Journal of information and communication convergence engineering 2022; 20(3): 195-203

Published online September 30, 2022

https://doi.org/10.56977/jicce.2022.20.3.195

© Korea Institute of Information and Communication Engineering

A Modified Steering Kernel Filter for AWGN Removal based on Kernel Similarity

Bong-Won Cheon 1 and Nam-Ho Kim2* , Member, KIICE

1Department of Intelligent Robot Engineering, Pukyong National University, Busan 48513, Republic of Korea
2School of Electrical Engineering, Pukyong National University, Busan 48513, Republic of Korea

Correspondence to : *Nam-Ho Kim (E-mail: nhk@pknu.ac.kr, Tel: +82-51-629-6328)
School of Electrical Engineering, Pukyong National University, Busan 48513, Republic of Korea

Received: December 24, 2021; Revised: August 22, 2022; Accepted: August 24, 2022

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Noise generated during image acquisition and transmission can negatively impact the results of image processing applications, and noise removal is typically a part of image preprocessing. Denoising techniques combined with nonlocal techniques have received significant attention in recent years, owing to the development of sophisticated hardware and image processing algorithms, much attention has been paid to; however, this approach is relatively poor for edge preservation of fine image details. To address this limitation, the current study combined a steering kernel technique with adaptive masks that can adjust the size according to the noise intensity of an image. The algorithm sets the steering weight based on a similarity comparison, allowing it to respond to edge components more effectively. The proposed algorithm was compared with existing denoising algorithms using quantitative evaluation and enlarged images. The proposed algorithm exhibited good general denoising performance and better performance in edge area processing than existing non-local techniques.

Keywords Image processing, Steering kernel, Noise removal, AWGN

The denoising of images is an important preprocessing step in systems that use image-based algorithms for detecting, recognizing, and tracking objects. Because high-frequency image details are mixed with noise in most cases, most existing image denoising methods fail to preserve edge and texture information while thoroughly eliminating noise [1,2].

Several traditional image-processing methods exploit the local structural regularity present in natural images. The rationale of de-noising algorithms is to use structural patterns to regularize the ill-posed restoration problem and smooth out texture blurry and flat regions [3,4].

Various filtering techniques to remove different types of noise and improve image quality and image recognition have been proposed [5]. Several denoising techniques, including bilateral total variation (BTV) [6] and split Bregman isotropic total variation denoising (SB-ITV) [7], have been demonstrated to effectively restore real scenes from noisy images. The local steering kernel (LSK) [8] is an effective method for removing noise while preserving the original image structures, because it solves the image noise and uncertainty problems by estimating the local structure. The steering kernel non-local means (SKNLM) [9] algorithm, which estimates original images based on the similarity of different image patches and produces superior noise removal performance has received considerable attention. However, most image denoising methods have a drawback in that it is difficult to preserve edge and text information while completely eliminating the noise because the information of the original image and noise are mixed in high-frequency regions.

In this study, we proposed a modified steering kernel filter algorithm to minimize the smoothing effects generated in the filtering process using a local region technique. The proposed algorithm set weights differently based on the similarity in a local region to improve upon the LSK techniques, which are based on the orientation of pixel values change in a local region. This study also incorporated the concept that the larger the size of the filtering window, the abler it is to remove noise, while the smaller the size of the filtering window, the abler it is to preserve details in general. Thus, in this study, we conducted a similarity comparison using an adaptive mask adjusted in size based on the noise intensity of the image. The modified steering kernel proposed in this study showed better image processing results and performance in the production of images by denoising and improving the details.

The remainder of this paper is organized as follows: In Section 2, LSK is briefly explained, and in Section 3, the proposed algorithm and its mechanism are described. Section 4 presents the simulation results and analysis. Finally, in Section 5, conclusions are presented.

A steering kernel is a weight frequently used in image processing, and is calculated using the analysis of the gradient and orientation of pixel values. Steering kernel regression (SKR) was used to calculate the steering weight, which is based on the position and intensity of the pixels, as well as the intrinsic local structure of the samples. The size and shape of the kernel significantly affected its spread and feature-extraction characteristics. LSK spreads the kernel to the highly correlated regions.

The steering kernel considers the gradient in a local region and analyzes the radiometric similarity of the pixels. Generally, it is expressed as follow [10]:

ωijLSK=detCij2πh2expixTCijiy2h2

where h represents the global smoothing constant. Filtering intensity is adjusted based on the value of h. The higher this value, the stronger the smoothing intensity. Variable ij refers to the coordinates of an input pixel, and xy refers to the corresponding pixel in a local region.

Matrix Cij denotes a symmetric gradient covariance matrix calculated in a square-shaped local window centered on pixel ij. The covariance matrix Cij is a 2 × 2 covariance matrix determined using the local gradient and represents the shape of a kernel set in a pixel. This is expressed as follows:

Cij=GijTGij=γijUijSijUθijT
Uij=cosθijsinθij-sinθijcosθij
Sij=δij00δij1

Here, Gij denotes a local gradient matrix indicating the gradient of the pixel values in the center window, γij denotes a scaling parameter, Uij and Sij denote rotation and stretching matrices, respectively, controlled by the rotation parameter θij and stretching parameter δij. Although this approach is simple and robust to noise, the resulting covariance estimate is generally either low or unstable. A parametric approach was adopted from [11] to obtain a stable covariance matrix estimate. When matrix Gij is an m × n matrix, Uij is an orthogonal matrix of size m × n composed of the left singular vector of the matrix Gij, and GijT is an orthogonal matrix of size n × n consisting of the right singular vector.

