Introduction
Breast cancer is the most common malignancy among women, accounting for 21.5% of all female cancers and ranking first in incidence and sixth in mortality (8.6%) according to the 2021 Cancer Incidence Statistics Report [1]. Mammography remains the primary screening modality and plays a crucial role in early detection, particularly for identifying microcalcifications and non-palpable tumors [2]. However, its diagnostic performance may be limited in postoperative breasts and is associated with exposure to ionizing radiation and patient discomfort.
To complement these limitations, additional imaging modalities such as computed tomography (CT) and magnetic resonance imaging (MRI) are used [3]. Although breast CT enables three-dimensional lesion assessment, it involves relatively high radiation exposure [4]. In contrast, MRI provides superior soft-tissue contrast without ionizing radiation or breast compression and has become an important diagnostic tool for evaluating therapeutic response and prognosis [5–7].
Despite these advantages, MRI is affected by Rician noise, which arises from charged particle motion induced by radiofrequency coils, fluctuations in preamplifiers, and interference between imaging signals. This noise degrades image quality and complicates interpretation. To address this, various denoising filters, including median, Gaussian, and Wiener filters, have been proposed [8–10]. The median filter replaces each pixel with the median of its neighbors, effectively reducing noise while preserving edges; however, it may blur fine structures or edges in areas with low noise [11, 12]. The Gaussian filter reduces normally distributed noise but can blur high-frequency features, such as tumor masses or microcalcifications [9, 13, 14]. The Wiener filter, a low-pass frequency-based approach, enhances signal restoration by processing noise components in the Fourier domain. Nevertheless, because it operates primarily in the frequency domain, it does not fully account for spatial image, characteristics which can result in edge blurring and loss of fine details [15, 16].
To overcome the limitations of conventional filters, a nonlocal means (NLM) algorithm has been proposed, which removes noise from individual pixels or locations using information from the entire image. As a non-local algorithm, NLM eliminates noise by exploiting the structural similarities and repetitive patterns inherent in an image [17, 18]. The algorithm extracts patches around each pixel and evaluates their similarity using Euclidean distance [19–21]. During similarity comparison, the patch around the reference pixel is compared with all patches within a search window, which defines the spatial range of the comparison [22, 23]. The computational cost of NLM depends on the search window size: smaller windows require less computation but may not remove noise effectively, while larger windows can enhance denoising at the cost of higher computational load and potential edge blurring [24, 25]. Therefore, optimizing the search window is essential to balance denoising performance and computational efficiency.
However, the NLM algorithm also requires long processing times owing to extensive pixel-by-pixel search window matching and high computational costs for weight calculations, which have limited its practical applicability [26, 27].
To address these drawbacks, the fast non-local means (FNLM) algorithm was developed. This method reduces computation time and improves time resolution by simplifying the weight calculation process from two-dimensional to one-dimensional operations [28]. However, the Euclidean distance-based matching used in FNLM is sensitive to directional changes and requires manual adjustment of the search window size to suit different imaging conditions. Therefore, the adaptive non-local means (ANLM) algorithm was proposed. ANLM adaptively applies a search window based on the gray level difference (GLD); a larger search window is used for regions with a small GLD, whereas a smaller search window is applied to regions with a large GLD, allowing for adaptive noise suppression. This approach effectively mitigates the limitations of FNLM, which requires manual optimization of the search window [29].
However, few studies have systematically compared the performance of a manually optimized FNLM algorithm with that of an adaptively adjusted ANLM algorithm using clinical breast MRI data. In particular, the impact of search window optimization on denoising performance and computational efficiency under realistic Rician noise conditions remains insufficiently investigated. Therefore, the present study aimed to determine the optimal search window size for the FNLM algorithm using root mean square error (RMSE)-based tuning on a dedicated subset of breast MRI images obtained from a publicly available clinical dataset. Rician noise with varying standard deviations (σ=0.05, 0.10, and 0.15) was synthetically introduced, and search window sizes ranging from 3×3 to 31×31 were evaluated. The optimized FNLM was then applied to the full dataset and compared with conventional NLM and ANLM algorithms in terms of quantitative image quality metrics and computational efficiency.
