Selftuning fast adaptive algorithm for impulsive noise suppression in color images
Abstract
In this paper, a selftuning version of the newly introduced Fast Adaptive Switching Trimmed Arithmetic Mean Filter, which is a very efficient technique for impulsive noise suppression, is elaborated. Most of the methods presented in the rich literature have numerous parameters, whose proper settings are crucial for efficient noise suppression. Although researchers often provide recommended values for their algorithms’ parameters, the actual choice remains in the hands of the user. Our goal is to free the operator from parameter selection dilemma and to propose an algorithm which includes required expert knowledge within itself. The only obligatory inputs of the proposed algorithm (from the user perspective) are the image itself and the size of the operating window.
Keywords
Impulsive noise reduction Color image enhancement and restoration Image quality Adaptive algorithm Switching filter1 Introduction
Rapid development of miniaturized highresolution, lowcost image sensors, dedicated to operate in various lighting conditions, makes image enhancement and noise suppression to be very important operations of digital image processing.

electric signal instabilities,

physical imperfections in sensors,

corrupted memory locations,

transmission errors,

aging of the storage material,

natural or artificial electromagnetic interferences.
Impulsive noise removal techniques are contextual processing schemes which estimate the channels of the processed pixel using information obtained from its neighborhood, represented by a sliding operational window. Many of them are based on a vectorordering scheme [10, 11, 12, 13, 14], and use cumulative distances between samples in a window as dissimilarity estimates. Those accumulated distances are then sorted and constitute the basis for further processing in various filtering algorithms.
One of the most basic filtering techniques, utilizing this ordering scheme, is the vector median filter (VMF) [10, 15]. The output of VMF is the pixel from operational window for which the sum of distances to other samples from the window is minimized. Although this filter does not introduce any new colors to the processed image, there is no guarantee that the output pixel is itself noisefree, and thus, numerous solutions were developed to solve this problem and improve filtering performance [16, 17, 18, 19, 20, 21].
The main reason, that the efficiency of vectorordering schemes is limited, lies in processing of every image pixel, regardless whether it is corrupted or not. Unnecessary processing of noisefree pixels results in inevitable degradation of the image quality. To address this issue, a significant improvement has been made by introduction of more sophisticated switching filters [22, 23, 24, 25, 26, 27, 28, 29, 30, 31], which focus on the restoration of corrupted pixels only.
The switching techniques use various approaches to determine if the processed pixel is corrupted or not. Then, only those classified as noisy are further processed by the output estimation algorithm. This way, not only the quality of output of restored image is preserved, but also a significant reduction of the computational cost is often achieved.
1.1 Notation

\(\varvec{X}\)—input image (corrupted),

\(\varvec{x}_{u,v}\)—input image pixel located at spatial coordinates (u, v),

\(\varvec{\hat{X}}\)—output image (restored),

\(\hat{\varvec{x}}_{u,v}\)—output image pixel located at (u, v),

\(\varvec{O}\)—original image (reference image),

\(\varvec{o}_{u,v}\)—original image pixel located at (u, v),

M—original map of noise acquired using artificial image contamination.

\(m_{u,v}\)—real state of pixel corruption located at (u, v) (0—noisy; 1—noisefree),

\(\hat{M}\)—final estimated map of noise acquired during noise detection phase.

\(\hat{m}_{u,v}\)—classification of pixel contamination located at u, v (0—noisy; 1—noisefree).

\(\varvec{W}\)—local operating window centered at \(\varvec{x}_{u,v}\), containing pixels from direct 8–neighborhood,

\(\varvec{x}_{i}\)—ith pixel of the local operating window \(\varvec{W}\) (the pixel \(\varvec{x}_{1}\) is the central pixel in \(\varvec{W}\)),

w—size of \(\varvec{W}\) (odd integer),

n—number of pixels in \(\varvec{W}\) (\(n=w \times w=w^2\)),

\(d(\varvec{x}_i,\varvec{x}_j)\)—distance between two pixels from \(\varvec{W}\),

\(\delta _{i}\)—distance between central pixel \(\varvec{x}_1\) and \(\varvec{x}_{i}\in \varvec{W}\),

\(\delta _{(r)}\)—rth smallest distance among all \(\delta _{i}\) computed for the same \(\varvec{W}\),

\(c_{u,v}\)—sum of \(\alpha\) smallest distances, representing raw impulsiveness of particular \(\varvec{x}_{u,v}\),

\(W_c\)—window containing values of raw impulsiveness computed for every pixel from local neighborhood of currently processed pixel \(\varvec{x}_{u,v}\),

\(c_{\min }\)—smallest accumulated distance in \(W_c\) representing simple estimate of image structure,

\(s_{u,v}\)—corrected impulsiveness measure of pixel at position u, v,

k—iteration number in selftuning (ST) procedure,

l—iteration number in multiple run test,

t—threshold value (filter parameter),

\(t_k\)—threshold value adjusted in the kth iteration of the selftuning,

\(\text {AMF}(\varvec{W})\)—output of the Arithmetic Mean Filter,

\(\rho\)—true noise density used in artificial image contamination.