The three parameters γij, Uij, and δij are determined using singular value decomposition (SVD) of the local gradient matrix Gij [9]. The adaptive property of the covariance matrix reflects the local structure of the image and leads to the kernel function spreading along the local edges. Therefore, the steering kernel can effectively learn the edge direction from the guidance image.

The SKR [11] is dependent not only on the position and intensity, but also on the intrinsic local structure of the samples. Thus, the size and shape of the regression kernel significantly affect its spread and feature extraction characteristics. The core of the SKR method is the LSK function, which accurately estimates the local structure even in the presence of large amounts of noise.

Digital images are acquired through sampling and quantization of images inputted through cameras and displayed after numerically expressing the brightness and color intensity in a two-dimensional (2D) space [12].

If the acquired images are damaged by noise, the quantized data includes not only the information of the original image but also noise signals. Assuming that the damaged image is Iij, a 2D digital image can be represented as follows:

Iij=Zij+εij, i1,R, j1,C

where Zij denotes the original image that is not damaged by noise and εij denotes the noise constant.

The estimation of the amount of noise in an image is important for adjusting the values of the parameters used in image processing algorithms and improving performance. Immerkær [13] proposed an algorithm to quickly and accurately estimate the noise variance in images. Fast noise variance estimation is based on the observation that a noise estimator is insensitive to the Laplacian of an image, because components, such as image edges, have a strong second-derivative component. Noise is estimated using the difference between two masks, L1 and L2, which approximates the Laplacian of an image, as expressed in the following equations:

L1=010141010, L2=12010141010
N=2L2L1=121242121

where N denotes the noise estimation operator. If the noise of each pixel has a standard deviation σest, it can be calculated with a mean of 0 and variance of 36σest2 using the noise estimation operator.

If the noise estimation operator is performed using the entire image, it can be calculated as follows:

σest2=136R2C2 image IIij*N2

where R and C represent the width and height of the image, respectively, and operator * denotes convolution.

The proposed algorithm starts by setting the center and matching windows. The center window is set to a square shape around a pixel coordinate p, which is set to a size of (2F + 1) × (2F + 1), using the constant F, which represents the size of the center window. The WijC center window can be expressed by the following equation:

WijC=Ii+k,j+l, k,lF,F

The size of the center window had a significant impact on the filtering result. The smaller the size of the mask, the better it preserves the image detail. Conversely, the larger the mask, the more advantageous it is for denoising. The proposed algorithm determines the size F of the center window based on the observation that a larger center window size is advantageous. The center window size set in the proposed algorithm can be expressed using the following equation:

F=maxroundρ,σest,0

where ρ is a constant that determines the size of the center window. The operator max [ ] is a function that returns the maximum value and round ( ) is a rounding function. The search window is an area centered on pixel coordinates p and is placed to define an area similar to the center window in the surrounding area. The search window WijS, can be obtained using the following equation:

WijS=Ii+x,j+y, x,yS,S

where x, y denotes the pixel coordinates inside the search window. WijS is set to a size of (2S + 1) × (2S + 1) using a constant S, which is the size of the search window. The search window determines the size of the local steering kernel and range of the matching mask.

A matching window of the same size as the center window was set around the pixel coordinates i + x, j + y located inside the search window, which was used to compare the similarity of the two regions. Matching window WxyM, can be obtained using the following equation:

WxyM=Ii+x+k,j+y+l, k,lF,F

Here, F represents the size of the matching window, which is the same size as the center window used in equation (9). The proposed algorithm compares the similarities of the two masks to identify the relationship between the filtering and matching masks. Similarity comparison involves a comparison of the pixel values located at the same internal coordinates of the two masks. The degree of difference between the two masks mxy after comparing the center and matching windows can be calculated using the following equation:

mxy=1F2 k,l=FFWijCk,lWxyMk,l2

The smaller the difference in the pixel values of the two masks, the lower the sxy. A low value indicates that the two masks have similar pixel arrangements or that they are very likely to be located on an extension line of continuous edges in an adjacent region. This difference improves the filtering result of the edge component of the image and text regions.

The lower the difference between the filtering and matching masks, the more similar the two masks are. If the two masks are dissimilar, the pixels of the coordinate must be excluded from the calculation. The proposed algorithm selected a smaller value for the difference than the threshold based on the differences calculated in equation (13), and the pixel values of the coordinates were used in the final calculation.

Weight Exy is calculated based on the difference between the two masks and the threshold, as follows:

Exy=1, if mxy<t1σest
Exy=34, if t1σestmxy<t2σest
Exy=12, if t2σestmxy<t3σest
Exy=14, if t3σestmxy<t4σest
Exy=0, if t4σestmxy

where t1, t2, t3, and t4 are constants used to set the value of the selection mask according to the threshold. The threshold value increases the weights based on the decision that the filtering and matching windows are similar, as indicated by the low value of mxy. Weight Exy, assigned based on the threshold, is applied to the local steering weight ωijLSK, and used to emphasize regions with higher similarity. The weight, Txy, calculated based on the two weights, can be expressed by equation (19) as follows:

Txy=ExyωijLSK

where ωijLSK denotes the modification of the local steering weight ωijLSK based on the internal coordinates of the WijSs.

Figure 1 shows regions that have different characteristics in the test image to verify the effect of the weight Txy.

Fig. 1. Original sample image to compare weight distribution.

Figures 2, 3, and 4 present enlarged images of regions I, II, and III in Figure 1, respectively.

Fig. 2. Schematic of enlarged image and weight distribution of sample image (area I). (a) Enlarged image (b) Local Steering weight (c) Weight of proposed algorithm.

Fig. 3. Schematic of enlarged image and weight distribution of sample image (area II). (a) Enlarged image (b) Local Steering weight (c) Weight of proposed algorithm.