Materials and Methods
1. Acquisition of Breast MRI Images and Addition of Rician Noise
This study utilized publicly available breast MRI data from the Duke-Breast-Cancer-MRI collection hosted by The Cancer Imaging Archive (https://www.cancerimagingarchive.net/collection/duke-breast-cancer-mri/). This dataset consists of dynamic contrast-enhanced breast MRI examinations acquired at Duke University Medical Center. A total of 50 patients were randomly selected for analysis. For each subject, the pre-contrast three-dimensional T1-weighted image was used as the reference image. All images were obtained in Digital Imaging and Communications in Medicine (DICOM) format and converted to grayscale intensity images for analysis. To ensure consistency across subjects, image intensities were normalized using 99th percentile scaling, such that all pixel values were rescaled to the range (0–1). This normalization reduces inter-subject variability in intensity distribution while preserving anatomical contrast.
To simulate realistic MRI noise characteristics, synthetic Rician noise was generated using the following Equation (1):
where I denotes the normalized noise-free image, and nr and ni are independent zero-mean Gaussian noise components with identical standard deviation σ.
Three noise levels were considered to represent increasing noise severity: σ=0.05, 0.10, and 0.15. For each subject and each noise level, five independent random realizations were generated to account for stochastic variability. The noise variance was controlled relative to the underlying signal intensity, ensuring consistent noise scaling across images. In total, the dataset comprised 50 subjects×3 noise levels×5 realizations, resulting in 750 noisy images, each paired with its corresponding noise-free reference image for quantitative evaluation.
2. Modeling and Implementation of the NLM and FNLM Algorithm
The NLM algorithm assigns weights based on the Euclidean distance between the target pixel region and its surrounding regions; the weighted sum is then applied to the pixel values within the region of interest. The mathematical formulation of the NLM algorithm is expressed in Equations (2) and (3):
where m and n denote reference and comparison pixels, respectively, λ represents the total number of pixels, Gσ(λ) is a Gaussian kernel size of σ, Z(m) is a normalization constant, and d indicates the reference angle during filtering.
However, the NLM algorithm requires substantial computational time because of the similarity in the measurement and weight computation between patches, which limits its applicability in clinical environments where fast image processing is essential. To address this limitation, the conventional two-dimensional weight calculation process of the NLM algorithm was simplified into a one-dimensional equation to reduce the overall computation time. The mathematical expressions for the FNLM algorithm are given in Equations (4) and (5). With this method, the computation time is independent of the patch size, resulting in a significant reduction in total processing time.
where m and n denote the reference and comparison pixels, respectively, p represents the patch radius in the one-dimensional vectorized image, and k is the cumulative summation index. Lambda denotes the relative distance between pixels, d is the filtering parameter controlling the similarity weighting, and Z(m) is a normalization factor.
3. Modeling and Implementation of the ANLM Algorithm
The ANLM algorithm is an improved version of the conventional NLM approach. This adaptive noise-reduction technique automatically adjusts the size of the search window according to the local characteristics of the image. ANLM preserves fine structural details while maintaining the denoising capability of the original NLM algorithm. In the present study, ANLM filtering was modeled as shown in Equation (6), where the final restored image X ^ f i n a l S n o p t
where X ^ f i n a l
4. FNLM Search Window Optimization
To ensure a fair and reproducible comparison, the search window size of the FNLM algorithm was determined through a dedicated parameter optimization procedure prior to final performance evaluation. A tuning subset consisting of 10 randomly selected subjects (five independent noise realizations per subject) was used exclusively for parameter selection. The search window size was varied from 3×3 to 31×31 with an increment of two pixels. For each noise level, the FNLM output was computed using each candidate window size. The optimal search window was selected by minimizing the mean RMSE between the denoised image and the corresponding noise-free reference image across the tuning subset. Other image quality metrics were not used for parameter selection and were instead evaluated independently in the final comparison to avoid metric-specific overfitting. The window size determined by RMSE minimization for each noise level was then fixed and applied to the full dataset (50 subjects) for subsequent comparative analysis against NLM and ANLM.
5. Quantitative Evaluation of Denoising Performance and Time Resolution
To quantitatively evaluate denoising performance, five full-reference image quality metrics were computed: RMSE, structural similarity index (SSIM), high-frequency error norm (HFEN), gradient magnitude similarity deviation (GMSD), and edge preservation index (EPI). RMSE measures the pixelwise intensity difference between the denoised image and the reference image and is defined as Equation (7):
where fi denotes the intensity of the ith pixel in the reference image, gi represents the intensity of the corresponding pixel in the compared image, and N indicates the total number of pixels.
