Nonlinear Vector Filtering for Impulsive Noise Removal from Color Images

I. INTRODUCTION

• The study addresses two gaps: a unified categorization of filters proposed since 1990 and an objective comparison across a large, diverse image set.A previous survey covered filters and applications but did not experimentally compare their performance.Source: §I · p. 4
• The survey presents 48 impulsive-noise filters in 8 families and compares effectiveness and efficiency across 100 images from varied domains.It also evaluates Minkowski, angular, and directional-distance measures and introduces a fast inverse-cosine approximation for fairer efficiency comparisons.Source: §I · p. 4

2.1.1 Vector Median Filter

• The Vector Median Filter (VMF) outputs the lowest-ranked vector after ordering window vectors by relative magnitude differences using a Minkowski distance.The commonly used distance choices are the L1 Manhattan and L2 Euclidean norms.Source: §2.1.1 · p. 6

2.1.2 Alpha-Trimmed Vector Median Filter

• The Alpha-Trimmed Vector Median Filter (ATVMF) averages the lowest-ranked 1 + α vectors, combining impulsive-noise trimming with averaging for Gaussian noise.The trimming operation supports impulsive-noise performance, while averaging helps with Gaussian noise.Source: §2.1.2 · p. 6

2.1.3 Basic Vector Directional Filter

• The Basic Vector Directional Filter (BVDF) ranks vectors by angle, favoring chromaticity preservation because vector direction corresponds to chromaticity.It identifies outliers as vectors with atypical directions.Source: §2.1.3 · p. 7

2.1.4 Generalized Vector Directional Filter

• The Generalized Vector Directional Filter (GVDF) selects several low-ranked vectors by angular distance, then filters their magnitudes using a grayscale operator.This study uses the arithmetic mean filter for magnitude processing.Source: §2.1.4 · p. 7

2.1.5 Directional Distance Filter

• The Directional Distance Filter (DDF) simultaneously minimizes the VMF and BVDF criteria, combining vector-magnitude and direction information.Its parameter γ controls the relative importance of magnitude and direction; γ = 0.5 gives them equal consideration.Source: §2.1.5 · pp. 7–8

2.1.6 Content Based Rank Filter

• The Content Based Rank Filter (CBRF) ranks vectors using a similarity ratio whose numerator and denominator represent commonality and totality, respectively.The corresponding vector difference and vector sum provide those two components.Source: §2.1.6 · p. 8

2.2 Adaptive Fuzzy Vector Filters

• Adaptive fuzzy vector filters transform a fuzzy weighted average of vectors in a window, with weights reflecting each vector’s contribution.The weights are determined by fuzzy transformations of cumulative distances associated with the input vectors.Source: §2.2 · pp. 8–9

2.2.1 Fuzzy Weighted Average Filters

• Fuzzy Weighted Average Filters use the identity function for the nonlinear transformation, so their output may fall outside the input-vector set.This averaging behavior can perform better with Gaussian noise than pure order-statistics filters that select an input vector.Source: §2.2.1 · p. 9

2.2.1.1 Fuzzy Vector Median Filter; 2.2.1.2 Fuzzy Vector Directional Filter

• The Fuzzy Vector Median Filter uses Minkowski distance with exponential fuzzy membership, whereas the Fuzzy Vector Directional Filter uses angular distance with sigmoidal membership.The FVMF parameters are γ = 0.5 and β = 1.0; for the FVDF, γ adjusts the weighting effect and β is a weight-scale threshold.Source: §2.2.1.1 · pp. 9–10, §2.2.1.2 · p. 10

2.2.1.3 Adaptive Nearest Neighbor Filter; 2.2.1.4 Adaptive Nearest Neighbor Multichannel Filter

• The Adaptive Nearest Neighbor Filter derives fuzzy weights from cumulative angular distances, while the multichannel variant replaces angular distance with a composite distance.Other distance measures, including Minkowski and directional-distance, can also be used for the ANNF weights.Source: §2.2.1.3 · pp. 10–11, §2.2.1.4 · p. 11