\(\hat{\rho }\)—estimated noise density obtained through noise detection phase,

\(\hat{\rho }_k\)—estimated noise density obtained in the kth iteration of self tuning procedure,

\(\rho _R, \rho _G, \rho _B\)—probability of contamination of channels in RGB color space,

\(\rho _A\)—probability of contamination of all pixel channels at once,

\(\hat{M}_k\)—estimated map of noise obtained during kth iteration of selftuning,

\(\mu\),\(\nu\) —height and width of the image (in pixels),

\(\theta\)—number of pixels in \(\varvec{X}\) (\(\varTheta = \mu \times \nu\)),

\(n_k\)—number of pixels designated as noisy during kth iteration of selftuning,

p—probability of error in statistical reasoning (result of statistical test which allows to hold or reject a null hypothesis).

\(k_{F}\)—final number of iterations of selftuning procedure, required to satisfy the convergence condition.
1.2 Impulsive noise models

Channel Together Random Impulse (CTRI)—if a pixel is noisy, all of its RGB channels are corrupted.

Channel Independent Random Impulse (CIRI)—if a pixel is contaminated, the alteration of every channel is independent.

Channel Correlated Random Impulse (CIRI)—if a pixel is contaminated, then the corruption of channels is correlated with fixed correlation coefficient.

Custom Probability Random Impulse (CPRI)—if a pixel is contaminated, there is a fixed set of probabilities that single RGB channels are corrupted (\(\rho _R,\rho _G,\rho _B\)) or that all channels are corrupted together (\(\rho _A=1(\rho _R+\rho _G+\rho _B)\)). The model does not take into account the corruption of two channels at once.
1.3 Performance measures

True positive (TP)—pixel was correctly recognized as being contaminated.

False positive (FP)—pixel was falsely classified as noisy—also known as TypeI error.

True negative (TN)—pixel was correctly recognized as not corrupted.

False negative (FN)—pixel was incorrectly classified as not noisy—also known as TypeII error.
In addition, the FeatureSIMilarity index for color images (FSIMc), [57] was used to provide additional information about noise suppression performance. In contrast to Peak SignaltoNoise Ratio (PSNR), Mean Absolute Error (MAE), and Normalized Color Difference (NCD), which operate on individual pixels, thus compare images in contextfree manner, the FSIMc index, is based on the properties of the human visual system.
2 Original FASTAMF algorithm
 1.
noise detection—the noise map (\(\hat{M}\)) is estimated upon input image (\(\varvec{X}\)), using reduced ordering scheme and two parameters provided by the user: operating window size (w) and threshold (t).
 2.
pixel replacement—the output image (\(\varvec{\hat{X}}\)) is obtained using input image (\(\mathbf {X}\)), and noise map (\(\hat{M}\)) provided by noise detection phase. Only pixels classified as noisy are processed by AMF with operating window size w.
2.1 Noise detection
 I.Evaluation of pixels impulsiveness begins with computation of dissimilarity measure \(d(\varvec{x}_1,\varvec{x}_j)\) between the central pixel and every other pixel contained in W, denoted as \(\delta _{i}\). Originally, the Euclidean distance was used, but many other dissimilarity measures can be used instead [11]. For example, in [58], the authors show that the use of Chebyshev distance (\(L_\infty\)) improves detection performance, because this way algorithm is then more sensitive to outliers occurring on individual channels. Next, distances \(\delta _i\) (excluding \(\delta _1\) which is equal to 0) are sorted in ascending order: \(\delta _{2}, \ldots , \delta _{n} \longrightarrow \delta _{(1)}, \ldots , \delta _{(n1)},\) and the trimmed sum of \(\alpha =2\) smallest distances is computed for pixel \(\varvec{x}_{u,v}\):The \(c_{u,v}\) can be interpreted as raw impulsiveness of the pixel (Fig. 3).$$\begin{aligned} c_{u,v} = \sum _{r=1}^{\alpha } \delta _{(r)}. \end{aligned}$$(7)
 II.Adaptation to local image variation is performed. For every \(c_{u,v}\), a window \(W_{c}\), containing n values \(c_{i}\), is taken, so that \(c_{1}=c_{u,v}\) is in the center of that window. The final corrected measure of corrected pixel impulsiveness (Fig. 4) assigned to pixel \(\varvec{x}_{u,v}\) is obtained:where \(c_{\min } = \min \{c \in W_c\}\). This correction normalizes impulsiveness based on the local image variation. In homogeneous image regions \(c_{\min }\) is close to 0 and it rises together with variation in local neighborhood. As a result, the pixels of high raw impulsiveness in harsh regions of the image are less likely to be classified as noisy pixels of the same raw impulsiveness in smooth areas.$$\begin{aligned} s_{u,v} = c_{u,v}c_{\min }, \end{aligned}$$(8)
 III.Noise map acquisition finalizes the noise detection phase, during which the estimated noise map \(\hat{M}\) is obtained. It is achieved by the comparison of \(s_{u,v}\) to the threshold t, provided by the user, for every pixel in the image \(\varvec{X}\) as follows:The labeling of noisy pixels as 0 in \(\hat{M}\) is needed for the subsequent pixel replacement phase.$$\begin{aligned} \hat{m}_{u,v}= \left\{ \begin{array}{ll} 0 &{} \text { if } \; s_{u,v}> t, \\ 1 &{} \; \text {otherwise}. \end{array} \right. \end{aligned}$$(9)
2.2 Pixel replacement
3 Selftuning
As it has been shown in Fig. 1, there are three inputs for the algorithm: processed image (\(\varvec{X}\)), threshold (t), and operational window size (w). As long as w is intuitive parameter to adjust, the proper choice of t may be a difficult one. It was shown in [54, 58] that optimal choice of t is dependent on impulsive noise density (\(\rho\)), which is mostly unknown in realcase scenarios, so the operator is forced to experimental search of adequate value of t.
To free the user from manual adjusting of this parameter, a selftuning modification is introduced. The main concept of this improvement is to use the estimated noise map \(\hat{M}\) (obtained during noise detection phase) to compute the estimated noise density \(\hat{\rho }\). Combining \(\hat{\rho }\) with proper tuning characteristics like provided in [58] enables to adjust the t value, which can be used to obtain more accurate noise map.
3.1 Algorithm
Optimal and recommended t values for CIRI noise model
\(\rho\) (%)  Mean (standard deviation) of optimal t obtained using  t  

