDE in the log-domain, enabling the reverse (denoising) process to be derived analytically via the Fokker-Planck and Anderson theorems (Vuong et al., 19 Aug 2024). The forward process increments in log-space represent Gaussian noise, thereby reducing the task to score-matching as in classic diffusion models. Probabilistic or ODE-based reverse sampling then yields denoised images.
• Universal Sampling Denoising (USD): For non-Cartesian MRI data, noise after regridding is spatially inhomogeneous and correlated. The USD pipeline first “whitens” channels via coil noise covariance estimation, then spatially decorrelates via a regridding covariance (obtained from the kernel), next applies PCA-based denoising (MPPCA) under restored i.i.d. conditions, and finally renormalizes for signal amplitude (Lee et al., 2023). This enables previously inapplicable matrix-projection denoising to be used in MRI with arbitrary $k$-space trajectories.

5. Noise Addition as an Information Preservation Technique

Counterintuitively, adding noise can in some cases enhance signal processing:

• Dithering in Digitization: When reducing quantization levels (e.g., to binary or ternary images), adding controlled randomization prior to quantization (dithering) smooths out contours and preserves perceived tones that would be lost by strict thresholding (Weinstein et al., 2016). The model $x_d = Q(x + \eta)$, with $Q$ quantization and $\eta$ dither noise, ensures that, in aggregates, the local mean matches the original grey level.
• Stochastic Resonance: In biological and physical threshold systems, added noise allows weak subthreshold signals to be amplified or become detectable. This principle is modeled by $dx/dt = -dU/dx + Asin(\omega t) + \xi(t)$, with transitions over energy barriers optimally driven by a finite noise “temperature”. Ant colony foraging and neural detection are cited as systems optimized for resonance-like information throughput (Weinstein et al., 2016).
• Intentional Noise Generation in Rendering/Compression: Physically and biologically inspired methods extract and store a compact, intensity-dependent noise model during image encoding (e.g., a power law in gamma-corrected color channels), and at decoding regenerate synthetic, high-frequency, perceptually plausible noise aligned with the original scene’s statistics (Khasanova et al., 2018). This recreates “grain” lost in lossy compression—user studies show preferences align with levels inferred from the model.

6. Pattern Recognition, Deep Learning, and Hybrid Solutions

Noise removal for complex or real-world signals often requires data-driven and adaptive techniques:

• Deep CNNs and Autoencoders: Fully convolutional denoising autoencoders (FCN-DAE), U-Nets, and encoder-decoder architectures have been applied to tasks such as removing coherent seismic noise (Agarwal et al., 2021), EEG artifact correction (Johari et al., 2023), and self-supervised hyperspectral denoising (Platt et al., 26 Mar 2024). In these, paired or self-supervised training regimes (e.g., Noise2Noise) allow effective learning even without access to clean ground truth.
• Canonical Correlation Analysis (CCA) with Reference Noise: The iCanClean algorithm applies CCA to find shared subspaces between multichannel data and explicit reference noise measurements, removing strongly correlated components via least-squares projection. The method’s efficiency makes it practical for real-time BCI preprocessing (Downey et al., 2022).
• Advanced Filtering in Biomedical and Remote Sensing: Adaptive approaches combine empirical mode decomposition (EMD), wavelet boost/thresholding, least squares regression with sparsity (DLSR), and Kalman filtering, sometimes in hybrid combinations tailored to artifact signature and measurement context (e.g., motion, muscle, line noise in EEG) (Johari et al., 2023).

7. Comparative Assessments and Application Domains

Table: Match of Filtering Algorithms to Noise Types and Application Domains

Noise Type	Optimal Filtering Strategy	Application Domains
Gaussian	Wiener, Gaussian	Medical imaging, FTIR spectroscopy, communications, radar
Salt & Pepper	Median, cascaded median–mean	Photography, surveillance, medical imaging, remote sensing
Impulse/Random	SCA, sparse signal reconstruction	Satellite, MRI, transmission with erasure (channel coding)
Speckle	SDE-based, bilateral, BM3D	SAR Imaging, laser/ultrasound, satellite, optical instrumentation
Mixed Gaussian+Imp	De-JASP, variational split-Bregman	General imaging, remote sensing, consumer photography
Coherent noise	CNN encoder-decoder, CCA	Seismic, EEG, BCI, aircraft/remote sensing telemetry
Quantum noise	Entropy Quantum Computing	LiDAR, astrophysics, quantum-limited sensor systems

Noise removal is highly context- and model-sensitive: for instance, transform-based SCA schemes are superior for low-rank, locally sparse impulse events; bilateral and non-local means tackle signal-dependent noise; perception-based approaches maximize fidelity in generative or simulation-driven visual tasks.

• Sparse Component Analysis (SCA) in Random-valued and Salt and Pepper Noise Removal (0812.2892)
• Salt-and-Pepper Noise Removal Based on Sparse Signal Processing (Kazerooni et al., 2011)
• High Density Noise Removal by Cascading Algorithms (Dash et al., 2015)
• Mixed Gaussian-Impulse Noise Removal from Highly Corrupted Images via Adaptive Local and Nonlocal Statistical Priors (Eslahi et al., 2015)
• Noise and Function (Weinstein et al., 2016)
• Noise generation for compression algorithms (Khasanova et al., 2018)
• A Variational Step for Reduction of Mixed Gaussian-Impulse Noise from Images (Islam et al., 2018)
• Image De-Noising For Salt and Pepper Noise by Introducing New Enhanced Filter (Kumar et al., 2019)
• Noise Removal of FTIR Hyperspectral Images via MMSE (Lee et al., 2019)
• Sentinel-1 Additive Noise Removal from Cross-Polarization Extra-Wide TOPSAR with Dynamic Least-Squares (Lee et al., 2021)
• Convolutional encoder decoder network for the removal of coherent seismic noise (Agarwal et al., 2021)
• The iCanClean Algorithm: How to Remove Artifacts using Reference Noise Recordings (Downey et al., 2022)
• Adaptive Non-linear Filtering Technique for Image Restoration (Satpathy et al., 2022)
• SVNR: Spatially-variant Noise Removal with Denoising Diffusion (Pearl et al., 2023)
• Noise removal methods on ambulatory EEG: A Survey (Johari et al., 2023)
• Universal Sampling Denoising (USD) for noise mapping and noise removal of non-Cartesian MRI (Lee et al., 2023)
• Noise2Noise Denoising of CRISM Hyperspectral Data (Platt et al., 26 Mar 2024)
• Perception-based multiplicative noise removal using SDEs (Vuong et al., 19 Aug 2024)
• Analyzing Noise Models and Advanced Filtering Algorithms for Image Enhancement (Akbar et al., 29 Oct 2024)
• Noise Reversal by Entropy Quantum Computing (Huang et al., 12 Feb 2025)