Multi-Objective Threshold Optimized Image De-Noising Algorithm for High Density Mixed Impulse Noise

Abstract

This paper proposes a novel Multi-Objective Optimization based Fuzzy Switching Median Filter (MOOFASMF) to remove high density Random Valued Impulse Noise (RVIN), “Salt & Pepper” Impulse Noise (SPIN) and Mixed Impulse Noise (MIN). In this work, multi-objective optimization technique is used to find out the fuzzy switching median filter threshold values for accurate detection of corrupted pixels. The proposed multi-objective framework uses Decomposition based Multi Objective Evolutionary Algorithm (MOEA/D) to obtain optimized fuzzy switching median filter drives the threshold values with the objectives Mean Square Error (MSE) and inverse of Structural Similarity Index Metrics (SSIM) as optimization objectives. Even though the MSE and SSIM are not closely related parameters, the optimized threshold value gives better results in terms of both PSNR and SSIM. The advantages of the proposed framework are that it works effectively on RVIN, SPIN, and MIN-affected images. The effectiveness of the proposed framework is outstanding for high-density RVIN, SPIN, and MIN, which makes it more advantageous over other existing methods. Experimental results in terms of visual and quantitative metrics such as Peak Signal to Noise Ratio (PSNR), Mean Square Error (MSE), Structural Similarity Index Metrics (SSIM), and Edge Preservation Index (EPI) clearly demonstrates the better performance of the proposed algorithm over the state of art techniques. The proposed framework performed 6.02% and 32.11% better than the best existing methods in terms of PSNR and SSIM for the mixture of 40% SPIN & 50% RVIN affected image.

Keywords

Fuzzy switching median filter RVIN SPIN multi-objective optimization threshold optimization

1. Introduction

Digital images and videos may undergo different types of degradation during acquisition, processing and transmission (Wang & Lu, 2011). Hence, there is a need for developing novel filters to remove such type of degradation before further processing. Impulse noise is a type of random noise which corrupts certain pixels in the image randomly, and follows a uniform distribution (Brownrigg, 1984). The impulse noise is further classified into fixed valued impulse noise or “Salt & Pepper” Impulse Noise (SPIN) and Random Valued Impulse Noise (RVIN) respectively (Ko & Lee, 1991). SPIN corrupts the image pixels with two different noise values such as 0 and 255 whereas RVIN corrupts the image by any random value in the range of 0 to 255 i.e., dynamic range of the image.

The detection of SPIN corrupted pixels is comparatively easier than the detection of RVIN corrupted pixel. Earlier, filters such as Standard Median Filter (SMF) (Wang & Lu, 2011), Weighted Median Filter (WMF) (Brownrigg, 1984) and Center Weighted Median Filter (CWMF) (Ko & Lee, 1991) etc., are applied directly on the noisy image without any detection of noisy pixel before filtering and those filters works only for low density noise level (Chen & Wu, 2001). Later, filters are proposed to incorporate detection stage which detects the noisy pixels and filtering process is applied only to the detected noisy pixels which leaves the remaining pixels unaltered. Those such filters are more popular because of better edge preservation and high density noise removal (Chen et al., 1999).

In recent days, high density impulse noise removal is more popular (Abreu et al., 1996). There are several state of art works such as Median Type Noise Detector (MTND) (Chan et al., 2005), Boundary Discriminative Noise Detector (BDND) (Ng & Ma, 2006), Decision Based Algorithm (DBA) (Srinivasan & Ebenezer, 2007), Adaptive Switching Median Filter (ASMF) (Akkoul et al., 2010), Modified Decision Based Unsymmetric Trimmed Median Filter (MDBUTMF) (Esakkirajan et al., 2011), Rank Order Absolute Difference (ROAD) (Garnett et al., 2005) and Rank Order Logarithmic Difference (ROLD) (Dong et al., 2007) etc., for the removal of impulse noise. In addition, there are two types of switching based filters namely fixed window based and adaptive window based filters. In order to remove very high noise densities, adaptive window algorithms such as MTND, BDND and ASMF are proposed. The above techniques for impulse noise removal somehow fail either in high noise densities or better edge preservation.

In this paper, an attempt to remove high density SPIN, RVIN and mixture of both noise (Wang et al., 2020). It employs two stage detection and filtering to remove both types of impulse noise Zhang and Li (2007) and Niu and Shen (2007). In the first stage, histogram approach (Toh & Isa, 2010) is used to detect SPIN and second stage a fuzzy membership function is used to detect the RVIN corrupted pixels based on multi objective optimized threshold values (Miettinen et al., 2008).

The threshold optimization itself a significantly wide domain. The wide range of classification in the optimization techniques provides flexibility to choose it based on the context and nature of the problem. The Genghis Khan Shark Optimizer (Hu et al., 2023), Geyser inspired optimization algorithm (Ghasemi et al., 2023), Prairie Dog Optimization Algorithm (Ezugwu et al., 2022), Dwarf Mongoose Optimization Algorithm (Agushaka et al., 2022), Gazelle optimization algorithm (Agushaka et al., 2022), and adaptive hybrid dandelion optimizer (Hu et al., 2023) are the recent popular optimization techniques. Even though, all these optimization techniques are effective they are based on single objective models. The constraint and effectiveness of the optimization are based on the problem. The main objective of our proposed approach is to determine the threshold value for distinguishing noisy pixels. Furthermore, the assessment of the denoised image quality relies on several parameters, including PSNR, MSE, and SSIM. PSNR serves as a conventional measure for evaluating image quality, while SSIM has emerged as a more accurate assessment metric in recent times. Thus, the problem addressed by our proposed scheme necessitates the utilization of a multi-objective optimization technique. Decomposition based Multi-Objective Evolutionary Algorithm (MOEA/D) (Zhang & Li, 2007) is used for optimizing the multiple objectives in finding the optimal threshold values of fuzzy switching filter.