Fig. 4. Schematic of enlarged image and weight distribution of sample image (area III). (a) Enlarged image (b) Local Steering weight (c) Weight of proposed algorithm.

Each region's local steering weight ωijLSK and weight Txy of the proposed algorithm for comparison are also shown.

Figure 2 shows the weight in a region that is judged to have a large difference from the pixel value in the center's black region and is therefore set to a low value. Figure 3 shows that the weight is modified into a sharply distributed shape along the edge component.

Figure 4 shows that the weight is evenly distributed overall, because it is the region whose change in pixel values is not very large.

It can be confirmed that the proposed weights provide high weights to pixel values close to the center pixel, and simultaneously distinguish regions with similar structural features.

The filtering of the output image Oij using weight Txy can be expressed as follows:

Oij=1c xyW ijSWijSTij, c= xyW ijSTxy

The proposed method is summarized in Algorithm 1.

Modified steering kernel filter algorithm

Inputs:
  • Noised input image (Iij);

  • Global smoothing parameter (h).

    • Initialization:

      • Calculate the noise estimation (σest)

    • Iteration:

      • Coordinates of pixels on the noisy image (Iij);

      • Calculate the steering kernel matrices (ωijLSK) according to Eq. (1);

      • Set the center window (Wijc), search window (WijS), and matching window (WxyM) according to Eqs. (9), (11), and (12);

      • Calculate the difference parameter (mxy) according to Eq. (13);

      • Set the weight matrices (Exy) according to Eqs. (14),-(18);

      • Calculate the modified steering kernel matrices (Txy) according to Eq. (19);

      • Output image (Oxy) according to Eq. (20);

      • Return to 2) (a).

    • Result:

      • Output the de-noised image (Oij).

In this case, the constants, such as the global smoothing parameter, are the same as those in the existing LSK method.

A. Experimental Setting

To investigate the efficacy of the proposed algorithm, a series of experiments were conducted with applications including edge-aware smoothing, detail enhancement, denoising, and de-hazing. Four well-known 512 × 512 8-bit gray images were used to evaluate the denoising function objectively, as shown in Fig. 5. Noisy images were prepared by setting the standard deviation of AWGN to five levels within a range of 5 to 30 to evaluate and analyze the denoising performance of the proposed algorithm based on the noise level.

Fig. 5. Original sample images. (a) Lena (b) Goldhill (c) Barbara (d) Boat

To compare and analyze the existing noise removal techniques, the denoising results of BTV, SB-ITV, SKNLM, and the proposed algorithm were compared with the original and noisy images. For objective evaluation, the performance was analyzed using the PSNR and SSIM of the denoised result images [14,15].

B. Parameter Analysis

The relationship between the parameters used in the proposed algorithm and those of the denoised images was analyzed. The global smoothing constant, h, a parameter used in the simulation, was used to calculate the steering kernel weight (equation 1). Constant h is an important factor that influences the denoising performance of the proposed algorithm. The denoising performance and image details were analyzed by changing the h value used in the proposed algorithm from 0.5 to 2.5.

Figure 6 shows the PSNR graph of the denoised image based on the change in h value in the proposed algorithm. The denoising performance was significantly lower in the high-noise image when the parameter was set to a low value, such as h(0.5). If the parameter was high, such as h(2.5), the PSNR tended to be lower than the other values. The PSNR comparison indicated that, when the smoothing constant h was 1.5, the best results were generally achieved, and the global smoothing constant in the proposed algorithm was set to h(1.5). The center window and search window were set to ρ = 0.1, S = 10.

Fig. 6. Comparisons of PSNR for different parameters. (a) Lena (b) Goldhill (c) Barbara (d) Boat

C. Comparison of Filtering Results

To explain the validity of the proposed method, denoised results are presented. Figures 7 and 8 show the enlarged images after denoising from the AWGN-damaged images with σ = 30 to demonstrate the basic denoising function of the proposed method. The simulation results demonstrated that the proposed algorithm had a clear advantage in denoising the edge area and better results in portraying details than the existing algorithms. Figures 9 and 10 show the denoised results for regions with strong edges.

Fig. 7. Simulation result of Lena image (σ = 30).

Fig. 8. Simulation result of Goldhill image (σ = 30).

Fig. 9. Simulation result of Barbara image (σ = 30).

Fig. 10. Simulation result of Boat image (σ = 30).

As shown in the figures, the proposed algorithm outperformed the other algorithms in terms of edge preservation. In particular, the proposed algorithm produced a clearer contrast in the stripes in the Barbara image than SKNLM and performed better at suppressing excessive blurring and preserving the details.

Tables 1 and 2 present a quantitative comparison of the results for PSNR and SSIM between the existing and proposed methods. The PSNR results in Table 1 indicate that the proposed algorithm suppresses excessive blurring, resulting in better results. The quantitative comparison demonstrated that the proposed algorithm produced better PSNR and SSIM values than existing methods.