2.2.2 Fuzzy Ordered Vector Filters

• Fuzzy Ordered Vector Filters select the input vectors with the largest fuzzy weights, with the retained count k chosen adaptively from weights greater than 1/n.This study considers fuzzy ordered median and directional variants.Source: §2.2.2 · pp. 11–12

2.3 Hybrid Vector Filters; 2.3.1 Extended Vector Median Filter; 2.3.2 Hybrid Directional Filter

• Hybrid vector filters combine sub-filters through linear or nonlinear combinations, so their output often is not an input vector.The Extended Vector Median Filter behaves like the VMF near edges and like the AMF in smooth areas, while HDF combines VMF and BVDF concepts.Source: §2.3 · p. 12, §2.3.1 · p. 12, §2.3.2 · p. 13

2.3.3 Adaptive Hybrid Directional Filter; 2.3.4 Vector Median-Rational Hybrid Filter

• The Adaptive Hybrid Directional Filter uses the AMF so output magnitude follows the mean in smooth regions and the median near edges.The Vector Median-Rational Hybrid Filter combines outputs from two VMFs and a center-weighted VMF in a rational function; its denominator parameters provide numerical stability and regulate nonlinearity.Source: §2.3.3 · p. 13, §2.3.4 · p. 13

2.3.6 Kernel Vector Median Filter

• The Kernel Vector Median Filter outputs a linear combination of the center pixel and VMF output, with weights determined by a kernel of their distance.When the center pixel is not noisy, the kernel approaches 1 and output stays near its original value; otherwise output approaches the VMF result.Source: §2.3.6 · p. 15

2.4 Adaptive Center-Weighted Vector Filters

• Weighted vector filters assign nonnegative integer weights to window pixels, and changing the distance function yields vector median, directional, or directional-distance variants.Their flexible weights can be optimized for a particular domain using training images.Source: §2.4 · pp. 15–16

2.4. Adaptive Center-Weighted Vector Filters

• In the Center-Weighted Vector Median Filter, increasing the center weight moves behavior from identity filtering toward the VMF, increasing smoothing.At k = 1 it performs no smoothing; at its maximum C it is equivalent to the VMF.Source: §2.4 · p. 16

2.4.2 Modified Center-Weighted Vector Median Filter

• The Modified Center-Weighted Vector Median Filter weights only the center pixel’s cumulative distance, requiring fewer multiplications than the CWVMF.Its center weight is a real number between 0 and 1, unlike the CWVMF’s nonnegative integer weight.Source: §2.4.2 · p. 17

2.5 Entropy Vector Filters

• Entropy Vector Filters extend grayscale local-contrast entropy filtering to multichannel images and form a family of adaptive switching filters.Noisy pixels contribute heavily to total local contrast entropy.Source: §2.5 · p. 18
• EVMF, EBVDF, and EDDF compare the center pixel’s fraction of local contrast entropy with an adaptive threshold to decide whether it is noisy.A detected noisy pixel is replaced by the VMF, BVDF, or DDF output, respectively; otherwise it remains unchanged.Source: §2.5 · p. 18

2.6 Peer Group Vector Filters

• Peer-group filters identify neighbors sufficiently similar to a pixel under a chosen measure and use that group for adaptive switching.The center pixel is either retained or replaced according to the filter’s noise decision.Source: §2.6 · p. 19

2.6.1 Peer Group Filter

• The Peer Group Filter sorts window pixels by distance to the center, then uses first-order differences and a threshold to decide whether to replace it with the VMF output.If no difference exceeds the user-specified threshold, the center pixel remains unchanged.Source: §2.6.1 · p. 19

2.6.2 Fast Peer Group Filter

• The Fast Peer Group Filter stops distance calculations once m sufficiently similar pixels are found, which can dramatically reduce computations when m and image noise are low.Its parameters are m = 3 and T = 45.Source: §2.6.2 · p. 20