The SSIM evaluates perceptual similarity by incorporating luminance, contrast, and structural information. SSIM values range from 0 to 1, with higher values indicating greater structural similarity and is defined as Equation (8):
where f and g denote the reference and denoised images, μf and μg represent the mean intensities of the reference and denoised images, σ f 2 σ g 2
HFEN quantifies the preservation of high-frequency components by applying a Laplacian-of-Gaussian filter to both reference and denoised images and computing the normalized difference in filtered outputs. Lower HFEN values indicate better edge and detail preservation and is defined as Equation (9).
where f and g denote the reference and denoised images, H(·) represents the Laplacian-of-Gaussian filtering operator applied to emphasize high-frequency components, and ||·||2 denotes the Euclidean (l2) norm computed over all image pixels.
GMSD evaluates structural distortion by measuring the standard deviation of gradient magnitude similarity maps between the reference and denoised images. Lower GMSD values indicate improved structural consistency. The mathematical expressions for the GMSD are given in Equations (10) and (11).
where Gf(x) and Gg(x) denote the gradient magnitudes of the reference image and the denoised image at pixel location x, C is a small positive constant introduced to prevent numerical instability when the denominator approaches zero, and N represents the total number of pixels in the image.
EPI measures the correlation between edge maps extracted from the reference and denoised images, thereby assessing edge preservation performance. Higher EPI values indicate better retention of edge structures, as Equation (12).
where Ef(i) and Eg(i) denote the edge map intensities of the reference image and the denoised image at the ith pixel, E − E −
All metrics were computed for each subject, noise level, and noise realization, and averaged across realizations to obtain robust performance estimates.
To evaluate computational efficiency, the processing time required for each denoising algorithm (NLM, FNLM, and ANLM) was recorded. All algorithms were implemented in MATLAB (MathWorks) and executed on the same hardware environment. The elapsed time was measured using the built-in tic and toc functions, capturing the duration from algorithm initialization to completion. Mean processing time per image was calculated for each method and noise level. All computations were performed using MATLAB (R2023a) on a Mac mini M4 Pro system (Apple Inc.) equipped with a 14-core central processing unit (CPU), 20-core integrated graphics processing unit (GPU), 64 GB of unified memory, and 1 TB of storage. No GPU acceleration was utilized in this study, and all denoising algorithms were executed using CPU-based processing only. Therefore, the reported computational times reflect pure CPU execution performance under identical hardware conditions for all methods.
6. Statistical Analysis
All statistical analyses were performed using JAMOVI version 2.3.28.0 software. Differences in denoising performance across algorithms and noise levels were evaluated using linear mixed-effects models (LMMs) to account for repeated measurements within subjects and inter-subject variability. For each image quality metric (RMSE, SSIM, HFEN, GMSD, and EPI), algorithm (NLM, FNLM, and ANLM) and noise level (σ=0.05, 0.10, and 0.15) were included as fixed effects, and subject was treated as a random intercept. The interaction between algorithm and noise level was additionally examined to determine whether denoising performance varied according to noise severity.
Noise realizations were averaged at the subject level prior to statistical testing to avoid pseudo-replication and to ensure independence of observations. When significant main effects or interaction effects were observed in the LMM, post-hoc pairwise comparisons were conducted using Bonferroni correction to adjust for multiple testing.
Computational efficiency was analyzed separately using LMMs with algorithm included as a fixed effect and subject as a random intercept for each noise level. Post-hoc pairwise comparisons were performed when appropriate.
Model assumptions were evaluated by inspecting residual normality and homogeneity of variance. For all analyses, statistical significance was set at p<0.05.
Results
1. Optimization of FNLM Search Window Size Based on RMSE
To identify the optimal search window size for the FNLM algorithm, a systematic window sweep was performed across sizes ranging from 3×3 to 31×31 under three Rician noise levels of 0.05, 0.10, and 0.15. For each noise condition, the mean RMSE was computed from 50 phantom images. To facilitate comparison across window sizes while accounting for differences in absolute noise magnitude, relative performance was expressed as ΔRMSE, defined as the difference between the RMSE at each window size and the minimum RMSE observed within the same noise level, as illustrated in Fig. 1.