Table 1 . Comparison of PSNR results

PSNR [dB]
ImageMethodσ = 5σ = 10σ = 15σ = 20σ = 25σ = 30
LenaBTV30.5831.3130.2629.4228.8028.21
SB-ITV36.5433.6231.9030.7629.9429.18
SKNLM33.5133.3832.6931.2129.0626.74
PFA37.8934.8933.2231.9930.9830.02
GoldhillBTV30.2029.2928.5127.8527.3226.86
SB-ITV34.4431.5530.0729.0628.3027.65
SKNLM30.4530.5230.4329.7028.2526.30
PFA36.1632.9330.8829.8128.8328.13
BarbaraBTV24.5524.4324.2524.0523.8423.65
SB-ITV32.8428.7426.7925.7124.9824.47
SKNLM27.8428.0128.0727.7426.7225.22
PFA37.1833.4231.3829.7328.5927.46
BoatBTV28.8928.1527.4226.8526.2925.87
SB-ITV34.5031.4429.7328.5927.7026.98
SKNLM30.5430.5530.3629.5227.9926.05
PFA35.8933.0530.9929.8228.7727.91


Table 2 . Comparison of SSIM results

SSIM
ImageMethodσ = 5σ = 10σ = 15σ = 20σ = 25σ = 30
LenaBTV0.81650.81460.81080.80630.80140.7924
SB-ITV0.64860.55180.49740.45590.42850.3992
SKNLM0.83550.83360.81510.75860.66220.5457
PFA0.91450.88020.85580.83160.80920.7800
GoldhillBTV0.67730.67430.67230.66940.66690.6615
SB-ITV0.89530.82290.77540.73970.71040.6863
SKNLM0.74580.75390.75560.73010.66790.5789
PFA0.92130.86120.79770.76580.73210.7103
BarbaraBTV0.66620.66390.66160.65820.65320.6468
SB-ITV0.74280.61690.53200.46980.42550.3908
SKNLM0.78120.78600.77980.74520.67670.5869
PFA0.93140.90210.87240.83610.80330.7618
BoatBTV0.68690.68460.68230.67960.67710.6733
SB-ITV0.69030.57860.50620.45400.41580.3816
SKNLM0.76770.77350.77030.73760.67020.5775
PFA0.89380.85410.80410.77420.74530.7174

A large number of studies have been conducted, and various techniques for balancing the denoising and detailed image preservation have been proposed. To solve this problem, a steering-kernel filter algorithm based on similarity comparison was proposed.

The simulation results showed that the proposed algorithm preserved the details of high-frequency regions and edge components and produced better denoising results than the existing methods. The performance of the proposed algorithm was evaluated using subjective visual effects and objective experiments based on enlarged images and PSNR comparisons, and improved results were obtained compared with those of existing methods.

The results processed using the proposed algorithm showed that a sharp picture was verified compared to other existing methods, and the differential image was smaller even in the edge components where the pixel value was significantly changed, thereby removing the AWGN.

The proposed algorithm produced 30.02 dB in the Lena image damaged by AWGN whose standard deviation was σ = 30, and the PSNR was improved by 1.81 dB, 0.84 dB, and 3.28 dB compared to BTV, SB-ITV, and SKNLM, respectively. The proposed algorithm produced 27.46 dB in the Barbara image, which improved the PSNR by 3.81 dB, 2.99 dB, and 2.24 dB compared to each existing method, respectively. In the boat image, the PSNR of the proposed algorithm was 27.91 dB, which was improved by 2.04 dB, 0.93 dB, and 1.86 dB compared to BTV, SB-ITV, and SKNLM, respectively.

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Bong-Won Cheon

received B.S., and M.S. degrees in Control and Instrumentation Engineering form Pukyong National University, Korea, in 2018 and 2020, respectively. He is currently pursuing Ph.D. at Pukyong National University. His major research interests include digital signal processing and high-frequency measurement.


Nam-Ho Kim

received B.S., M.S., and Ph.D. degrees in Electronics Engineering from Yeungnam University, Korea in 1984, 1986, and 1991, respectively. Since 1992, he has been with Pukyong National University (PKNU), Korea, where he is currently a professor in the School of Electrical Engineering. From 2004 to 2006, he was Vice Dean of the College of Engineering, PKNU. His research interests include circuits and systems, high-frequency measurement, sensor systems, image and signal processing with wavelet and adaptive filters, and communications theory.


Article

Journal of information and communication convergence engineering 2022; 20(3): 195-203

Published online September 30, 2022 https://doi.org/10.56977/jicce.2022.20.3.195

Copyright © Korea Institute of Information and Communication Engineering.

A Modified Steering Kernel Filter for AWGN Removal based on Kernel Similarity

Bong-Won Cheon 1 and Nam-Ho Kim2* , Member, KIICE

1Department of Intelligent Robot Engineering, Pukyong National University, Busan 48513, Republic of Korea
2School of Electrical Engineering, Pukyong National University, Busan 48513, Republic of Korea

Correspondence to:*Nam-Ho Kim (E-mail: nhk@pknu.ac.kr, Tel: +82-51-629-6328)
School of Electrical Engineering, Pukyong National University, Busan 48513, Republic of Korea

Received: December 24, 2021; Revised: August 22, 2022; Accepted: August 24, 2022

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Noise generated during image acquisition and transmission can negatively impact the results of image processing applications, and noise removal is typically a part of image preprocessing. Denoising techniques combined with nonlocal techniques have received significant attention in recent years, owing to the development of sophisticated hardware and image processing algorithms, much attention has been paid to; however, this approach is relatively poor for edge preservation of fine image details. To address this limitation, the current study combined a steering kernel technique with adaptive masks that can adjust the size according to the noise intensity of an image. The algorithm sets the steering weight based on a similarity comparison, allowing it to respond to edge components more effectively. The proposed algorithm was compared with existing denoising algorithms using quantitative evaluation and enlarged images. The proposed algorithm exhibited good general denoising performance and better performance in edge area processing than existing non-local techniques.

Keywords: Image processing, Steering kernel, Noise removal, AWGN

I. INTRODUCTION

The denoising of images is an important preprocessing step in systems that use image-based algorithms for detecting, recognizing, and tracking objects. Because high-frequency image details are mixed with noise in most cases, most existing image denoising methods fail to preserve edge and texture information while thoroughly eliminating noise [1,2].

Several traditional image-processing methods exploit the local structural regularity present in natural images. The rationale of de-noising algorithms is to use structural patterns to regularize the ill-posed restoration problem and smooth out texture blurry and flat regions [3,4].