ACC  PSNR  MAE  NCD (\(10^{4}\))  FSIMc  
0.1  111.82 (29.95)  114.57 (38.26)  117.88 (36.22)  100.57 (28.85)  110.63 (36.86)  111 
1  79.73 (21.57)  82.10 (24.91)  85.61 (23.58)  68.75 (19.59)  78.13 (24.55)  79 
5  56.87 (15.77)  63.92 (18.25)  65.65 (17.60)  50.04 (13.51)  62.29 (17.57)  60 
10  47.73 (13.11)  56.99 (15.44)  58.75 (15.19)  43.11 (11.58)  56.10 (14.85)  53 
15  42.51 (11.07)  53.34 (14.33)  54.29 (14.45)  39.26 (9.60)  52.59 (13.81)  48 
20  39.22 (10.06)  49.90 (12.83)  51.39 (12.75)  36.70 (9.24)  49.88 (12.38)  45 
25  36.43 (8.90)  47.44 (12.28)  49.00 (12.07)  34.58 (8.18)  48.21 (11.97)  43 
30  34.52 (8.21)  45.50 (11.40)  46.97 (11.11)  32.77 (7.38)  46.20 (11.95)  41 
35  32.70 (7.25)  43.12 (10.30)  44.98 (10.60)  31.10 (6.75)  44.21 (11.18)  39 
40  31.04 (6.79)  41.10 (9.73)  43.17 (9.72)  29.56 (6.28)  42.27 (10.67)  37 
45  29.73 (6.09)  38.28 (9.39)  41.32 (9.44)  27.63 (5.91)  40.02 (10.72)  35 
50  28.43 (5.40)  35.44 (8.64)  38.74 (9.00)  25.38 (5.46)  37.07 (10.38)  33 
55  26.99 (4.72)  31.83 (8.40)  36.08 (8.46)  22.69 (5.25)  33.87 (10.56)  30 
60  25.24 (4.00)  27.65 (8.38)  32.52 (8.73)  19.41 (4.80)  29.74 (10.65)  27 
65  22.60 (3.31)  22.25 (7.77)  27.78 (8.45)  15.66 (4.54)  25.67 (10.53)  23 
70  19.10 (2.63)  17.56 (7.03)  23.12 (7.87)  12.19 (4.06)  21.44 (10.57)  19 
75  14.45 (2.09)  12.90 (6.24)  18.42 (7.29)  9.55 (3.45)  17.06 (9.89)  14 
80  9.24 (1.57)  9.07 (4.68)  14.25 (6.44)  7.49 (2.78)  13.54 (9.33)  11 
Optimal and recommended t values for CCRI noise model
\(\rho\) (%)  Mean (standard deviation) of optimal t obtained using  t  

