Impulsive noise removal for color images usually employs vector median filter, switching median filter, the total variation
Noise reduction is a fundamental issue in image processing. In order to reduce noise, a great number of works have been presented over the past several decades. The overwhelming majority of these works cope with the gray-scale images. With the increase in use of color images, however, more and more works to reduce noise for color images are rapidly growing.
Indeed, color images often are corrupted by various types of noise. In this paper, impulsive noise is only considered, containing salt-and-pepper noise and random-valued impulses where they present themselves as occurring isolated chromatic points. To yield better images from their noisy versions, a series of various methods have been proposed.
Vector median filter (VMF) [
To overcome this limitation, various kinds of switching vector filters have been proposed in the rich literature, which aim to only replace the corrupted pixels. This type of filters usually consists of noise detection and noise removal. The former identifies the corrupted pixels, and the latter applies some algorithm to those corrupted pixels. The noise detection is crucial. If a detector fails to identify corrupted pixels, then they will be left unchanged, resulting in poor filtered images. If the detector classifies a corrupted pixel correctly but also identifies noise-free pixels as corrupted, the tiny details may be lost. Therefore, accurate and complete identification determines the quality of filtered images.
Currently, there exist many techniques to identify noisy pixels. Peris-Fajarnes et al. [
Apart from the filters mentioned above, there exist a family of variational and partial differential equations. This type of methods usually adopts a variational energy minimization model to obtain the solution, such as the well-known Rudin-Osher-Fatemi (ROF) model [
Inspired by switching filters and variational methods, in this paper a decision-based marginal diffusion method is proposed to reduce impulsive noise for color images. The proposed method, in contrast to vectorial methods emphasizing the correlation between the channels, independently treats each channel in a color image and implements diffusion operations on every corrupted component rather than every vector. In addition, the proposed method divides components into different categories based on different noise characteristics. If an image is corrupted by salt-and-pepper noise, the components are divided into the corrupted and the noise-free components; if the image is corrupted by random-valued impulses, the components are divided into the corrupted, noise-free, and the possibly corrupted components. Components falling into different categories are processed differently. If a component is corrupted, a modified total variation diffusion is applied; if the component is possibly corrupted, a scaled total variation diffusion is applied; otherwise, the component is left unchanged. Experimental results show that the proposed method is robust to different noise strengths and suitable for different images, with strong noise removal capability as shown by PSNR/MSSIM/FSIM results as well as the visual quality of restored images.
The rest of this paper is organized as follows. In Section
In this section, the details of reducing salt-and-pepper noise and random-valued impulses are described, containing denoising structures, noise models, noise detection, and noise removal.
Two denoising structures are devised based on two noise characteristics, respectively. As shown in Figure
Two structures of the noise removal. The operations surrounded by the rectangle with dashed lines are repeated.
The denoising structure associated with salt-and-pepper noise
The denoising structure associated with random-valued impulses
RGB images to be processed are only considered in this paper, which contain three channels, the red, green, and blue, and a color image is defined as a two-dimensional matrix consisting of a certain number of pixels. Let
The probability of a noise-free component processed incorrectly is analyzed in the vectorial and marginal methods. Assuming detectors can correctly judge whether every component is corrupted or not, then the marginal methods can correctly process all components in the removal stage. The vectorial methods, however, may identify noise-free components as noisy to process, because the processing is based on the pixel unit rather than the component unit. If a component is corrupted with probability
Illustration of the relationship between two probabilities, where a component is corrupted and a noise-free component is processed incorrectly, assuming a vectorial method is applied.
Two detectors are used in the SPM and RDM, respectively. Both work in separate channels, based on the features of local neighborhood. In other words, a component is detected within the channel it lies in. Let
The number
In the removal stage, diffusion operations are implemented in separate channels. The main reason is twofold. First, each channel in a color image is contaminated independently, and contaminative components are independent and identically distributed. Second, in a natural image the gradients of different components of a pixel are similar to one another, where Figure
Illustration of RGB three channels and gradients of component in separate channels.