Various filtering techniques to remove different types of noise and improve image quality and image recognition have been proposed [5]. Several denoising techniques, including bilateral total variation (BTV) [6] and split Bregman isotropic total variation denoising (SB-ITV) [7], have been demonstrated to effectively restore real scenes from noisy images. The local steering kernel (LSK) [8] is an effective method for removing noise while preserving the original image structures, because it solves the image noise and uncertainty problems by estimating the local structure. The steering kernel non-local means (SKNLM) [9] algorithm, which estimates original images based on the similarity of different image patches and produces superior noise removal performance has received considerable attention. However, most image denoising methods have a drawback in that it is difficult to preserve edge and text information while completely eliminating the noise because the information of the original image and noise are mixed in high-frequency regions.

In this study, we proposed a modified steering kernel filter algorithm to minimize the smoothing effects generated in the filtering process using a local region technique. The proposed algorithm set weights differently based on the similarity in a local region to improve upon the LSK techniques, which are based on the orientation of pixel values change in a local region. This study also incorporated the concept that the larger the size of the filtering window, the abler it is to remove noise, while the smaller the size of the filtering window, the abler it is to preserve details in general. Thus, in this study, we conducted a similarity comparison using an adaptive mask adjusted in size based on the noise intensity of the image. The modified steering kernel proposed in this study showed better image processing results and performance in the production of images by denoising and improving the details.

The remainder of this paper is organized as follows: In Section 2, LSK is briefly explained, and in Section 3, the proposed algorithm and its mechanism are described. Section 4 presents the simulation results and analysis. Finally, in Section 5, conclusions are presented.

II. LOCAL STEERING KERNEL

A steering kernel is a weight frequently used in image processing, and is calculated using the analysis of the gradient and orientation of pixel values. Steering kernel regression (SKR) was used to calculate the steering weight, which is based on the position and intensity of the pixels, as well as the intrinsic local structure of the samples. The size and shape of the kernel significantly affected its spread and feature-extraction characteristics. LSK spreads the kernel to the highly correlated regions.

The steering kernel considers the gradient in a local region and analyzes the radiometric similarity of the pixels. Generally, it is expressed as follow [10]:

ωijLSK=detCij2πh2expixTCijiy2h2

where h represents the global smoothing constant. Filtering intensity is adjusted based on the value of h. The higher this value, the stronger the smoothing intensity. Variable ij refers to the coordinates of an input pixel, and xy refers to the corresponding pixel in a local region.

Matrix Cij denotes a symmetric gradient covariance matrix calculated in a square-shaped local window centered on pixel ij. The covariance matrix Cij is a 2 × 2 covariance matrix determined using the local gradient and represents the shape of a kernel set in a pixel. This is expressed as follows:

Cij=GijTGij=γijUijSijUθijT
Uij=cosθijsinθij-sinθijcosθij
Sij=δij00δij1

Here, Gij denotes a local gradient matrix indicating the gradient of the pixel values in the center window, γij denotes a scaling parameter, Uij and Sij denote rotation and stretching matrices, respectively, controlled by the rotation parameter θij and stretching parameter δij. Although this approach is simple and robust to noise, the resulting covariance estimate is generally either low or unstable. A parametric approach was adopted from [11] to obtain a stable covariance matrix estimate. When matrix Gij is an m × n matrix, Uij is an orthogonal matrix of size m × n composed of the left singular vector of the matrix Gij, and GijT is an orthogonal matrix of size n × n consisting of the right singular vector.

The three parameters γij, Uij, and δij are determined using singular value decomposition (SVD) of the local gradient matrix Gij [9]. The adaptive property of the covariance matrix reflects the local structure of the image and leads to the kernel function spreading along the local edges. Therefore, the steering kernel can effectively learn the edge direction from the guidance image.

The SKR [11] is dependent not only on the position and intensity, but also on the intrinsic local structure of the samples. Thus, the size and shape of the regression kernel significantly affect its spread and feature extraction characteristics. The core of the SKR method is the LSK function, which accurately estimates the local structure even in the presence of large amounts of noise.

III. PROPOSED ALGORITHM

Digital images are acquired through sampling and quantization of images inputted through cameras and displayed after numerically expressing the brightness and color intensity in a two-dimensional (2D) space [12].

If the acquired images are damaged by noise, the quantized data includes not only the information of the original image but also noise signals. Assuming that the damaged image is Iij, a 2D digital image can be represented as follows:

Iij=Zij+εij, i1,R, j1,C

where Zij denotes the original image that is not damaged by noise and εij denotes the noise constant.

The estimation of the amount of noise in an image is important for adjusting the values of the parameters used in image processing algorithms and improving performance. Immerkær [13] proposed an algorithm to quickly and accurately estimate the noise variance in images. Fast noise variance estimation is based on the observation that a noise estimator is insensitive to the Laplacian of an image, because components, such as image edges, have a strong second-derivative component. Noise is estimated using the difference between two masks, L1 and L2, which approximates the Laplacian of an image, as expressed in the following equations:

L1=010141010, L2=12010141010
N=2L2L1=121242121

where N denotes the noise estimation operator. If the noise of each pixel has a standard deviation σest, it can be calculated with a mean of 0 and variance of 36σest2 using the noise estimation operator.

If the noise estimation operator is performed using the entire image, it can be calculated as follows:

σest2=136R2C2 image IIij*N2

where R and C represent the width and height of the image, respectively, and operator * denotes convolution.