ACC  PSNR  MAE  NCD (\(10^{4}\))  FSIMc  
0.1  111.19 (29.58)  115.77 (38.76)  119.12 (36.95)  100.27 (29.95)  109.24 (37.77)  111 
1  78.33 (21.76)  82.59 (24.95)  84.97 (23.81)  66.86 (18.62)  77.47 (24.69)  78 
5  54.62 (15.90)  63.08 (18.69)  65.42 (18.34)  47.88 (13.56)  61.22 (17.73)  58 
10  44.80 (12.70)  55.26 (15.60)  57.06 (15.67)  40.81 (10.95)  54.54 (15.19)  50 
15  39.58 (10.96)  51.05 (14.56)  53.09 (14.58)  36.98 (9.78)  50.92 (14.01)  46 
20  36.23 (9.75)  47.86 (13.38)  49.94 (13.35)  34.34 (9.02)  48.69 (13.05)  43 
25  33.33 (8.74)  45.53 (12.14)  47.63 (12.42)  32.08 (7.93)  45.96 (12.46)  41 
30  31.07 (7.69)  42.49 (11.53)  45.30 (11.79)  30.04 (6.93)  43.65 (11.78)  39 
35  29.14 (6.82)  40.28 (10.63)  43.13 (11.12)  28.62 (6.63)  41.78 (11.58)  37 
40  27.56 (6.25)  38.17 (10.30)  41.49 (10.69)  26.83 (6.23)  39.54 (11.20)  35 
45  26.04 (5.65)  35.96 (9.97)  39.58 (9.94)  25.18 (5.56)  37.84 (10.59)  33 
50  24.56 (5.06)  33.15 (9.10)  37.06 (9.46)  23.01 (5.18)  35.18 (10.96)  31 
55  23.14 (4.50)  30.12 (8.89)  34.97 (8.97)  20.61 (4.88)  32.80 (10.61)  28 
60  21.56 (3.62)  27.18 (8.50)  32.18 (8.75)  18.17 (4.69)  29.91 (10.91)  26 
65  19.30 (3.04)  23.34 (8.12)  29.12 (8.74)  15.72 (4.65)  26.41 (10.89)  23 
70  16.41 (2.50)  19.31 (7.34)  25.56 (8.28)  12.86 (4.17)  23.41 (10.74)  20 
75  12.57 (1.88)  15.43 (6.83)  22.16 (8.30)  10.53 (3.71)  20.33 (10.98)  16 
80  8.27 (1.35)  11.77 (5.78)  18.78 (7.85)  8.39 (3.16)  17.39 (10.64)  13 
 (a)
The estimated map of noise for the current iteration \(\hat{M}_k\) is obtained by (9) using \(t_k\).
 (b)
The estimated noise density for the current iteration \(\hat{\rho }_k\) is evaluated: \(\hat{\rho }_k = n_k/\theta\), where \(n_k\) is the number of pixels designated as corrupted (in kth iteration) and \(\theta\) is the number of pixels in image X.
 (c)
\(t_{k+1}\) value is interpolated (simple linear interpolation between two closest values) using tuning tables (see Table 4), which were obtained using procedure presented in Sect. 3.2.
Optimal and recommended t values for CTRI noise model
\(\rho\) (%)  Mean (standard deviation) of optimal t obtained using  t  

ACC  PSNR  MAE  NCD (\(10^{4}\))  FSIMc  
0.1  114.48 (29.20)  115.07 (34.46)  115.96 (32.98)  104.66 (26.34)  110.09 (33.29)  112 
1  87.78 (19.69)  85.11 (22.69)  87.03 (21.39)  76.35 (18.30)  81.69 (22.64)  84 
5  68.94 (14.99)  68.53 (16.50)  69.89 (15.53)  60.62 (13.01)  66.30 (15.47)  67 
10  61.22 (12.57)  61.71 (13.63)  62.60 (12.98)  54.39 (11.36)  60.90 (13.85)  60 
15  57.18 (11.21)  57.87 (12.01)  58.95 (11.70)  50.66 (9.61)  56.67 (11.66)  56 
20  54.24 (9.93)  54.38 (10.23)  55.74 (10.16)  47.90 (8.95)  53.69 (10.73)  53 
25  52.10 (8.88)  51.83 (9.15)  53.48 (9.37)  46.03 (7.43)  51.35 (9.87)  51 
30  50.58 (8.25)  49.74 (8.36)  51.38 (8.25)  44.13 (7.23)  49.02 (9.20)  49 
35  49.49 (7.27)  46.84 (8.28)  49.05 (8.06)  42.01 (6.66)  46.63 (9.23)  47 
40  48.58 (6.61)  43.01 (7.56)  46.00 (7.81)  39.15 (6.30)  43.14 (9.36)  44 
45  47.91 (5.35)  37.86 (8.48)  42.05 (8.13)  34.57 (6.14)  38.01 (10.30)  40 
50  46.89 (4.81)  30.52 (8.83)  36.15 (8.48)  28.72 (6.53)  31.71 (10.34)  35 
55  44.54 (4.21)  22.62 (7.83)  28.61 (8.59)  21.78 (5.89)  24.26 (9.87)  28 
60  39.82 (4.78)  15.16 (5.74)  20.22 (7.42)  15.37 (4.62)  17.63 (8.46)  22 
65  31.72 (4.73)  9.81 (3.81)  13.43 (5.11)  10.70 (3.25)  12.27 (6.01)  16 
70  22.66 (4.48)  6.29 (2.16)  8.93 (3.17)  7.59 (2.10)  8.36 (4.51)  11 
75  14.52 (3.14)  4.47 (1.29)  6.42 (1.92)  5.89 (1.42)  5.73 (3.18)  7 
80  8.20 (1.83)  3.46 (0.78)  4.96 (1.20)  4.65 (1.03)  3.00 (2.64)  5 
Tuning values for threshold t
\(\rho\) (%)  0.1  1  5  10  20  25  30  35  40  45  50  55  60  65  70  75  80 

t  111  80  61  54  50  47  45  43  41  38  36  33  28  25  20  16  12 
Final estimated map of noise, denoted as \(\hat{M}\), is then taken as an input to the pixel replacement phase. It is important that only the final step of the entire noise detection phase (9) has to be recursively repeated, so the increase in computational cost is not significant.

It is very intuitive to choose w value, and using windows larger than \(3\times 3\) is reasonable for high noise intensities only (\(\rho>50\%\)).

Window size w has a critical impact on computational cost of the algorithm, so its automatic onthefly tuning certainly makes its execution time extremely unpredictable.

Alteration of the w during algorithm’s execution requires repetition of the entire noise detection phase, which is very costly form computational point of view. Therefore, such tuning algorithm would be inapplicable for realtime image processing tasks.