2.7 Vector Sigma Filters

• Vector Sigma Filters use approximations of multivariate variance to detect noisy center pixels, replacing them with VMF, BVDF, or DDF output while leaving other pixels unchanged.They are adaptive switching, multichannel extensions of the grayscale sigma filter.Source: §2.7 · p. 20
• Covariance-matrix eigenvalue sums or products can measure multivariate variance, but are computationally expensive, so these filters approximate variance using the mean vector or lowest-ranked vector.Non-adaptive variants use λ as a switching threshold; adaptive variants determine the threshold adaptively, and λ is set to 4.0.Source: §2.7 · pp. 20–21

2.8 Miscellaneous Vector Filters

• The miscellaneous family contains filters outside the preceding categories, including similarity-based Adaptive Multichannel Non-Parametric Filters that do not combine multiple sub-filters.The AMNFs resemble the KVMF in using similarity rather than distance.Source: §2.8 · p. 21

2.8.1 Vector Signal-Dependent Rank Order Mean Filter

• The Vector Signal-Dependent Rank Order Mean Filter sorts pixels by cumulative distance, tests the lowest-ranked candidates against increasing thresholds, and replaces a noisy center with the VMF output.Its thresholds are 35, 40, 45, and 50.Source: §2.8.1 · p. 22

2.8.2 Adaptive Multichannel Non-Parametric Filters

• Adaptive Multichannel Non-Parametric Filters use non-parametric kernel density estimation, with multivariate exponential or Gaussian kernels and k = 0.33.The general form is referenced in Table 8.Source: §2.8.2 · p. 22

2.8.3 Fast Modified Vector Median Filter

• The Fast Modified Vector Median Filter replaces the center with the lowest cumulative-distance window pixel only when the center’s distance exceeds the minimum by more than a threshold.The center is privileged in the distance calculation because it uses n - 1 terms, while other pixels use n - 2; the threshold is 0.75.Source: §2.8.3 · pp. 22–23

2.8.4 Adaptive Vector Median Filter and Adaptive Basic Vector Directional Filter

• The Adaptive Vector Median Filter tests the center against the mean of the lowest-ranked vectors and substitutes VMF output when noisy; the Adaptive Basic Vector Directional Filter is its angular counterpart.Their thresholds are 100 and 0.16, respectively, and both use k = ⌈n / 2⌉.Source: §2.8.4 · p. 23

2.8.5 Fast Fuzzy Noise Reduction Filter

• The fuzzy similarity metric’s precomputed values make its computation faster than the L1 norm.The Fast Fuzzy Noise Reduction Filter selects the window pixel with maximum cumulative similarity to the others, excluding the center pixel; K = 1024 and α = 3.5.Source: §2.8.5 · pp. 23–24

3.1 Image Set Description

• The evaluation uses 100 high-quality RGB images spanning people, animals, plants, buildings, maps, natural scenes, art, scientific and biomedical images, and common test images.The image set also includes man-made objects, sketches, and synthetic images.Source: §3.1 · p. 25

3.2 Noise Models

• Two impulsive-noise models are considered, and a noise configuration means a particular combination of model and noise level.The models are identified as uncorrelated and correlated impulsive noise.Source: §3.2 · pp. 25–26

3.3 Filtering Performance Criteria

• Effectiveness is assessed with MAE for detail preservation, MSE for noise suppression, and perceptually oriented NCD for color preservation; efficiency is execution time.The inverse-cosine approximation is used to make runtime comparisons involving angular and directional-distance filters fairer.Source: §3.3 · pp. 27–28
• 1428 seconds versus 102 seconds: the BVDF’s approximate-acos implementation ran about 13x–14x faster across 100 images, with virtually the same MAE, MSE, and NCD as standard acos.Similar runtime gains were observed for other angular and directional-distance filters; speeding up the VMF itself was outside the study’s scope.Source: §3.3 · p. 29