The proposed algorithm starts by setting the center and matching windows. The center window is set to a square shape around a pixel coordinate p, which is set to a size of (2F + 1) × (2F + 1), using the constant F, which represents the size of the center window. The WijC center window can be expressed by the following equation:

WijC=Ii+k,j+l, k,lF,F

The size of the center window had a significant impact on the filtering result. The smaller the size of the mask, the better it preserves the image detail. Conversely, the larger the mask, the more advantageous it is for denoising. The proposed algorithm determines the size F of the center window based on the observation that a larger center window size is advantageous. The center window size set in the proposed algorithm can be expressed using the following equation:

F=maxroundρ,σest,0

where ρ is a constant that determines the size of the center window. The operator max [ ] is a function that returns the maximum value and round ( ) is a rounding function. The search window is an area centered on pixel coordinates p and is placed to define an area similar to the center window in the surrounding area. The search window WijS, can be obtained using the following equation:

WijS=Ii+x,j+y, x,yS,S

where x, y denotes the pixel coordinates inside the search window. WijS is set to a size of (2S + 1) × (2S + 1) using a constant S, which is the size of the search window. The search window determines the size of the local steering kernel and range of the matching mask.

A matching window of the same size as the center window was set around the pixel coordinates i + x, j + y located inside the search window, which was used to compare the similarity of the two regions. Matching window WxyM, can be obtained using the following equation:

WxyM=Ii+x+k,j+y+l, k,lF,F

Here, F represents the size of the matching window, which is the same size as the center window used in equation (9). The proposed algorithm compares the similarities of the two masks to identify the relationship between the filtering and matching masks. Similarity comparison involves a comparison of the pixel values located at the same internal coordinates of the two masks. The degree of difference between the two masks mxy after comparing the center and matching windows can be calculated using the following equation:

mxy=1F2 k,l=FFWijCk,lWxyMk,l2

The smaller the difference in the pixel values of the two masks, the lower the sxy. A low value indicates that the two masks have similar pixel arrangements or that they are very likely to be located on an extension line of continuous edges in an adjacent region. This difference improves the filtering result of the edge component of the image and text regions.

The lower the difference between the filtering and matching masks, the more similar the two masks are. If the two masks are dissimilar, the pixels of the coordinate must be excluded from the calculation. The proposed algorithm selected a smaller value for the difference than the threshold based on the differences calculated in equation (13), and the pixel values of the coordinates were used in the final calculation.

Weight Exy is calculated based on the difference between the two masks and the threshold, as follows:

Exy=1, if mxy<t1σest
Exy=34, if t1σestmxy<t2σest
Exy=12, if t2σestmxy<t3σest
Exy=14, if t3σestmxy<t4σest
Exy=0, if t4σestmxy

where t1, t2, t3, and t4 are constants used to set the value of the selection mask according to the threshold. The threshold value increases the weights based on the decision that the filtering and matching windows are similar, as indicated by the low value of mxy. Weight Exy, assigned based on the threshold, is applied to the local steering weight ωijLSK, and used to emphasize regions with higher similarity. The weight, Txy, calculated based on the two weights, can be expressed by equation (19) as follows:

Txy=ExyωijLSK

where ωijLSK denotes the modification of the local steering weight ωijLSK based on the internal coordinates of the WijSs.

Figure 1 shows regions that have different characteristics in the test image to verify the effect of the weight Txy.

Figure 1. Original sample image to compare weight distribution.

Figures 2, 3, and 4 present enlarged images of regions I, II, and III in Figure 1, respectively.

Figure 2. Schematic of enlarged image and weight distribution of sample image (area I). (a) Enlarged image (b) Local Steering weight (c) Weight of proposed algorithm.

Figure 3. Schematic of enlarged image and weight distribution of sample image (area II). (a) Enlarged image (b) Local Steering weight (c) Weight of proposed algorithm.

Figure 4. Schematic of enlarged image and weight distribution of sample image (area III). (a) Enlarged image (b) Local Steering weight (c) Weight of proposed algorithm.

Each region's local steering weight ωijLSK and weight Txy of the proposed algorithm for comparison are also shown.

Figure 2 shows the weight in a region that is judged to have a large difference from the pixel value in the center's black region and is therefore set to a low value. Figure 3 shows that the weight is modified into a sharply distributed shape along the edge component.

Figure 4 shows that the weight is evenly distributed overall, because it is the region whose change in pixel values is not very large.

It can be confirmed that the proposed weights provide high weights to pixel values close to the center pixel, and simultaneously distinguish regions with similar structural features.

The filtering of the output image Oij using weight Txy can be expressed as follows:

Oij=1c xyW ijSWijSTij, c= xyW ijSTxy

The proposed method is summarized in Algorithm 1.

Modified steering kernel filter algorithm

Inputs:
  • Noised input image (Iij);

  • Global smoothing parameter (h).

    • Initialization:

      • Calculate the noise estimation (σest)

    • Iteration:

      • Coordinates of pixels on the noisy image (Iij);

      • Calculate the steering kernel matrices (ωijLSK) according to Eq. (1);

      • Set the center window (Wijc), search window (WijS), and matching window (WxyM) according to Eqs. (9), (11), and (12);

      • Calculate the difference parameter (mxy) according to Eq. (13);

      • Set the weight matrices (Exy) according to Eqs. (14),-(18);

      • Calculate the modified steering kernel matrices (Txy) according to Eq. (19);

      • Output image (Oxy) according to Eq. (20);

      • Return to 2) (a).

    • Result:

      • Output the de-noised image (Oij).

In this case, the constants, such as the global smoothing parameter, are the same as those in the existing LSK method.