Preliminary tests (omitted in the paper) revealed that w has stronger impact on noise detection phase performance than on pixel replacement phase. Therefore, partial solution, assuming the use of altered onthefly w for pixel replacement phase only, resulted in lack of restored image quality improvement.
3.2 Tuning tables
The core of the selfadjusting threshold t modification is the tuning Table 4 which provides the t values for interpolation step (c). Originally, this table was proposed for CTRI and CIRI noise models in [58]; however, in this paper, more thorough experiments were performed, to obtain more general insight into the problem.
The set of 100 color images has been taken as the training set (Fig. 6) [59]. Each of those images was artificially contaminated with CIRI, CCRI (correlation coefficient set to 0.5) and CTRI model for noise densities \(\rho \in \left\{ 0.1, 1, 5, 10, 15, \ldots , 80\%\right\}\). Finally, for each contaminated image, the optimization has been performed to find optimal value of t for which ACC, PSNR, and FSIMc are maximal and MAE and NCD is minimal.
The mean values (and standard deviations) of optimal t, computed upon entire set of training images for a chosen performance measure, noise model, and noise density are presented in Tables 1, 2, and 3. The final proposition of general tuning values obtained as an average of results from all experiments is shown in Table 4.
4 Noise suppression performance
Our aim was to provide the most objective noise suppression performance test; therefore, a new set of ten images was taken as validation set (Fig. 7) [60]. In addition, all of those validation images were contaminated with CPRI noise (\(p_R=p_G=p_B=p_A=0.25\)), which has not been used for obtaining the tuning Table 4. This way, we provided an independent test input, further minimizing the possibility that the tuning values are optimized for particular image set or noise model.
4.1 FASTAMF compared with stateoftheart algorithms

H0: There is no evidence that results for all algorithms are significantly heterogeneous.

H1: There is evidence that results for all algorithms are significantly heterogeneous.

H2: There is no evidence that FASTAMF performs significantly better than the compared algorithm.

H3: There is evidence that FASTAMF performs significantly better than compared algorithm.
Friedman’s test results—FASTAMF compared with stateoftheart filters
\(\rho\) (\(\%\))  Mean ranks for  H1  

FASTAMF  ACWVMF  FAPGF  FFNRF  FPGF  
Friedman’s test for PSNR  
10  5.0  3.8  2.9  1.5  1.8  \(p<0.001\) 
20  4.9  3.0  3.0  1.4  2.7  \(p<0.001\) 
30  4.9  2.0  3.1  1.7  3.3  \(p<0.001\) 
40  5.0  1.4  3.7  2.0  2.9  \(p<0.001\) 
50  4.9  1.2  4.0  2.0  2.9  \(p<0.001\) 
Friedman’s test for MAE  
10  1.1  2.0  3.4  3.7  4.8  \(p<0.001\) 
20  1.0  2.2  3.7  3.5  4.6  \(p<0.001\) 
30  1.0  3.5  3.6  2.9  4.0  \(p<0.001\) 
40  1.0  4.2  3.2  2.8  3.8  \(p<0.001\) 
50  1.0  4.4  2.8  2.7  4.1  \(p<0.001\) 
Friedman’s test for NCD  
10  1.1  2.1  3.5  4.2  4.1  \(p<0.001\) 
20  1.0  2.8  3.6  4.3  3.3  \(p<0.001\) 
30  1.0  4.2  3.1  3.8  2.9  \(p<0.001\) 
40  1.0  4.7  2.9  3.6  2.8  \(p<0.001\) 
50  1.0  4.9  2.8  3.6  2.7  \(p<0.001\) 
Friedman’s test for FSIMc  
10  5.0  3.3  3.5  1.5  1.7  \(p<0.001\) 
20  4.9  2.7  3.5  1.6  2.3  \(p<0.001\) 
30  4.9  1.7  3.3  2.3  2.8  \(p<0.001\) 
40  5.0  1.6  3.3  2.5  2.6  \(p<0.001\) 
50  4.9  1.1  3.4  3.1  2.5  \(p<0.001\) 
Post hoc test results—FASTAMF compared with stateoftheart filters (emboldened results speak against FASTAMF superiority)
\(\rho\) (\(\%\))  Hypothesis (p) for  