At a noise level of 0.05, the lowest RMSE value of 0.0329 was obtained with a 3×3 search window. Increasing the window size resulted in a progressive rise in RMSE, reaching 0.0406 at 31×31. When the noise level was 0.10, the minimum RMSE of 0.0719 was achieved at a window size of 5×5. Although performance differences between adjacent window sizes were relatively small, RMSE consistently increased beyond this point.
Under the highest noise condition of 0.15, the optimal window size shifted toward a substantially larger neighborhood. The minimum RMSE of 0.1160 was observed at 19×19, whereas a 3×3 window produced a higher RMSE of 0.1208, reflecting insufficient noise suppression at smaller window sizes. RMSE decreased progressively as the window size expanded up to 19×19 and subsequently plateaued, indicating a stable operating region near the optimum.
Accordingly, window sizes of 3×3 for a noise level of 0.05, 5×5 for 0.10, and 19×19 for 0.15 were selected for subsequent comparative analyses.
2. Quantitative Comparison of Denoising Performance across Algorithms
The performances of the FNLM, NLM, and ANLM algorithms with optimized search window sizes for different noise intensities, varied according to the standard deviation of the Rician noise. Fig. 2 presents the visual comparison of the noisy images and the corresponding denoised outputs.
LMMs identified significant main effects of noise level and denoising method across all five metrics (all p<0.001), as well as significant noise-by-method interactions (all p<0.001), indicating that relative algorithm performance depended on noise severity. The distributions of RMSE, SSIM, HFEN, GMSD, and EPI across algorithms and noise levels are summarized in Fig. 3. Detailed pairwise comparisons derived from the LMMs are provided in Table 1.
For RMSE, at noise level 0.05, FNLM achieved significantly lower RMSE than NLM (β=–0.009; 95% confidence interval [CI], –0.012 to –0.005; p<0.001). No significant difference was observed between FNLM and ANLM (p>0.05). At noise level 0.10, no significant differences were observed among the three methods (all p>0.05), suggesting comparable reconstruction accuracy across FNLM, ANLM, and NLM at this noise level. At noise level 0.15, FNLM showed significantly lower RMSE than ANLM (β=–0.004; 95% CI, –0.008 to 0.001; p=0.004). However, no significant difference was observed between FNLM and NLM (β=–0.002; 95% CI, –0.005 to 0.002; p=1.000).
For SSIM, at noise level 0.05, FNLM achieved significantly higher SSIM than both ANLM (β=0.053; 95% CI, 0.031 to 0.074; p<0.001) and NLM (β=0.123; 95% CI, 0.101 to 0.144; p<0.001). At noise level 0.10, FNLM remained significantly superior to ANLM (β=0.043; 95% CI, 0.022 to 0.065; p<0.001), whereas no significant difference was observed between FNLM and NLM (p=1.000). At noise level 0.15, no significant differences were observed among the three methods.
For HFEN, at noise level 0.05, FNLM achieved significantly lower HFEN than ANLM (β=–0.068; 95% CI, –0.083 to –0.053; p<0.001), whereas no significant difference was observed between FNLM and NLM (p=1.000). At noise level 0.10, FNLM remained significantly lower than ANLM (β=–0.059; 95% CI, –0.073 to –0.044; p<0.001), with no significant difference relative to NLM (p=1.000). At noise level 0.15, FNLM demonstrated significantly lower HFEN than both ANLM (β=–0.037; 95% CI, –0.052 to –0.023; p<0.001) and NLM (β=–0.050; 95% CI, –0.065 to –0.036; p<0.001).
For GMSD, at noise level 0.05, FNLM achieved significantly lower GMSD than both ANLM (β=–0.033; 95% CI, –0.037 to –0.028; p<0.001) and NLM (β=–0.065; 95% CI, –0.069 to –0.060; p<0.001). At noise level 0.10, FNLM remained significantly lower than ANLM (β=–0.018; 95% CI, –0.022 to –0.013; p<0.001) and also demonstrated significantly lower GMSD than NLM (β=–0.013; 95% CI, –0.018 to –0.009; p<0.001). At noise level 0.15, no significant difference was observed between FNLM and ANLM (p=1.000), whereas FNLM showed significantly higher GMSD than NLM (β=0.020; 95% CI, 0.015 to 0.024; p<0.001).