The rest of the paper is structured as the literature review of the related works in Section 2, the multi-objective optimization technique is proposed in 3, the Multi-Objective Optimization based Fuzzy Switching Median Filter (MOOFSMF) is proposed in Section 4, the visual and quantitative results are discussed in Section 5, and concluding remarks are mentioned in Section 6.

2. Related Works

There are different nonlinear and linear filters have been proposed in the literature to remove either SPIN or RVIN. Among these algorithms, some of the popular and related algorithms are outlined in this section. Fuzzy based filters are working well in discriminating noisy and noiseless pixels than the other types of filters.

A two stage median filter to remove high density SPIN is proposed by Thanh et al. (2020). It classifies the noise density initially into low and high. In the first stage, to remove low density noise pixels, the median of highly recurring pixels in the window is used. To remove high noise density, the median of uncorrupted pixels in the window is used. It uses fixed and adaptive window size for filtering. It shows better performance on high density SPIN, but it is poor on RVIN corrupted images.

In paper Srinivasan and Ebenezer (2007), a popular fixed window, Decision Based Algorithm (DBA) for image denoising to remove SPIN using a small $3 \times 3$ filtering window is proposed. Though it works well upto 70% noise level and produces streaking effect in higher noise densities more than 70% due to repeated replacement of the previously filtered pixel. It fails to remove impulse noise at high noise density level such as 90%. Since it uses mean filtering concept to replace the corrupted pixel, if all pixels in the $3 \times 3$ window are noisy, it will replace the corrupt pixel with a noisy pixel value again.

In paper Veerakumar et al. (2018), an Iterative Adaptive Unsymmetric Trimmed Shock Filter (IAUTSF) based on Partial Differential Equation (PDE) for the removal of SPIN is proposed. This algorithm consists of two steps namely detection and recovery of the noisy pixels. This algorithm works based on a morphological image analysis method. The performance of IAUTSF is better for low density SPIN removal.

In paper Mahalakshmi and Sreenivas (2020), an image denoising method which have three stages namely noise identification, noise correction and image enhancement is proposed. Initially, a type-II fuzzy system is used to identify the noisy pixels and the noise correction is done with the help of Adaptive Non-Local Mean Filter (ANLMF). Finally, the image enhancement is done with the help of Jaya optimization algorithm. The image enhancement is based on the single objective optimization framework. This denoising work also considers three stage image denoising framework. In this work, the noise identification and correction are included during the multi-objective optimization process. MOEA/D is used to identify the threshold values of a fuzzy system which is responsible for noise identification.

A Weighted Schatten p-Norm Minimization (WSNM) (Wang et al., 2021) algorithm is proposed for the removal of impulse noise. Here, the anisotropic Total Variation (TV) regularization was incorporated to preserve edge information and The Alternating Direction Method of Multipliers (ADMM) algorithm was adopted for solving the formulated non-convex optimization problem. This work is mainly for the natural clinical images. The main objective of this algorithm is achieved for upto 40% of RVIN removal.

The performance of recent related works is consolidated and observed. The removal of high density impulse noise is required. The proposed work mainly focus on the high density impulse noise removal. It is described in the following sections.

3. Proposed Multi-objective Optimization Framework

The design of fuzzy switching median filter is formulated as multi objective problem by considering the minimization of Mean Square Error (MSE) and maximization of Structural Similarity Index Metrics (SSIM) and represented mathematically in equation (1).

\begin{aligned} Minimize: & F (X) = (M S E, \frac{1}{S S I M})^{T} \\ Subject to: & X_{l k} \leq Z \leq X_{u k}, k = 1 t o n . \end{aligned}

(1)

where, the solution vector

X = [t_{1}, t_{2}]

n

is the dimension of the problem,

X_{l k}

and

X_{u k}

are the lower and upper bounds of the element solution vectors. Here, for the purpose of visualizing the Pareto front, maximization of SSIM is converted in to minimization of 1/SSIM. The optimization algorithms namely MOEA/D (Zhang & Li, 2007) finds the optimal non-dominated solutions for the formulated multi-objective fuzzy switching median filter design. The steps involved in the proposed optimization technique is discussed in algorithm 1.

3.1. Decomposition Based Multi-Objective Evolutionary Algorithm (MOEA/D)

Decomposition based MOEA (MOEA/D) decomposes the Multi-Objective Problems (MOP) into $N_{p}$ number of Single Objective Sub-Problems (SOSPs). The SOSPs are evaluated using weighted combination of all the objectives in the MOP. MOEA/D optimize all the SOSPs using neighborhood information between the sub-problems. Many decomposition approaches are available in the literature and among them Tchebycheff decomposition approach is used in this MOEA/D algorithm (Zhang & Li, 2007). In MOEA/D, SOSPs can be represented using $N_{p}$ evenly spread weight vectors as ${λ^{1}, λ^{2}, λ^{3}, \dots, λ^{N_{p}}}$ . Each weight vector is generated as ${λ_{1}^{i}, λ_{2}^{i}, λ_{3}^{i}, \dots, λ_{m}^{i}}$ which satisfies $\sum_{k = 1}^{m} λ_{k}^{i} = 1$ and $λ_{k}^{i} \geq 0$ , where, $m$ represents the number of objectives in the MOP.