ACWVMF  FAPGF  FFNRF  FPGF  
Post hoc tests for PSNR  
10  H2 (\(>0.05\))  H3 (\(<0.01\))  H3 (\(<0.01\))  H3 (\(<0.01\)) 
20  H3 (\(<0.05\))  H3 (\(<0.05\))  H3 (\(<0.01\))  H3 (\(<0.01\)) 
30  H3 (\(<0.01\))  H3 (\(<0.05\))  H3 (\(<0.01\))  H3 (\(<0.05\)) 
40  H3 (\(<0.01\))  H2 (\(>0.05\))  H3 (\(<0.01\))  H3 (\(<0.01\)) 
50  H3 (\(<0.01\))  H2 (\(>0.05\))  H3 (\(<0.01\))  H3 (\(<0.01\)) 
Post hoc tests for MAE  
10  H2 (\(>0.05\))  H3 (\(<0.01\))  H3 (\(<0.01\))  H3 (\(<0.01\)) 
20  H2 (\(>0.05\))  H3 (\(<0.01\))  H3 (\(<0.01\))  H3 (\(<0.01\)) 
30  H3 (\(<0.01\))  H3 (\(<0.01\))  H3 (\(<0.05\))  H3 (\(<0.01\)) 
40  H3 (\(<0.01\))  H3 (\(<0.01\))  H3 (\(<0.05\))  H3 (\(<0.01\)) 
50  H3 (\(<0.01\))  H3 (\(<0.05\))  H3 (\(<0.05\))  H3 (\(<0.01\)) 
Post hoc tests for NCD  
10  H2 (\(>0.05\))  H3 (\(<0.01\))  H3 (\(<0.01\))  H3 (\(<0.01\)) 
20  H3 (\(<0.05\))  H3 (\(<0.01\))  H3 (\(<0.01\))  H3 (\(<0.01\)) 
30  H3 (\(<0.01\))  H3 (\(<0.01\))  H3 (\(<0.01\))  H3 (\(<0.05\)) 
40  H3 (\(<0.01\))  H3 (\(<0.05\))  H3 (\(<0.01\))  H3 (\(<0.05\)) 
50  H3 (\(<0.01\))  H3 (\(<0.05\))  H3 (\(<0.01\))  H3 (\(<0.05\)) 
Post hoc tests for FSIMc  
10  H3 (\(<0.05\))  H2 (\(>0.05\))  H3 (\(<0.01\))  H3 (\(<0.01\)) 
20  H3 (\(<0.01\))  H2 (\(>0.05\))  H3 (\(<0.01\))  H3 (\(<0.01\)) 
30  H3 (\(<0.01\))  H3 (\(<0.05\))  H3 (\(<0.01\))  H3 (\(<0.01\)) 
40  H3 (\(<0.01\))  H3 (\(<0.05\))  H3 (\(<0.01\))  H3 (\(<0.01\)) 
50  H3 (\(<0.01\))  H2 (\(>0.05\))  H3 (\(<0.05\))  H3 (\(<0.01\)) 
The results of above tests are summarized in Table 5 (Friedman’s tests) and in Table 6 (Post hoc tests). All emboldened values in the tables do not support the superiority of the FASTAMF (as those are in minority). In case of PSNR and FSIMc measures, the better value is the higher one, so the higher mean rank values support the superiority of particular algorithm. The opposite situation is for MAE and NCD measures which are better if smaller.

The results obtained from all algorithms (represented by quality measures) were always heterogeneous (H0 was discarded in favor of H1 in every case). In addition, p in each Friedman’s test was very low, so the differences in results are unquestionably significant.

For every measure and noise density, the best mean ranks were observed for FASTAMF algorithm, which means that it was the best or almost the best for every tested image.

Only very few of Post hoc tests resulted in favor of H2 hypothesis. For those rare cases, we can state that FASTAMF is not significantly better than the compared algorithm. For each other case, however, it is the best performing algorithm among tested.

For lownoise densities, the ACWVMF tends to be a competitive choice for FASTAMF, while, for higher noise contamination ratios, the FAPGF provides the most similar results.
4.2 Selftuning FASTAMF against original FASTAMF

H0: There is not enough evidence that STF provides significantly better results.

H1: There is enough evidence that STF provides significantly better results.

Larger Positive Sums of Ranks for PSNR ad FSIMc quality measures indicate that STF algorithm performs better (values of measure are more frequently higher). Such outcome can be observed for \(\rho \ge 20\%\), for both measures.

Smaller Positive Sums of Ranks for NCD and MAE measures indicate that STF algorithm performs better (values of measure are more frequently lower). Such outcome can be observed for MAE when \(\rho \ge 40\%\) and for all results evaluated with NCD.

For \(\rho \ge 40\%\) the STF algorithm performs significantly better than OF form NCD and PSNR point of view.
4.3 Multirun and visual comparison
One of the common approaches to achieve good noise suppression performance is to repeat the processing of the noisy picture several times, using output image as an input for next algorithm’s execution. This way noisy pixels omitted during first filtering may be detected and restored during subsequent runs. However, this approach may lead to stronger degradation of image details, especially if the algorithm has adaptive features.
The noise suppression scheme (further referenced as multirun or MR) was performed for three iterations (\(l=1,2,3\)) upon all the validating images for noise densities \(\rho \in \left\{ 10, 30, 50\right\}\), and the four representative images were selected for detailed comparison (validation images 1, 7, 8, and 9).
The efficiency of both algorithms in terms of PSNR and FSIMc measures is presented in Table 8. In addition, the visual comparison of performance for both filtering schemes for \(l=1\), \(l=3\) and \(\rho = 30\%\) is depicted in Figs. 10, 11, 12, and 13.
Wilcoxon’s tests for STF and OF comparison (emboldened results denotes a significant superiority of ST)
\(\rho\) (\(\%\))  Sum of ranks  Hypothesis (p)  