For EPI, at noise level 0.05, FNLM achieved significantly higher EPI than ANLM (β=0.059; 95% CI, 0.046 to 0.072; p<0.001), whereas no significant difference was observed between FNLM and NLM (p=1.000). At noise level 0.10, FNLM remained significantly higher than ANLM (β=0.045; 95% CI, 0.032 to 0.058; p<0.001), while no significant difference was observed between FNLM and NLM (p=0.886). At noise level 0.15, FNLM demonstrated significantly higher EPI than ANLM (β=0.023; 95% CI, 0.009 to 0.036; p<0.001), whereas NLM showed significantly higher EPI than FNLM (β=–0.023; 95% CI, –0.036 to –0.009; p<0.001).
3. Comparisons of Computational Time between the FNLM and ANLM Algorithms
The computational time required by each algorithm across noise levels is summarized in Fig. 4. Detailed pairwise comparisons from the LMM are presented in Table 2. For processing time, at noise level 0.05, the mean processing time was 11.90±0.88 seconds for ANLM, 1.41±0.09 seconds for NLM, and 0.48±0.03 seconds for FNLM. FNLM was approximately 25.1 times faster than ANLM and 3.0 times faster than NLM under this condition. Consistent with the post-hoc analysis, FNLM required significantly less computation time than ANLM (β=–10.521; 95% CI, –10.754 to –10.290; p<0.001) and also significantly less time than NLM (β=–0.932; 95% CI, –1.164 to –0.700; p<0.001). At noise level 0.10, the mean processing times were 11.90±0.82 seconds for ANLM, 1.64±0.093 seconds for NLM, and 0.518±0.038 seconds for FNLM. FNLM was approximately 23.0 times faster than ANLM and 3.2 times faster than NLM. FNLM remained significantly faster than ANLM (β=–10.286; 95% CI, –10.518 to –10.054; p<0.001) and faster than NLM (β=–1.126; 95% CI, –1.358 to –0.894; p<0.001). At noise level 0.15, the mean processing times increased to 12.00±0.83 seconds for ANLM, 4.20±0.20 seconds for NLM, and 1.53±0.07 seconds for FNLM. FNLM was approximately 7.8 times faster than ANLM and 2.7 times faster than NLM at this noise level. FNLM continued to demonstrate significantly shorter processing time than ANLM (β=–7.838; 95% CI, –8.069 to –7.606; p<0.001) and also required significantly less time than NLM (β=–2.664; 95% CI, –2.896 to –2.432; p<0.001).
Discussion
This study systematically evaluated the performance of the FNLM algorithm across multiple Rician noise levels and compared it with NLM and ANLM using quantitative metrics including RMSE, SSIM, HFEN, GMSD, and EPI. For each noise condition, the optimal search window size of the FNLM algorithm was determined through a window sweep analysis and subsequently used for comparative evaluation across algorithms.
The optimal search window size for FNLM varied according to noise intensity. At a noise level of 0.05, the minimum RMSE was achieved with a 3×3 window, whereas at 0.10, the optimal size increased to 5×5, and at 0.15, it further expanded to 19×19. These findings demonstrate that the search window size required for optimal denoising increases as noise severity becomes greater. Notably, RMSE progressively decreased as the window size increased under high-noise conditions up to 19×19, after which performance plateaued. This trend highlights the importance of noise-dependent parameter selection when applying FNLM for image denoising. Overall, the comparative analysis indicated that FNLM achieved competitive or superior performance relative to ANLM and NLM depending on the evaluation metric and noise level, while consistently demonstrating a substantial advantage in computational efficiency.
According to Park et al. [30], who evaluated the effect of smoothing strength in the block matching and 3D filtering (BM3D) algorithm on brain MRI images with Rician noise, higher noise levels required stronger smoothing; however, beyond a certain threshold, excessive smoothing degraded image quality indicators such as RMSE and contrast to noise ratio. A similar pattern was observed in the present study, where increasing the FNLM search window size improved denoising performance up to an optimal level, after which further expansion did not yield additional benefit and, in some cases, led to performance degradation. These findings reinforce the importance of noise-dependent parameter optimization rather than simply enlarging the similarity search range.