Moreover, components falling into different categories are processed in different diffusion ways. In SPM, the components are divided into the corrupted and noise-free components. If a component is corrupted, the MTV diffusion is applied; otherwise, the component is left unchanged. Let
In RDM, the components are divided into the corrupted, noise-free, and the possibly corrupted components, and a hierarchical scheme is adopted that first processes the corrupted and then the possibly corrupted ones. As shown in inset (b) in Figure
The representation of discretization at half-pixel resolution in a channel.
At the half-pixel
In this section, three metrics used to evaluate denoising performance are first introduced containing Peak Signal to Noise Ratio (PSNR), Mean Structural Similarity Index Measure (MSSIM), and extended Feature Similarity index (FSIMc), then the setting parameters are discussed, and finally the experimental results and comparisons are reported.
PSNR, given in decibels (dB), is a measurement based on mean pixel intensity errors between the noise-free and the restored images. Higher PSNR means better denoising capability. Let
The desirable number of iterations
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Dependence of PSNR measures on the parameters
Illustration of the dependence of PSNR and MSSIM results on the number of iterations.
The value of the parameter
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Comparison of the efficiency with those of other methods for the Lena image in SPM.
Measure | Method | Salt-and-pepper noise |
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PSNR | MMF | 28.81 | 28.09 | 25.57 | 21.90 | 18.09 | 14.60 | 11.86 | 9.61 | 7.82 | 6.37 |
AMMF | 28.81 | 28.09 | 25.03 | 22.09 | 21.11 | 19.35 | 18.47 | 16.91 | 15.04 | 11.30 | |
VMF | 28.52 | 27.81 | 24.92 | 20.69 | 16.89 | 13.66 | 11.21 | 9.28 | 7.68 | 6.37 | |
TVL1 [ |
26.37 | 26.31 | 26.15 | 26.00 | 25.81 | 25.45 | 25.09 | 23.94 | 17.89 | 12.40 | |
TVL1 [ |
27.44 | 27.38 | 27.18 | 23.74 | 11.00 | 5.35 | 3.57 | 3.16 | 3.01 | 3.01 | |
Ours |
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MSSIM | MMF | 0.83 | 0.82 | 0.78 | 0.66 | 0.45 | 0.25 | 0.13 | 0.06 | 0.03 | 0.01 |
AMMF | 0.83 | 0.82 | 0.75 | 0.69 | 0.68 | 0.63 | 0.61 | 0.54 | 0.45 | 0.21 | |
VMF | 0.82 | 0.80 | 0.72 | 0.54 | 0.33 | 0.18 | 0.10 | 0.06 | 0.03 | 0.01 | |
TVL1 [ |
0.72 | 0.72 | 0.72 | 0.71 | 0.70 | 0.69 | 0.68 | 0.64 | 0.42 | 0.18 | |
TVL1 [ |
0.78 | 0.78 | 0.77 | 0.69 | 0.22 | 0.03 | 0.01 | 0.00 | 0.01 | 0.00 | |
Ours |
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FSIMc | MMF | 0.95 | 0.94 | 0.93 | 0.88 | 0.79 | 0.65 | 0.53 | 0.43 | 0.37 | 0.33 |
AMMF | 0.95 | 0.94 | 0.88 | 0.83 | 0.82 | 0.77 | 0.75 | 0.70 | 0.66 | 0.57 | |
VMF | 0.94 | 0.94 | 0.90 | 0.84 | 0.73 | 0.61 | 0.50 | 0.42 | 0.37 | 0.33 | |
TVL1 [ |
0.85 | 0.85 | 0.84 | 0.84 | 0.83 | 0.82 | 0.81 | 0.79 | 0.71 | 0.59 | |
TVL1 [ |
0.87 | 0.87 | 0.87 | 0.83 | 0.49 | 0.33 | 0.30 | 0.29 | 0.29 | 0.29 | |
Ours |
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Seven noise-free images.
Visual results from the proposed SPM method for the Lena image.
Visual zoom-in result comparison between different methods on the Pepper image.
Visual zoom-in result comparison between different methods on the Baboon image.