The SOSP $g^{t e}$ of the $i^{t h}$ sub problem $λ_{i}$ using Tchebycheff approach is given in equation (2).

\begin{aligned} Min g^{t e} (X | λ^{i}, z^{*}) = max_{1 \leq j \leq m} {{λ_{j}}^{i} | f_{j} (X) - {z_{j}}^{*} |}; \\ i = 1, 2, \dots, N_{p}; j = 1, 2, \dots, m \\ Subject to: X \in ϕ \subset R^{n} \end{aligned}

(2)

where

ϕ

is represented for search space.

Equation (2) is used only for normalized objectives in the MOP. However, real-time problems are disparately scaled objectives, hence, equation (3) is derived from equation (2), for disparately scaled objectives.

\begin{aligned} Min g^{t e} (X | λ^{i}, z^{*}) = max_{1 \leq j \leq m} {{λ_{j}}^{i} | \frac{f_{j} (X) - {z_{j}}^{*}}{{z_{j}}^{n a d} - {z_{j}}^{*}} |}; \\ i = 1, 2, \dots, N_{p}; j = 1, 2, \dots, m \\ Subject to: X \in ϕ \subset R^{n} \end{aligned}

(3)

Here, $f_{j}$ denotes the $j^{t h}$ objective of MOP. The reference point ${z_{j}}^{*}$ and nadir point ${z_{j}}^{n a d}$ for each objectives in the MOP can be calculated using equation (4).

\begin{aligned} {z_{j}}^{*} = m i n {f_{j} (X) | X \in ϕ \subset R^{n}} \\ {z_{j}}^{n a d} = m a x {f_{j} (X) | X \in ϕ \subset R^{n}}, \forall j = 1, 2, \dots, m \end{aligned}

(4)

The

N_{p}

numbers SOSPs are converged towards optimal Pareto solution by applying evolutionary procedure and sharing the neighborhood information among the sub-problems. With the help of Euclidean distance, the neighborhood relation among the sub-problems can be evaluated.

4. Proposed Multi-Objective Optimization Based Fuzzy Switching Median Filter (MOOFSMF)

This section introduces the principle and pseudo-code of the proposed filter. It employs a multi-objective optimization based two stage impulse noise detection algorithm. The first stage uses the noisy image histogram to mark pixels that are corrupted by SPIN. The second stage assigns a Fuzzy flag value that pertains to the probability of RVIN corruption for remaining pixels. The filtered output pixel is a weighted linear combination of the original center pixel value and median value of uncorrupted pixels in the window, with the weight based on its flag value. It removes RVIN upto a noise density of 70% and also a mixture of RVIN & SPIN . The main feature of this algorithm is that it has good detail preserving capability. The steps involved in the proposed technique is discussed in algorithm 2.

Let $X_{i j}$ be the current pixel to be processed. Let $S_{i j}$ be the sliding window of size $W \times W$ centred at $X_{i j}$ , where $W = (2 L + 1)$ and $L$ is a positive integer. $S_{i j}$ comprises of pixels defined by the set $X_{i - u, j - v}, - L \leq u, v \leq L$ . The detection and correction algorithm is iterated $I C$ (Iteration Count) number of times with the restored image $Y$ fed back as noisy image. The iteration count $I C$ is defined based on the noise density $N D$ .

5. Results and Discussion

In this section, the proposed multi-objective denoising framework is evaluated using Matlab installed computer with configuration of Intel core i3 processor and 4GB RAM. In order to verify the performance of proposed framework, the following test images namely Lena, Bridge and Peppers are considered. Similarly, the performance of proposed work is compared with the recent state-of-art algorithms. The quantitative metrics namely PSNR, MSE, SSIM, and EPI are considered for evaluating the denoising performance.

5.1. Parameter Configuration

The window size, $W$ used in the algorithm and the iteration count, $I C$ to be carried out as given in Tables 1 and 2. The $I C$ depends only on the RVIN where as the window size $W$ depends on both SPIN and RVIN.

Table 1.
Parameter Specification for PSDBMF.

Noise type Noise range (%) Iteration Count, $I C$ Window size, $W$

SPIN $\leq 40$ $1$ $3$

$40$ to $65$ $1$ $5$

$65$ to $75$ $1$ $7$

$> 75$ $1$ $9$

RVIN $< 20$ $1$ $3$

$20$ to $40$ $2$ $3$

$> 40$ $3$ $3$

Noise type	Noise range (%)	Iteration Count, $I C$	Window size, $W$
SPIN	$\leq 40$	$1$	$3$
	$40$ to $65$	$1$	$5$
	$65$ to $75$	$1$	$7$
	$> 75$	$1$	$9$
RVIN	$< 20$	$1$	$3$
	$20$ to $40$	$2$	$3$
	$> 40$	$3$	$3$

Table 2.

Parameter Setting for MOEA/D and DE Operator.