IV. EXPERIMENTAL RESULT

A. Experimental Setting

To investigate the efficacy of the proposed algorithm, a series of experiments were conducted with applications including edge-aware smoothing, detail enhancement, denoising, and de-hazing. Four well-known 512 × 512 8-bit gray images were used to evaluate the denoising function objectively, as shown in Fig. 5. Noisy images were prepared by setting the standard deviation of AWGN to five levels within a range of 5 to 30 to evaluate and analyze the denoising performance of the proposed algorithm based on the noise level.

Figure 5. Original sample images. (a) Lena (b) Goldhill (c) Barbara (d) Boat

To compare and analyze the existing noise removal techniques, the denoising results of BTV, SB-ITV, SKNLM, and the proposed algorithm were compared with the original and noisy images. For objective evaluation, the performance was analyzed using the PSNR and SSIM of the denoised result images [14,15].

B. Parameter Analysis

The relationship between the parameters used in the proposed algorithm and those of the denoised images was analyzed. The global smoothing constant, h, a parameter used in the simulation, was used to calculate the steering kernel weight (equation 1). Constant h is an important factor that influences the denoising performance of the proposed algorithm. The denoising performance and image details were analyzed by changing the h value used in the proposed algorithm from 0.5 to 2.5.

Figure 6 shows the PSNR graph of the denoised image based on the change in h value in the proposed algorithm. The denoising performance was significantly lower in the high-noise image when the parameter was set to a low value, such as h(0.5). If the parameter was high, such as h(2.5), the PSNR tended to be lower than the other values. The PSNR comparison indicated that, when the smoothing constant h was 1.5, the best results were generally achieved, and the global smoothing constant in the proposed algorithm was set to h(1.5). The center window and search window were set to ρ = 0.1, S = 10.

Figure 6. Comparisons of PSNR for different parameters. (a) Lena (b) Goldhill (c) Barbara (d) Boat

C. Comparison of Filtering Results

To explain the validity of the proposed method, denoised results are presented. Figures 7 and 8 show the enlarged images after denoising from the AWGN-damaged images with σ = 30 to demonstrate the basic denoising function of the proposed method. The simulation results demonstrated that the proposed algorithm had a clear advantage in denoising the edge area and better results in portraying details than the existing algorithms. Figures 9 and 10 show the denoised results for regions with strong edges.

Figure 7. Simulation result of Lena image (σ = 30).

Figure 8. Simulation result of Goldhill image (σ = 30).

Figure 9. Simulation result of Barbara image (σ = 30).

Figure 10. Simulation result of Boat image (σ = 30).

As shown in the figures, the proposed algorithm outperformed the other algorithms in terms of edge preservation. In particular, the proposed algorithm produced a clearer contrast in the stripes in the Barbara image than SKNLM and performed better at suppressing excessive blurring and preserving the details.

Tables 1 and 2 present a quantitative comparison of the results for PSNR and SSIM between the existing and proposed methods. The PSNR results in Table 1 indicate that the proposed algorithm suppresses excessive blurring, resulting in better results. The quantitative comparison demonstrated that the proposed algorithm produced better PSNR and SSIM values than existing methods.

Table 1 . Comparison of PSNR results.

PSNR [dB]
ImageMethodσ = 5σ = 10σ = 15σ = 20σ = 25σ = 30
LenaBTV30.5831.3130.2629.4228.8028.21
SB-ITV36.5433.6231.9030.7629.9429.18
SKNLM33.5133.3832.6931.2129.0626.74
PFA37.8934.8933.2231.9930.9830.02
GoldhillBTV30.2029.2928.5127.8527.3226.86
SB-ITV34.4431.5530.0729.0628.3027.65
SKNLM30.4530.5230.4329.7028.2526.30
PFA36.1632.9330.8829.8128.8328.13
BarbaraBTV24.5524.4324.2524.0523.8423.65
SB-ITV32.8428.7426.7925.7124.9824.47
SKNLM27.8428.0128.0727.7426.7225.22
PFA37.1833.4231.3829.7328.5927.46
BoatBTV28.8928.1527.4226.8526.2925.87
SB-ITV34.5031.4429.7328.5927.7026.98
SKNLM30.5430.5530.3629.5227.9926.05
PFA35.8933.0530.9929.8228.7727.91


Table 2 . Comparison of SSIM results.

SSIM
ImageMethodσ = 5σ = 10σ = 15σ = 20σ = 25σ = 30
LenaBTV0.81650.81460.81080.80630.80140.7924
SB-ITV0.64860.55180.49740.45590.42850.3992
SKNLM0.83550.83360.81510.75860.66220.5457
PFA0.91450.88020.85580.83160.80920.7800
GoldhillBTV0.67730.67430.67230.66940.66690.6615
SB-ITV0.89530.82290.77540.73970.71040.6863
SKNLM0.74580.75390.75560.73010.66790.5789
PFA0.92130.86120.79770.76580.73210.7103
BarbaraBTV0.66620.66390.66160.65820.65320.6468
SB-ITV0.74280.61690.53200.46980.42550.3908
SKNLM0.78120.78600.77980.74520.67670.5869
PFA0.93140.90210.87240.83610.80330.7618
BoatBTV0.68690.68460.68230.67960.67710.6733
SB-ITV0.69030.57860.50620.45400.41580.3816
SKNLM0.76770.77350.77030.73760.67020.5775
PFA0.89380.85410.80410.77420.74530.7174

V. CONCLUSION

A large number of studies have been conducted, and various techniques for balancing the denoising and detailed image preservation have been proposed. To solve this problem, a steering-kernel filter algorithm based on similarity comparison was proposed.

The simulation results showed that the proposed algorithm preserved the details of high-frequency regions and edge components and produced better denoising results than the existing methods. The performance of the proposed algorithm was evaluated using subjective visual effects and objective experiments based on enlarged images and PSNR comparisons, and improved results were obtained compared with those of existing methods.