To augment the evaluation, the proposed method is compared with other five methods including MMF [
In addition, the mean PSNR, MSSIM, and FSIM results for fixed noise are calculated for noisy images, the MMF, AMMF, VMF, TVL1 in [
The mean PSNR/MSSIM/FSIM results from different methods in SPM.
Comparison of the efficiency with those of other methods for the London image in RDM.
Measure | Method | Random valued impulses |
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PSNR | MMF | 33.70 | 32.92 | 30.89 | 27.86 | 24.71 | 21.72 | 19.29 |
AMMF | 33.70 | 28.33 | 28.89 | 26.01 | 25.57 | 23.67 | 22.21 | |
VMF | 33.51 | 32.49 | 29.38 | 25.58 | 22.27 | 19.47 | 17.33 | |
AVMF | 33.51 | 32.49 | 28.61 | 27.13 | 24.70 | 22.76 | 20.58 | |
TVL1 | 31.07 | 31.00 | 30.87 | 30.63 | 29.50 | 25.38 | 20.49 | |
Ours |
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MSSIM | MMF | 0.91 | 0.91 | 0.88 | 0.80 | 0.68 | 0.51 | 0.36 |
AMMF | 0.91 | 0.83 | 0.80 | 0.75 | 0.71 | 0.67 | 0.62 | |
VMF | 0.91 | 0.90 | 0.82 | 0.67 | 0.49 | 0.32 | 0.21 | |
AVMF | 0.91 | 0.90 | 0.79 | 0.74 | 0.66 | 0.59 | 0.53 | |
TVL1 | 0.85 | 0.85 | 0.84 | 0.84 | 0.82 | 0.77 | 0.66 | |
Ours |
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FSIMc | MMF | 0.99 | 0.98 | 0.97 | 0.95 | 0.90 | 0.83 | 0.76 |
AMMF | 0.99 | 0.93 | 0.94 | 0.88 | 0.86 | 0.80 | 0.76 | |
VMF | 0.98 | 0.98 | 0.96 | 0.92 | 0.86 | 0.78 | 0.71 | |
AVMF | 0.98 | 0.98 | 0.93 | 0.91 | 0.83 | 0.79 | 0.72 | |
TVL1 | 0.97 | 0.97 | 0.97 | 0.96 | 0.95 | 0.90 | 0.81 | |
Ours |
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Visual results from different methods on the Pepper image in RDM. (PSNR/MSSIM/FSIM).
Visual results from different methods on the Baboon image in RDM (A/B/C = PSNR/MSSIM/FSIM).
For comparisons, other five methods, MMF, AMMF, VMF, AVMF [
Similar to the comparisons in SPM, the mean PSNR, MSSIM, and FSIM results for fixed noise are calculated for noisy images, MMF, AMMF, VMF, AVMF, TVL1 [
The mean PSNR/MSSIM/FSIM results from different methods in SPM.
In this paper, a decision-based marginal total variation diffusion is proposed for impulsive noise removal. In contrast to vectorial methods, the proposed method only treats distorted components rather than distorted vectors. Furthermore, the proposed method divides components into different categories based on different noise characteristics. Components are divided into the corrupted and noise-free components in SPM and divided into the corrupted, noise-free, and the possibly corrupted components in RDM. Components falling into different categories are processed differently. If a component is corrupted, modified total variation diffusion is applied; if it is possibly corrupted, scaled total variation diffusion is applied; otherwise, the component is left unchanged. To achieve better results, a hierarchical scheme is adopted in RDM; the corrupted components are first processed and then the possibly corrupted ones. A total of 119 noisy images are tested. Experimental results show that the proposed method is robust to different noise strengths and suitable for different images, with strong noise removal capability as shown by PSNR/SSIM/FSIM results as well as the visual quality of restored images.
The authors declare that they have no conflicts of interest.
This work is partially supported by Foundation Science and Forefront Technology of Chongqing Science & Technology Commission under Grant no. cstc2016jcyjA0571 and also supported by Ph.D. Cultivation Foundation of Chongqing University of Posts and Telecommunications under Grant no. RC 2016002.