Parameters for MOEAD
Number of sub-problems ( $N_{p}$ ) (i.e Population size)	$100$
Maximum function evaluation	$3000$
Lower Bound $X_{L}$	$[1, 1]$
Upper Bound $X_{U}$	$[255, 255]$
Number of closest weight vectors( $T$ )	$0.1 \times N u m b e r o f s u b - p r o b l e m s$
Parameters for DE crossover and mutation operator
Mutation parameter ( $F$ )	$0.5$
Crossover Rate ( $C R$ )	$1$
$η$	$20$

The proposed MOEA/D algorithm is executed for solving the multi-objective de-noising problem as given in equation (1) for different images. In this work, three different images are considered and they are: 1) Lena, 2) Bridge, and 3) Pepper. MOEA/D is executed for each image separately to find the optimal threshold by solving the multi-objective de-noising problem. The parameter setting for MOEA/D and DE mutation operator is given in the Table 2.

After the Maximum Function Evaluations (MFE), MOEA/D is stopped and the points in the objective space are filtered by a Pareto filter to identify the non-dominated solutions. In the filtered non-dominated solutions, three points are considered and they are: 1) Anchor point-1, 2) Compromised solution, and 3) Anchor point-2. Anchor point-1 and -2 are solutions at the ends of Pareto front graph which provides extremely good denoising result in terms of SSIM and MSE over each other respectively. Compromised solutions are having balance between the two objectives. Hence, compromised solution is considered for image de-noising analysis. A fuzzy system proposed by Abido (2006) is used to identify the compromised solution in the Pareto Front. After MFE, MOEA/D is identified totally 15, 7, and 7 non-dominated solutions in the Pareto front for Lena, Bridge and Peppers images respectively. The Pareto front for Lena, Bridge and Peppers image is given in Figure 1 to 3 respectively.

Figure 1.

Pareto Front Graph for Lena Image.

Figure 2.

Pareto Front Graph for Bridge Image.

Figure 3.

Pareto Front Graph for Peppers Image.

The Anchor point-1, Anchor point-2, and compromised solutions are identified from the Pareto front for each image and they are shown in the Tables 3 to 5 respectively.

Table 3.

Threshold Values for Lena Image from Pareto Front Chart.

	$t_{1}$	$t_{2}$	$M S E$	$\frac{1}{S S I M}$	$S S I M$
Anchor point $1$	$13$	$101$	$435.1097$	$1.76431$	$0.5667$
Compromised solution	$4$	$79$	$439.7292$	$1.6998$	$0.5882$
Anchor point $2$	$3$	$70$	$470.1517$	$1.6828$	$0.5942$

Table 4.

Threshold Values for Bridge Image from Pareto Front Chart.

	$t_{1}$	$t_{2}$	$M S E$	$\frac{1}{S S I M}$	$S S I M$
Anchor point $1$	$1$	$119$	$889.5726$	$2.30037$	$-$
Compromised solution	$1$	$136$	$900.8193$	$2.284699$	$-$
Anchor point $2$	$4$	$136$	$898.6989$	$2.289919$	$-$

Table 5.

Threshold Values for Peppers Image from Pareto Front Chart.

	$t_{1}$	$t_{2}$	$M S E$	$\frac{1}{S S I M}$	$S S I M$
Anchor point $1$	$2$	$106$	$520.6389$	$1.8562$	$0.5387$
Compromised solution	$5$	$66$	$542.3865$	$1.8018$	$0.5550$
Anchor point $2$	$14$	$71$	$520.6844$	$1.8205$	$0.5492$

5.2. Performance Analysis of Proposed Algorithm in RVIN & SPIN

In order to test the performance of proposed framework for RVIN noise corrupted “Lena” image of size $512 \times 512$ (8-bit/pixel) is used. The proposed algorithm is compared with the recent state-of-art algorithms such as SMF (Wang & Lu, 2011), ACWMF (Chen & Wu, 2001), TSMF (Chen et al., 1999), PSMF (Wang & Zhang, 1999), 2-State SD ROM (Abreu et al., 1996), Trilateral filter (Chen et al., 2019), BDND (Ng & Ma, 2006), ROLD (Dong et al., 2007), IMF (Erkan et al., 2019), NAFSM (Toh & Isa, 2010), TSM (Thanh et al., 2020), MMPAPF (Satti et al., 2020), IAUTSF (Veerakumar et al., 2018), BCNNDM (Chen et al., 2019), LSTSA (Lin et al., 2020) and WSNM (Wang et al., 2021). Figure 4 shows the performance of proposed algorithm on the 50% of RVIN. In Figure 5, the filter outputs of fifteen different algorithms have been displayed for 70% RVIN corrupted image. The visual quality among the restored images using different algorithms are depicted in Figure 5. Along with proposed method, the performance of few other algorithms specifically designed for RVIN is compared. It is evident that proposed framework provided better visual quality of denoised image when compared to other algorithms. Table 6 shows the quantitative results of Lena image with low density RVIN (i.e., 10% to 40%). Meanwhile, the proposed method compromise only 1.2% in PSNR to achieve 5% improvement in SSIM when compared to BCNNDM for 10% RVIN corrupted Lena image. Likewise, with respect to EPI, the proposed work scored 2.55% more than LSTSA. This data clearly shows the good performance of the proposed framework in terms of the visual quality, over BCNNDM & LSTSA. Consistent performance of the proposed algorithm is demonstrated by using three images like Lena, Bridge and Peppers images are considered with high density RVIN (i.e., 50%, 60% and 70%). The quantitative metrics(i.e., SSIM, MSE, PSNR and EPI) for the algorithms with high density RVIN for the Lena, Bridge and Pepper images are presented in Tables 7 to 9, respectively. From the Tables 7 to 9, it can be inferred that the two algorithms such as BCNNDM, LSTSA and WSNM are competitive with our proposed framework.