Positive  Negative  
Wilcoxon’s test for PSNR  
10  18  37  H0 (\(>0.05\)) 
20  31  24  H0 (\(>0.05\)) 
30  39  16  H0 (\(>0.05\)) 
40  46  9  H1 (\(<0.05\)) 
50  51  4  H1 (\(<0.05\)) 
Wilcoxon’s test for MAE  
10  43  12  H0 (\(>0.05\)) 
20  36  19  H0 (\(>0.05\)) 
30  31  24  H0 (\(>0.05\)) 
40  25  30  H0 (\(>0.05\)) 
50  21  34  H0 (\(>0.05\)) 
Wilcoxon’s test for NCD  
10  26  29  H0 (\(>0.05\)) 
20  19  36  H0 (\(>0.05\)) 
30  11  44  H0 (\(>0.05\)) 
40  2  53  H1 (\(<0.05\)) 
50  0  55  H1 (\(<0.05\)) 
Wilcoxon’s test for FSIMc  
10  16  39  H0 (\(>0.05\)) 
20  28  27  H0 (\(>0.05\)) 
30  32  23  H0 (\(>0.05\)) 
40  39  16  H0 (\(>0.05\)) 
50  43  12  H0 (\(>0.05\)) 
Multirun test (emboldened results are superior in each contamination rate)
l  Filter  Image 1  Image 7  Image 8  Image 9  

t  PSNR  FSIMc  t  PSNR  FSIMc  t  PSNR  FSIMc  t  PSNR  FSIMc  
\(\rho\)=10%  
1  OF  60.0  37.90  0.9975  60.0  26.99  0.9849  60.0  40.09  0.9979  60.0  24.05  0.9645 
2  37.83  0.9976  26.50  0.9824  40.40  0.9981  23.34  0.9571  
3  37.73  0.9975  26.37  0.9818  40.41  0.9981  23.12  0.9548  
1  STF  56.3  38.01  0.9976  52.8  26.37  0.9822  56.3  40.31  0.9980  52.3  23.41  0.9597 
2  111.0  38.06  0.9977  111.0  26.29  0.9817  111.0  40.55  0.9982  109.1  23.20  0.9573  
3  111.0  38.04  0.9977  111.0  26.29  0.9817  111.0  40.55  0.9982  111.0  23.16  0.9569  
\(\rho\)=30%  
1  OF  60.0  31.19  0.9862  60.0  24.05  0.9672  60.0  32.25  0.9848  60.0  21.51  0.9378 
2  32.16  0.9886  23.96  0.9661  34.11  0.9898  21.21  0.9332  
3  32.18  0.9887  23.90  0.9654  34.15  0.9898  21.10  0.9312  
1  STF  44.7  31.90  0.9883  42.9  23.32  0.9596  44.7  33.89  0.9905  42.4  20.68  0.9256 
2  111.0  32.37  0.9894  109.6  23.33  0.9595  111.0  35.04  0.9925  108.0  20.62  0.9243  
3  111.0  32.38  0.9894  111.0  23.33  0.9596  111.0  35.05  0.9926  111.0  20.61  0.9242  
\(\rho\)=50%  
1  OF  60.0  26.31  0.9562  60.0  21.45  0.9374  60.0  26.84  0.9375  60.0  19.63  0.9049 
2  27.96  0.9656  21.86  0.9397  29.16  0.9583  19.70  0.9050  
3  28.07  0.9661  21.87  0.9393  29.34  0.9594  19.66  0.9044  
1  STF  36.4  27.74  0.9640  33.2  21.16  0.9265  36.3  29.42  0.9674  32.5  19.07  0.8902 
2  105.7  28.52  0.9676  100.9  21.27  0.9271  105.4  30.80  0.9742  101.9  19.09  0.8900  
3  111.0  28.54  0.9677  111.0  21.28  0.9272  111.0  30.84  0.9744  111.0  19.10  0.8901 
The number of iterations (\(k_{\text {F}}\)), required to achieve the ST convergence
\(k_{\text {F}}\)  \(\rho\)  

0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  
Min.  3  3  3  3  4  4  4  4 
Med.  3  3  3  3  4  4  5  4 
Max.  3  3  3  4  5  6  6  5 
We can observe that although the selftuning feature of the algorithm is very convenient and may achieve a better statistical performance, especially if the noise density is unknown or nonstationary, it may achieve slightly inferior efficiency than fixed t value version for more complicated images.

the OF removes less noisy pixels for \(l=1\) than its STF counterpart (Fig. 10c, d), due to higher value of t. If the reason behind those leftovers is high variance of the local area, those are mostly removed in subsequent iterations (Fig. 10e). If a too high t value caused this omission, those will not be restored, no matter how many iterations will be performed. In addition, fixed t value makes OF completely insensitive to less explicit noisy pixels, which is reflected in numerical (PSNR) and structural (FSIMc) measures;

the STF scheme removes more noisy pixels in the first execution (Fig. 10d), strongly decreasing the local variance of the image. Consequently, it is easier to remove omitted noisy pixels in subsequent iterations (Fig. 10f), and there is also lower count of less explicit noisy pixels due to lower t in the first run;

the STF scheme tends to remove more details (Fig. 11d, f) from the image (it is more blurry), than OF (Fig. 11c, e). As long as it is hard to be noticed visually without zoom, it clearly affects numerical (PSNR) and also structural (FSIMc) measures;

undetected by OF noisy pixels during the first execution (Fig. 12c), may cause lowlevel distortions around them, which will not be repaired in subsequent iterations (Fig. 12e). Such phenomenon is far less noticeable if STF scheme is used (Fig. 12d, f).