IV. DISCUSSION AND CONCLUSIONS

• The section identifies filters that offer a good effectiveness-efficiency compromise and compares Minkowski, angular, and directional-distance measures.The experimental discussion first assesses effectiveness and efficiency, then identifies compromise filters and compares distance measures.Source: §IV · p. 29

4.1 Discussion

• Tables 9–12 present filter rankings based on MAE, MSE, NCD, and execution time for uncorrelated and correlated impulsive-noise models at 5%, 10%, and 15%; average rankings average individual filter rankings over the entire image set.Source: §4.1 · p. 30
• Average rankings summarize each filter's performance across the full image set.The rankings average individual filter rankings over the image set.Source: §4.1 · p. 30

INSERT TABLE 12 HERE

• Adaptive Center-Weighted and Vector Sigma families stand out in effectiveness; their noise tests use, respectively, a robust but computationally expensive iterative scheme and approximations of multivariate variance.In general, non-adaptive Vector Sigma filters perform better than their adaptive counterparts.Source: INSERT TABLE 12 HERE · pp. 30–31

INSERT TABLE 13 HERE

• ACWDDF consistently ranks highest among filters effective across noise configurations; PGF and ACWVMF have relatively stable rankings, while SDDF_rank fluctuates.The filters effective across all Table 13 rows are ACWDDF, PGF, SDDF_rank, and ACWVMF.Source: INSERT TABLE 13 HERE · p. 31

INSERT FIGURE 3 HERE

• Except for VMF, every filter in the ten most efficient list uses switching between the identity and filter operations.The efficient-filter list includes FPGF, ASVMF_mean, PGF, SVMF_mean, FFNRF, MCWVMF, ASBVDF_mean, SVMF_rank, FMVMF, and VMF.Source: INSERT FIGURE 3 HERE · p. 32
• FPGF is the most efficient filter, and all but FFNRF and ASBVDF_mean among the ten most efficient filters use Minkowski distance.ASBVDF_mean ranks seventh among angular filters; ASDDF_mean, the most efficient directional-distance filter, ranks twelfth.Source: INSERT FIGURE 3 HERE · pp. 31–33
• PGF is the only filter ranked very high in both effectiveness and efficiency, while ACWDDF is among the slowest despite its effectiveness.Other filters noted as offering a good compromise include MCWVMF, FMVMF, FFNRF, SVMF_rank, SVMF_mean, and FPGF.Source: INSERT FIGURE 3 HERE · p. 32
• No distance measure dominates overall: two of the four most effective filters use directional-distance, while Minkowski-based filters are inherently more efficient.Only eight of the 48 filters use directional-distance.Source: INSERT FIGURE 3 HERE · p. 32
• Hybrid and adaptive fuzzy filters perform unsatisfactorily in these comparisons, which the authors attribute to color artifacts from combining input vectors; they can be more effective for Gaussian noise.Their averaging nature is cited in the Gaussian-noise qualification.Source: INSERT FIGURE 3 HERE · p. 33

4.2 Conclusions

• The study systematically surveys 48 impulsive-noise-removal filters in unified notation, categorizes them into families, compares them on a large image set, and introduces a fast inverse-cosine approximation for more even efficiency comparisons.Some literature filters were omitted due to time constraints, including FANRF, FIVF, and VRMKNNF.Source: INSERT FIGURE 3 HERE · p. 33, §4.2 · p. 34

• Figure 1 is captioned as showing representative images from the image set.Source: Figure 1 · pp. 61, 63
• Figures 2 and 3 show sample filtering results for the baboon and Native American images, respectively.Source: Figure 2 · p. 61, Figure 3 · p. 62, FIGURE LEGEND · pp. 64–65
• The listed metric triplets are MAE 3.017, MSE 124.358, NCD 0.019; MAE 3.031, MSE 123.020, NCD 0.019; and MAE 3.726, MSE 171.007, NCD 0.023.Source: FIGURE LEGEND · p. 62