Figure 4.

Performance of the Proposed Algorithm (a) Original Image, (b) Noisy Image (50% RVIN), (c) Denoised Image (PSNR=25.68dB).

Figure 5.

Visual Image Comparison of all Algorithms (a) Original Image, (b) Noisy Image (with 70% RVIN), (c) SMF, (d) ACWMF, (e) TSMF, (f) PSMF, (g) 2-state SD ROM, (h) Trilateral Filter, (i) BDND, (j) ROLD, (k) IMF, (l) NAFSM, (m) TSF, (n) MMPAPF, (o) IAUTSF, (p) BCNNDM, (q) LSTSA and (r) Proposed.

Table 6.

Comparison of Performance Metric Values (PSNR, MSE, SSIM & EPI) of Restored Lena Image Using Various Algorithms for the Different Low Density RVIN.

	PSNR				MSE				SSIM				EPI
	Noise Density (in %)				Noise Density (in %)				Noise Density (in %)				Noise Density (in %)
Methods	10	20	30	40	10	20	30	40	10	20	30	40	10	20	30	40
SMF	34.08	31.51	28.16	24.46	25	46	99	233	0.944	0.914	0.826	0.671	0.752	0.616	0.419	0.258
ACWMF	35.50	31.83	28.80	26.15	18	43	86	158	0.959	0.891	0.796	0.677	0.830	0.682	0.523	0.385
TSMF	36.40	32.91	28.95	25.41	15	33	83	187	0.973	0.939	0.852	0.719	0.850	0.715	0.515	0.344
PSMF	29.92	27.43	26.17	25.43	66	118	157	186	0.755	0.634	0.568	0.526	0.609	0.477	0.401	0.330
2-State SD ROM	37.32	33.37	33.37	27.72	12	30	60	110	0.974	0.941	0.876	0.786	0.884	0.751	0.597	0.456
Trilateral	32.91	31.93	30.90	29.30	33	42	53	76	0.922	0.914	0.901	0.872	0.774	0.729	0.673	0.591
BDND	23.56	20.41	18.36	16.98	287	591	949	1304	0.557	0.361	0.245	0.181	0.348	0.237	0.178	0.144
ROLD	21.40	18.84	15.43	13.96	471	849	1862	2610	0.648	0.539	0.303	0.216	0.295	0.200	0.129	0.102
IMF	19.23	16.30	14.54	13.28	776	1524	2285	3053	0.310	0.173	0.116	0.083	0.218	0.148	0.115	0.093
NAFSM	19.29	16.26	14.48	13.26	765	1538	2317	3067	0.310	0.172	0.113	0.083	0.217	0.150	0.112	0.094
TSF	19.30	16.24	14.53	13.29	764	1544	2291	3050	0.310	0.170	0.113	0.083	0.219	0.148	0.112	0.093
MMPAPF	28.75	28.23	26.72	24.51	87	98	138	230	0.866	0.849	0.783	0.684	0.507	0.484	0.427	0.364
IAUTSF	19.30	16.24	14.53	13.29	764	1544	2291	3050	0.310	0.170	0.113	0.083	0.219	0.148	0.112	0.093
BCNNDM	39.68	36.67	34.51	32.02	7	14	23	41	0.929	0.901	0.864	0.843	0.907	0.883	0.812	0.719
LSTSA	38.04	35.13	33.61	31.12	10	20	28	50	0.956	0.931	0.914	0.901	0.922	0.910	0.864	0.799
WSNM	35.34	34.15	28.64	24.31	19	25	89	241	0.962	0.951	0.922	0.902	0.933	0.912	0.868	0.806
Proposed	39.21	35.96	34.12	31.91	8	17	25	42	0.975	0.958	0.936	0.910	0.956	0.920	0.882	0.830

Table 7.

Comparison of Performance Metric Values (PSNR, MSE, SSIM & EPI) of Restored Lena Image Using Various Algorithms for the Different High Density RVIN.

	PSNR			MSE			SSIM			EPI
	Noise Density (in %)			Noise Density (in %)			Noise Density (in %)			Noise Density (in %)
Methods	50	60	70	50	60	70	50	60	70	50	60	70
SMF	21.47	18.87	16.75	463	843	1373	0.505	0.340	0.222	0.160	0.102	0.068
ACWMF	23.75	21.32	18.76	274	480	865	0.546	0.410	0.285	0.270	0.191	0.121
TSMF	22.08	19.27	16.94	403	770	1317	0.540	0.371	0.230	0.216	0.133	0.077
PSMF	24.41	22.94	20.10	235	331	636	0.489	0.445	0.364	0.270	0.216	0.166
2-State SD ROM	25.05	22.35	19.45	203	378	738	0.656	0.509	0.358	0.309	0.204	0.126
Trilateral	26.99	23.76	20.00	130	273	651	0.801	0.662	0.453	0.473	0.358	0.250
BDND	15.77	14.71	13.77	1724	2200	2731	0.138	0.101	0.078	0.118	0.099	0.078
ROLD	12.59	11.56	10.83	3580	4537	5367	0.116	0.063	0.038	0.080	0.068	0.056
IMF	12.28	11.51	10.83	3844	4595	5370	0.061	0.046	0.034	0.080	0.062	0.055
NAFSM	12.29	11.51	10.84	3838	4593	5362	0.062	0.046	0.034	0.081	0.063	0.055
TSF	12.31	11.51	10.82	3822	4594	5389	0.061	0.046	0.033	0.074	0.064	0.056
MMPAPF	22.02	19.87	17.98	408	670	1035	0.563	0.458	0.378	0.309	0.271	0.239
IAUTSF	12.31	11.51	10.82	3822	4594	5389	0.061	0.046	0.033	0.074	0.064	0.056
BCNNDM	30.91	28.11	24.62	53	100	224	0.811	0.799	0.685	0.685	0.568	0.466
LSTSA	29.98	28.87	25.19	65	84	197	0.851	0.809	0.708	0.703	0.609	0.473
WSNM	22.56	20.91	18.54	361	527	911	0.856	0.811	0.721	0.712	0.631	0.501
Proposed	31.03	29.56	25.55	51	72	181	0.872	0.819	0.736	0.759	0.659	0.517