The results processed using the proposed algorithm showed that a sharp picture was verified compared to other existing methods, and the differential image was smaller even in the edge components where the pixel value was significantly changed, thereby removing the AWGN.

The proposed algorithm produced 30.02 dB in the Lena image damaged by AWGN whose standard deviation was σ = 30, and the PSNR was improved by 1.81 dB, 0.84 dB, and 3.28 dB compared to BTV, SB-ITV, and SKNLM, respectively. The proposed algorithm produced 27.46 dB in the Barbara image, which improved the PSNR by 3.81 dB, 2.99 dB, and 2.24 dB compared to each existing method, respectively. In the boat image, the PSNR of the proposed algorithm was 27.91 dB, which was improved by 2.04 dB, 0.93 dB, and 1.86 dB compared to BTV, SB-ITV, and SKNLM, respectively.

Fig 1.

Figure 1.Original sample image to compare weight distribution.
Journal of Information and Communication Convergence Engineering 2022; 20: 195-203https://doi.org/10.56977/jicce.2022.20.3.195

Fig 2.

Figure 2.Schematic of enlarged image and weight distribution of sample image (area I). (a) Enlarged image (b) Local Steering weight (c) Weight of proposed algorithm.
Journal of Information and Communication Convergence Engineering 2022; 20: 195-203https://doi.org/10.56977/jicce.2022.20.3.195

Fig 3.

Figure 3.Schematic of enlarged image and weight distribution of sample image (area II). (a) Enlarged image (b) Local Steering weight (c) Weight of proposed algorithm.
Journal of Information and Communication Convergence Engineering 2022; 20: 195-203https://doi.org/10.56977/jicce.2022.20.3.195

Fig 4.

Figure 4.Schematic of enlarged image and weight distribution of sample image (area III). (a) Enlarged image (b) Local Steering weight (c) Weight of proposed algorithm.
Journal of Information and Communication Convergence Engineering 2022; 20: 195-203https://doi.org/10.56977/jicce.2022.20.3.195

Fig 5.

Figure 5.Original sample images. (a) Lena (b) Goldhill (c) Barbara (d) Boat
Journal of Information and Communication Convergence Engineering 2022; 20: 195-203https://doi.org/10.56977/jicce.2022.20.3.195

Fig 6.

Figure 6.Comparisons of PSNR for different parameters. (a) Lena (b) Goldhill (c) Barbara (d) Boat
Journal of Information and Communication Convergence Engineering 2022; 20: 195-203https://doi.org/10.56977/jicce.2022.20.3.195

Fig 7.

Figure 7.Simulation result of Lena image (σ = 30).
Journal of Information and Communication Convergence Engineering 2022; 20: 195-203https://doi.org/10.56977/jicce.2022.20.3.195

Fig 8.

Figure 8.Simulation result of Goldhill image (σ = 30).
Journal of Information and Communication Convergence Engineering 2022; 20: 195-203https://doi.org/10.56977/jicce.2022.20.3.195

Fig 9.

Figure 9.Simulation result of Barbara image (σ = 30).
Journal of Information and Communication Convergence Engineering 2022; 20: 195-203https://doi.org/10.56977/jicce.2022.20.3.195

Fig 10.

Figure 10.Simulation result of Boat image (σ = 30).
Journal of Information and Communication Convergence Engineering 2022; 20: 195-203https://doi.org/10.56977/jicce.2022.20.3.195

Table 1 . Comparison of PSNR results.

PSNR [dB]
ImageMethodσ = 5σ = 10σ = 15σ = 20σ = 25σ = 30
LenaBTV30.5831.3130.2629.4228.8028.21
SB-ITV36.5433.6231.9030.7629.9429.18
SKNLM33.5133.3832.6931.2129.0626.74
PFA37.8934.8933.2231.9930.9830.02
GoldhillBTV30.2029.2928.5127.8527.3226.86
SB-ITV34.4431.5530.0729.0628.3027.65
SKNLM30.4530.5230.4329.7028.2526.30
PFA36.1632.9330.8829.8128.8328.13
BarbaraBTV24.5524.4324.2524.0523.8423.65
SB-ITV32.8428.7426.7925.7124.9824.47
SKNLM27.8428.0128.0727.7426.7225.22
PFA37.1833.4231.3829.7328.5927.46
BoatBTV28.8928.1527.4226.8526.2925.87
SB-ITV34.5031.4429.7328.5927.7026.98
SKNLM30.5430.5530.3629.5227.9926.05
PFA35.8933.0530.9929.8228.7727.91

Table 2 . Comparison of SSIM results.

SSIM
ImageMethodσ = 5σ = 10σ = 15σ = 20σ = 25σ = 30
LenaBTV0.81650.81460.81080.80630.80140.7924
SB-ITV0.64860.55180.49740.45590.42850.3992
SKNLM0.83550.83360.81510.75860.66220.5457
PFA0.91450.88020.85580.83160.80920.7800
GoldhillBTV0.67730.67430.67230.66940.66690.6615
SB-ITV0.89530.82290.77540.73970.71040.6863
SKNLM0.74580.75390.75560.73010.66790.5789
PFA0.92130.86120.79770.76580.73210.7103
BarbaraBTV0.66620.66390.66160.65820.65320.6468
SB-ITV0.74280.61690.53200.46980.42550.3908
SKNLM0.78120.78600.77980.74520.67670.5869
PFA0.93140.90210.87240.83610.80330.7618
BoatBTV0.68690.68460.68230.67960.67710.6733
SB-ITV0.69030.57860.50620.45400.41580.3816
SKNLM0.76770.77350.77030.73760.67020.5775
PFA0.89380.85410.80410.77420.74530.7174

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