Although ANLM approaches are theoretically expected to provide performance advantages by adjusting the search window size according to local structural characteristics [31], previous studies have reported that the effectiveness of adaptive strategies strongly depends on the stability of local noise estimation and parameter tuning. Manjon et al. [32] demonstrated that NLM-based denoising performance is highly sensitive to noise-level-dependent parameter selection, particularly under Rician noise conditions, and that inappropriate local adaptation may lead to suboptimal smoothing behavior. Similarly, Coupe et al. [31] emphasized that increased algorithmic complexity and adaptive weighting do not necessarily guarantee improved reconstruction quality, especially when local similarity estimation becomes unstable in noisy regions. In this context, the optimized FNLM implementation used in the present study may have benefited from stable, noise-specific parameter tuning, resulting in more consistent similarity estimation across noise levels. These findings suggest that adaptive window strategies do not universally outperform simplified NLM variants when noise conditions are controlled and parameters are carefully optimized.
When examining quantitative metrics in detail, the relative performance of FNLM varied depending on the evaluation index and noise level. FNLM generally demonstrated lower RMSE and HFEN than ANLM, indicating improved noise suppression and high-frequency error reduction. Improvements in SSIM suggested enhanced structural similarity preservation under low and moderate noise conditions. For GMSD and EPI, performance differences were more nuanced, with FNLM showing competitive performance relative to NLM across most noise levels, although slight advantages for NLM were observed under the highest noise condition for specific edge-sensitive metrics. These findings indicate that algorithm performance depends on the chosen quality metric and noise severity. Importantly, despite the substantial reduction in computational time, FNLM did not exhibit systematic performance degradation compared with NLM. Across most noise levels and evaluation metrics, FNLM achieved comparable or improved quantitative results relative to NLM while reducing processing time by approximately 2.7- to 3.2-fold. This suggests that the simplified similarity calculation in FNLM improves computational efficiency without sacrificing denoising effectiveness.
Although the ANLM algorithm required longer processing time than FNLM, it employs an adaptive search strategy that may facilitate preservation of fine structural details. ANLM applies larger search windows in homogeneous regions and smaller windows near structural boundaries or areas with high spatial variation [33], which can contribute to maintaining edge information in anatomically complex regions such as vessels or lesion margins [34]. In the present study, although ANLM generally showed lower quantitative performance compared with FNLM across several metrics, its adaptive mechanism remains conceptually advantageous for structure-sensitive denoising [35]. However, the pixel-wise optimization of search windows substantially increases computational complexity [36], resulting in markedly prolonged processing times relative to FNLM.
In conclusion, FNLM achieved competitive quantitative denoising performance while markedly improving computational efficiency. These findings suggest that FNLM may offer practical advantages in imaging environments where both processing speed and stable reconstruction performance are required.
However, this study has several limitations. Comparisons were limited to NLM-based algorithms, and other advanced denoising approaches, including probabilistic NLM variants or deep learning-based methods, were not evaluated [19, 21]. Therefore, future research should include quantitative comparisons with various NLM variants and deep learning-based denoising techniques to further validate and extend the findings of this study. Although this study utilized real clinical breast MRI data, variability across scanner manufacturers, field strengths, and acquisition protocols may influence optimal parameter selection. Therefore, the FNLM search window optimization procedure should be regarded as a dataset-specific tuning framework rather than a universally fixed configuration. Future studies involving multi-center datasets would further strengthen the generalizability of the proposed approach.
Conclusion
This study determined the optimal search window sizes for the FNLM algorithm under three Rician noise levels (σ=0.05, 0.10, and 0.15) and quantitatively compared FNLM, ANLM, and NLM using RMSE, SSIM, HFEN, GMSD, EPI, and computational time. The optimal window sizes for FNLM were 3×3 at noise level 0.05, 5×5 at 0.10, and 19×19 at 0.15. LMMs revealed significant main effects of noise level and denoising method, as well as significant noise-by-method interactions across all image quality metrics. FNLM showed statistically significant differences compared with ANLM and/or NLM depending on the metric and noise level. For computational time, FNLM required significantly less processing time than both ANLM and NLM at all noise levels. In conclusion, the optimized FNLM algorithm achieved competitive or superior denoising performance across multiple quantitative metrics while consistently demonstrating significantly reduced computational time compared with ANLM and NLM across all tested Rician noise levels.
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