From Table 7, for 70% RVIN corrupted Lena image, 3.8% high PSNR and 7.5% high SSIM than the BCNNDM algorithm. From Table 8, for 70% RVIN corrupted Bridge image, 0.5% high PSNR and 4.6% high SSIM than the LSTSA algorithm. Similarly, from Table 9, for 70% RVIN corrupted Peppers image, 7.2% high PSNR and 3.8% high SSIM than the WSNM algorithm. This confirms that proposed de-noising framework can preserve the structure of the image even in high noise density. Eventhough, the trade-off characteristics between SSIM and PSNR, proposed method also produce good results based on PSNR or MSE.

Table 8.

Comparison of Performance Metric Values (PSNR, MSE, SSIM & EPI) of Restored Bridge Image Using Various Algorithms for the Different High Density RVIN.

	PSNR			MSE			SSIM			EPI
	Noise Density (in %)			Noise Density (in %)			Noise Density (in %)			Noise Density (in %)
Methods	50	60	70	50	60	70	50	60	70	50	60	70
SMF	19.20	17.21	15.34	780	1233	1898	0.451	0.336	0.230	0.107	0.068	0.047
ACWMF	20.63	18.71	16.81	561	875	1352	0.528	0.405	0.280	0.300	0.208	0.131
TSMF	19.53	17.39	15.47	724	1187	1844	0.50	0.36	0.24	0.19	0.12	0.07
PSMF	21.77	20.02	17.80	432	647	1078	0.576	0.478	0.358	0.315	0.225	0.154
2-State SD ROM	21.51	19.51	17.20	459	727	1237	0.613	0.478	0.334	0.318	0.216	0.131
Trilateral	22.62	20.48	17.69	355	581	1106	0.607	0.504	0.364	0.298	0.215	0.145
BDND	14.69	13.70	12.75	2208	2775	3450	0.212	0.156	0.113	0.163	0.126	0.094
ROLD	11.80	11.03	10.35	4292	5134	5993	0.118	0.085	0.062	0.120	0.095	0.073
IMF	12.45	11.65	11.02	3696	4449	5145	0.063	0.048	0.035	0.071	0.062	0.051
NAFSM	11.83	11.06	10.39	4262	5097	5946	0.118	0.086	0.062	0.120	0.095	0.073
TSF	11.97	11.18	10.53	4130	4955	5752	0.116	0.084	0.062	0.104	0.079	0.066
MMPAPF	22.02	19.87	17.98	409	670	1034	0.559	0.456	0.377	0.317	0.282	0.249
IAUTSF	11.97	11.18	10.53	4130	4957	5758	0.116	0.084	0.062	0.105	0.079	0.066
BCNNDM	26.23	23.88	21.67	155	266	443	0.631	0.544	0.403	0.431	0.397	0.302
LSTSA	25.91	23.75	22.04	167	274	406	0.681	0.562	0.412	0.487	0.411	0.321
WSNM	26.14	23.68	21.18	158	279	496	0.672	0.565	0.418	0.499	0.425	0.339
Proposed	26.61	24.87	22.13	142	212	398	0.694	0.592	0.431	0.523	0.458	0.391

Table 9.

Comparison of Performance Metric Values (PSNR, MSE, SSIM & EPI) of Restored Peppers Image Using Various Algorithms for the Different High Density RVIN.