STF scheme tends to remove more details in the most difficult cases (Fig. 13d, f), which is reflected mostly by the PSNR measure.
The visual comparison shows that STF algorithm seems to always achieve a better noise suppression efficiency (less explicit leftovers can be noticed). Therefore, lower PSNR values might be caused by very subtle differences, which can be detected on the numerical level only.
5 Efficiency
5.1 Computational complexity
 1.
Estimation of map of noise \(\hat{M}_k\) which needs \(\mu \times \nu\) comparisons (COMPS). This step has linear complexity.
 2.
Estimation of noise density \(\hat{\rho }_k\) for which \(\mu \times \nu\) additions (ADDS) and one division (DIVS) are necessary. This step also has linear complexity.
 3.Linear interpolation of t:where A and B are the nearest indicates of values in Table 4 for which \(\rho _A \le \hat{\rho }_k \le \rho _B\). It demands 5 subtractions (SUBS), 2 divisions (DIVS), 2 multiplications (MULTS), and 1 addition (ADDS) and up to 18 COMPS (required for determination for A and B). This step is not image sizedependent, so it can be treated as step with constant computational complexity.$$\begin{aligned} t_k=\frac{t_At_B}{\rho _A\rho _B}\hat{\rho }_k+\left( t_A \frac{t_At_B}{\rho _A\rho _B}\rho _A\right) , \end{aligned}$$(11)
 4.
Convergence condition check which also has constant computational complexity and requires one subtraction, and two comparisons.
The computational cost of single iteration of ST modification is not very heavy and is linearly dependent on image size. The number of iteration required to archive final t values is fairly low and predictable, so this modification is a suitable addition to FASTAMF algorithm in terms of realtime image processing requirements.
5.2 Experimental comparison
The execution time and noise suppression efficiency of FASTAMF with ST modification has been compared to the original FASTAMF (with recommended \(t=60\)). In tests, all ten images’ validation set (Fig. 7) contaminated with CPRI model and noise densities \(\rho \in \left\{ 10, 20, 30\%\right\}\) is used.
The noise suppression efficiency has been evaluated by PSNR, MAE, NCD, and FSIMc measures. Since the tuning tables were obtained as the tradeoff between those measures, in Fig. 14, the most favorable (NCD) and most adverse (PSNR) outcomes of using ST modification were presented. On vertical axis, the difference in particular measure is presented, while, on horizontal one, the change of execution time (in percentages) is exhibited. The point (100,0) refers to all results obtained using original FASTAMF algorithm, and marked points represent results obtained for ST version using ten test images.
Interestingly, in individual cases for \(\rho = 10\%\), the ST version might be even faster than original algorithm, because the threshold value is calculated to be higher then \(t=60\). As a consequence, fewer pixels are recognized as noisy, and noise suppression (AMF) has less work to do.
Also the major conclusion is that results for ST modification become better, along with increasing noise density.
6 Summary
The achieved denoising results are very satisfactory, since the reduction of the number of FASTAMF parameters and a better overall performance have been the main goal of this research. The new selftuning FASTAMF, achieves slightly better or at least not worse overall performance than the original algorithm, yet it has no parameters which require experimental adjustment.
Also the computational cost of the selftuning is not significantly higher, since it works after the most computationally expensive part of the algorithm (estimation of the pixel impulsiveness).
The major virtue of the selftuning FASTAMF is its adaptability to noise density. As it can be useful for filtering of images contaminated by impulsive noise of unknown density, it might be even more advantageous for processing of video sequences distorted by noise with timedependent parameters. The initial value of t (for selftuning mechanism) can be carried out from frame to frame, in such implementations, decreasing the number of potential iterations, required to achieve required convergence. The application of proposed filtering scheme to the video enhancement will be the subject of future work.
Footnotes
 1.
The notation of parameters used in the respective papers was adopted.
Notes
Acknowledgements
This work was supported by a research Grant 2017/25/B/ ST6/02219 from the National Science Centre, Poland, and was also funded by the Silesian University of Technology, Poland (Grants BK 2018).
Author contributions
LBM is an Assistant Professor at the Division of Industrial Informatics. He graduated from Silesian University of Technology and received M.Sc. degree in Automatic Control And Robotics in June 2009. After that he started the Ph.D. studies and performed research in field of identification of bilinear timeseries models. He received his Ph.D. in April 2014. His current main scientific interest is focused on digital image proccesing but also he explores field of multidimentional optimisation for nonlinear problems. BS received the Diploma in Physics degree from the Silesian University, Katowice, Poland, in 1986 and the Ph.D. degree in computer science from the Department of Automatic Control, Silesian University of Technology, Gliwice, Poland, in 1998. From 1986 to 1989, he was a Teaching Assistant at the Department of Biophysics, Silesian Medical University, Katowice, Poland. From 1992 to 1994, he was a Teaching Assistant at the Technical University of Esslingen, Germany. Since 1994, he has been with the Silesian University of Technology. In 1998, he was appointed as an Associate Professor in the Department of Automatic Control. He has also been an Associate Researcher with the Multimedia Laboratory, University of Toronto, Canada since 1999. In 2007, Dr. BS was promoted to Professor at the Silesian University of Technology. He has published over 200 papers on digital signal and image processing in refereed journals and conference proceedings. His current research interests include lowlevel color image processing, humancomputer interaction, and visual aspects of image quality.
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