	PSNR			MSE			SSIM			EPI
	Noise Density (in %)			Noise Density (in %)			Noise Density (in %)			Noise Density (in %)
Methods	50	60	70	50	60	70	50	60	70	50	60	70
SMF	20.59	18.05	15.98	568	1018	1639	0.459	0.309	0.199	0.132	0.085	0.056
ACWMF	20.63	18.71	16.81	561	875	1352	0.528	0.405	0.280	0.300	0.208	0.131
TSMF	21.27	18.29	16.18	485	963	1566	0.509	0.331	0.213	0.195	0.119	0.078
PSMF	23.90	21.68	18.98	265	442	822	0.487	0.428	0.347	0.200	0.132	0.097
2-State SD ROM	24.64	21.72	18.67	223	438	884	0.645	0.490	0.335	0.290	0.178	0.114
Trilateral	26.39	22.82	19.18	149	340	786	0.806	0.652	0.442	0.328	0.203	0.121
BDND	15.30	14.21	13.18	1921	2467	3124	0.125	0.093	0.070	0.079	0.061	0.049
ROLD	12.23	11.22	10.48	3893	4905	5817	0.095	0.054	0.035	0.062	0.050	0.039
IMF	11.95	11.13	10.49	4155	5010	5815	0.056	0.042	0.032	0.061	0.049	0.038
NAFSM	11.90	11.09	10.44	4200	5056	5875	0.055	0.041	0.032	0.058	0.046	0.038
TSF	12.04	11.29	10.63	4061	4831	5625	0.056	0.042	0.032	0.056	0.045	0.034
MMPAPF	20.78	18.69	16.93	543	878	1317	0.536	0.441	0.370	0.241	0.217	0.193
IAUTSF	12.04	11.29	10.62	4063	4832	5632	0.056	0.042	0.032	0.056	0.045	0.034
BCNNDM	28.35	26.97	23.12	95	131	317	0.798	0.678	0.513	0.345	0.301	0.274
LSTSA	28.17	27.06	23.49	99	128	291	0.803	0.711	0.527	0.384	0.332	0.291
WSNM	27.52	26.40	22.93	115	149	331	0.811	0.706	0.521	0.402	0.319	0.281
Proposed	28.31	27.76	24.57	96	109	227	0.831	0.736	0.541	0.413	0.374	0.335

In order to test the performance of the proposed algorithm for SPIN corrupted image, Bridge image of size $512 \times 512$ is used. The proposed algorithm is compared with the recent state-of-art algorithms such as SMF (Wang & Lu, 2011), WMF (Brownrigg, 1984), CWMF (Ko & Lee, 1991), ACWMF (Chen & Wu, 2001), AMF (Hwang & Haddad, 1995), PSMF (Wang & Zhang, 1999), FIDT (Luo, 2006), BDND (Ng & Ma, 2006), DBA (Srinivasan & Ebenezer, 2007), IMF (Erkan et al., 2019), NAFSM (Toh & Isa, 2010), TSM (Thanh et al., 2020), MMPAPF (Satti et al., 2020), and IAUTSF (Veerakumar et al., 2018) for visual results. In Figure 6, the filter outputs of fifteen different algorithms as been displayed for 95% SPIN noise corrupted image. Figure 6 depicts the visual quality among the restored images using different algorithms. Figure 7 shows the performance of proposed algorithm on the 90% of SPIN. From the image comparisons, it is evident that the proposed algorithm performs very well for high density SPIN too.

Figure 6.

Visual Image Comparison of all Algorithms (a) Original Image, (b) Noisy Image (with 95% SPIN), (c) SMF, (d) WMF, (e) CWMF, (f) ACWMF, (g) AMF, (h) PSMF, (i) FIDT, (j) BDND, (k) DBA, (l) IMF, (m) NAFSM, (n) TSF, (o) MMPAPF, (p) IAUTSF and (q) Proposed.

Figure 7.

Performance of the Proposed Algorithm (a) Original Image, (b) Noisy Image (90% SPIN), (c) Denoised Image (PSNR=27.67dB).

5.3. Performance Analysis of Proposed Algorithm in MIN

The mixture of SPIN and RVIN are applied in few combinations of noise densities to Peppers image of size $512 \times 512$ and 8-bit per pixel. In this paper, SPIN and RVIN are mixed in the ratios of proportions such as (10%, 10%), (20%, 20%), (30%, 30%), (30%, 40%), (30%, 50%), (30%, 60%), and (40%, 50%) respectively. In Figure 8, the images are restored from 40% of SPIN & 50% of RVIN corrupted image using different existing algorithms. Figure 8 shows that the proposed method having better visual results compared to other algorithms. Figure 9 shows the performance of proposed algorithm on the mixed impulse noise (30% of SPIN & 30% of RVIN). The quantitative metric PSNR, MSE and SSIM are presented from Tables 10 to 11. In Table 10, eventhough the proposed framework under performs than few other algorithm for (10%, 10%),(20%, 20%), (30%, 30%) and (30%, 40%) of SPIN and RVIN mixed noise respectively, in Table 11, SSIM for these noise densities are better than other algorithms. For (10%,10%) MIN, the proposed framework have 1.4% low PSNR than MMPAPF, but SSIM is 8.8% higher than MMPAPF. In another case, for (40%,50%) MIN, PSNR is 32.4% higher than MMPAPF. From Table 11, the performance of 2-state SD ROM and PSMF algorithms are higher than the proposed method. But, the objective of our work is to restore high noise density corrupted image. The proposed framework scored 22.8% and 6% high PSNR than 2-state SD ROM and PSMF respectively for (40%,50%) MIN. Also, in terms of SSIM the proposed algorithm outperforms all other algorithms for high noise densities of mixed impulse noise.

Figure 8.

Visual Image Comparison of all Algorithms (a) Original Image, (b) Noisy Image (Mixed noise (40% SPIN and 50% RVIN)), (c) SMF, (d) ACWMF, (e) PSMF, (f) 2-State SD ROM, (g) BDND, (h) ROLD, (i) Trilateral, (j) IMF, (k) NAFSM, (l) TSF, (m) MMP, (n) IAUTSF, (o) Proposed.

Figure 9.

Performance of the Proposed Algorithm (a) Original Image, (b) Noisy Image (Mixed Impulse Noise (30% SPIN & 30% RVIN)), (c) Denoised Image (PSNR=24.86dB).

Table 10.

Comparison of PSNR & MSE Values of Restored Peppers Image using Various Algorithms for the Different Mixed Noise Densities.

	Noise Density (in %) – Percentage of SPIN and RVIN respectively
	10,10		20,20		30,30		30,40		30,50		30,60		40,50
Methods	PSNR	MSE	PSNR	MSE	PSNR	MSE	PSNR	MSE	PSNR	MSE	PSNR	MSE	PSNR	MSE
SMF	31.48	46	24.14	251	18.33	955	16.85	1342	15.42	1865	14.18	2483	13.58	2850
ACWMF	32.60	36	27.59	113	23.93	263	22.09	402	20.03	646	18.32	958	18.83	852
PSMF	29.36	75	27.41	118	26.24	155	25.10	201	23.24	308	20.86	534	22.59	359
2-State SD ROM	34.62	22	28.96	83	24.79	216	23.19	312	21.11	504	19.09	803	19.50	729
BDND	23.23	309	20.45	587	18.82	854	17.42	1177	16.16	1575	14.90	2102	16.47	1466
ROLD	17.05	1283	11.84	4252	9.49	7307	8.98	8220	8.76	8647	8.43	9333	7.64	11186
Trilateral	31.68	44	27.81	108	20.79	542	18.77	863	16.48	1463	14.45	2334	12.85	3376
IMF	19.29	766	16.69	1393	15.31	1916	13.97	2607	13.01	3253	12.13	3982	13.34	3013
NAFSM	19.30	765	16.88	1332	15.61	1785	14.36	2383	13.37	2995	12.52	3644	13.99	2596
TSF	19.45	738	16.80	1359	15.45	1853	14.13	2511	13.04	3233	12.14	3976	13.41	2963
MMPAPF	29.02	81	27.30	121	23.85	268	21.46	464	19.45	739	17.75	1092	18.09	1008
IAUTSF	19.31	762	16.84	1346	15.51	1826	14.20	2471	13.15	3152	12.22	3902	13.64	2810
Proposed	28.60	90	27.19	124	25.80	171	24.71	220	23.72	276	22.53	363	23.95	262

Table 11.

Comparison of SSIM Values of Restored Peppers Image.

Noise Density (in %) – Percentage of SPIN and RVIN respectively
Methods	10,10	20,20	30,30	30,40	30,50	30,60	40,50
SMF	0.917	0.700	0.363	0.262	0.181	0.129	0.118
ACWMF	0.932	0.810	0.653	0.539	0.425	0.321	0.379
PSMF	0.757	0.706	0.697	0.646	0.590	0.495	0.598
2-State SD ROM	0.962	0.879	0.740	0.637	0.514	0.384	0.439
BDND	0.544	0.364	0.268	0.199	0.148	0.111	0.157
ROLD	0.474	0.286	0.064	0.037	0.030	0.019	0.049
Trilateral	0.924	0.865	0.610	0.483	0.349	0.222	0.159
IMF	0.311	0.181	0.133	0.095	0.072	0.054	0.079
NAFSM	0.309	0.186	0.134	0.099	0.075	0.057	0.085
TSF	0.317	0.187	0.138	0.099	0.074	0.057	0.083
MMPAPF	0.882	0.814	0.643	0.525	0.425	0.357	0.321
IAUTSF	0.314	0.188	0.137	0.101	0.076	0.057	0.086
Proposed	0.960	0.924	0.874	0.838	0.794	0.748	0.790

6. Conclusion

This paper introduces a new approach to fix the fuzzy switching median filter threshold values using multi-objective optimization frame work. It uses two different multi-objective functions in order to fix the threshold values. The main highlight of the research is to detect and restore the impulse corrupted images. Not many techniques in literature are capable of removing the mixture of high density RVIN or mixed noise. This problem has been overcome in the proposed techniques by using an efficient two stage detection procedure with the use of optimized threshold using MOEA/D. The objectives of optimization methods are MSE and SSIM. So, both the quantitative result and visual quality are predominantly good for high noise density. The proposed framework performed 6.02% and 32.11% better than the best existing methods in terms of PSNR and SSIM for the mixture of 40% SPIN & 50% RVIN affected image. In future, the framework can be improved further, by increasing the number of non-dominated solutions in optimization.

Footnotes

ORCID iDs

Suresh Babu V

Vijaykumar V R

Mohaideen Abdul Kadhar K

Sudhakar R

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

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Multi-Objective Threshold Optimized Image De-Noising Algorithm for High Density Mixed Impulse Noise

Abstract

Keywords

1. Introduction

2. Related Works

3. Proposed Multi-objective Optimization Framework

5. Results and Discussion

5.1. Parameter Configuration

Table 1. Parameter Specification for PSDBMF. Noise type Noise range (%) Iteration Count, I C Window size, W SPIN ≤ 40 1 3 40 to 65 1 5 65 to 75 1 7 > 75 1 9 RVIN < 20 1 3 20 to 40 2 3 > 40 3 3

Footnotes

ORCID iDs

Funding

Declaration of Conflicting Interests

References

Table 1.
Parameter Specification for PSDBMF.

Noise type Noise range (%) Iteration Count, $I C$ Window size, $W$

SPIN $\leq 40$ $1$ $3$

$40$ to $65$ $1$ $5$

$65$ to $75$ $1$ $7$

$> 75$ $1$ $9$

RVIN $< 20$ $1$ $3$

$20$ to $40$ $2$ $3$

$> 40$ $3$ $3$