A New Image Denoising Method Based on Adaptive Multiscale Morphological Edge Detection

Wavelet transform is an effective method for removal of noise from image. But traditional wavelet transform cannot improve the smooth effect and reserve image’s precise details simultaneously; even false Gibbs phenomenon can be produced. This paper proposes a new image denoising method based on adaptive multiscale morphological edge detection beyond the above limitation. Firstly, the noisy image is decomposed by using one wavelet base.Then, the image edge is detected by using the adaptive multiscale morphological edge detection based on thewavelet decomposition.On this basis, wavelet coefficients belonging to the edge position are dealt with with the improved wavelet domain wiener filtering, and the others are dealt with with the improved Bayesian threshold and the improved threshold function. Finally, wavelet coefficients are inversely processed to obtain the denoised image. Experimental results show that this method can effectively remove the image noise without blurring edges and highlight the characteristics of image edge at the same time. The validation results of the denoised images with higher peak signal to noise ratio (PSNR) and structural similarity (SSIM) demonstrate their robust capability for real applications in the future.


Introduction
The image will be contaminated by random noise in the process of collection and transmission, which would inevitably lead to the degradation of the image quality in the subsequent process such as image compression and feature extraction.Hence, it is important to estimate the original image from the noisy image [1].
Image denoising is the basic problem of signal recovery in image process and is required to reduce or eliminate the noise of the observed images, while preserving the texture, corner, and edge details of the original image as much as possible is needed.Many scholars have devoted a lot of energy to the study of image denoising and put forward many effective methods over the past few decades.Wavelet analysis has good localization properties and multiresolution characteristics in either time domain or frequency domain, which makes it more effective to distinguish useful signal and noise.Therefore, wavelet has become a very effective method for the image denoising [2][3][4][5][6][7][8][9][10].At present, there are mainly three kinds of wavelet denoising techniques to remove noise from image data.The first is to separate signal from noise through the singularity detection with wavelets [11][12][13]; the second is to reduce image noise through the wavelet coefficient thresholding method [14][15][16]; and the third is to reduce image noise through the wavelet domain Bayesian threshold criterion coefficient of shrinkage method [17][18][19].Among them, wavelet coefficient thresholding method is the most widely used method for the image denoising because of its simpleness and effectiveness.The basic idea of wavelet coefficient thresholding method is derived from theory of Donoho.The general threshold denoising method based on orthogonal wavelet transform is initially presented by Donoho, which is based on the assumption that the complex noise reduction problem can be solved by simple coefficient process.However, the estimation of Donoho threshold does not have the adaptability for different scale spaces and will zero the wavelet coefficients in excess, which will lead to the loss in image details.So many scholars have proposed different scales of wavelet coefficients using the adaptive threshold to reduce noise, such as VisuShrink threshold, SureShrink threshold, and NormalShrink threshold.Although these algorithms can obtain better denoising effect to some extent, more details are eliminated so that the image quality is severely reduced; even false Gibbs phenomenon can be produced.
In this paper, we propose a new image denoising method based on adaptive multiscale morphological edge detection.Traditional or modern image edge detection method is accurate for the edge location of the nonnoisy image.However, the edge of the noisy image cannot be detected by it.In contrast, adaptive multiscale morphological edge detection method is able to locate the edge location of noisy image accurately.This is because the adaptive multiscale morphological edge detection method uses mathematical morphology theory based on the multistructuring elements and is constructed by using the improved morphological edge detection operator.Therefore, the adaptive multiscale morphological edge detection method can preserve more edge information by changing the size of structuring elements [20][21][22].We decompose the noisy image by using wavelet firstly.Then, the edge of the image is detected via adaptive multiscale morphological edge detection according to the characteristics of the wavelet decomposition.On this basis, the wavelet coefficients belonging to the edge position are dealt with with the improved wavelet domain wiener filtering and the others are dealt with with the improved Bayesian threshold and the improved threshold function.Finally, wavelet coefficients are inversely processed to obtain the denoised image.The experimental results show that this method can not only remove the noise without blurring of edges and important characteristics of images but also highlight the characteristics of image edge compared with the existing methods.The denoised images obtain higher PSNR and higher SSIM; what is more, the denoising effect is better than the previous article [23][24][25][26][27][28]; hence, the method is of great application value.

Adaptive Multiscale Morphological Edge Detection
Suppose that the model of the image with Gauss white noise is as follows: where (, ) denotes the location of a pixel in the whole image space:  = 1, 2, . . ., , and  = 1, 2, . . ., ; (, ) is the original image (not containing Gauss white noise) and its size is  × ; (, ) is Gauss white noise.The wavelet transform of the noisy image   (, ) is as follows: where   is the wavelet coefficient of the noise,   is the wavelet coefficient of the original image, and   is the wavelet coefficient of the noisy image.Assume that (, ) represents a gray-scale image and (, ) is the given structuring element.  and   are the domains of definition of (, ) and (, ), respectively.
The opening operation of (, ) to the gray-scale image (, ) is as follows: The closing operation of (, ) to the gray-scale image (, ) is as follows: Assume that (, ) represents the image edge detection operator.Thus, an antinoise edge detection operator is obtained by using morphological dilation, erosion, opening, and closing operation.
Antinoise dilation operator is as follows: Antinoise erosion operator is as follows: Antinoise dilation erosion operator is as follows: Let Since the edge of the image is weak and not continuous, we propose the following improved morphological edge detection operator to improve the image effect: where 0 <  < 1 is an adjustable parameter.Because the improved morphological edge detection operator can add image edge information, the discontinuity in image is reduced to some extent.When  becomes bigger, more edge information can be detected, but antinoise performance will drop.In order to strike a balance between more edge information and antinoise performance, usually  is between 0.3 and 0.6.
In order to reduce image noise and detect the detailed information of image edge in different directions, the choosing of structuring elements is a key factor in morphological image processing.The size and shape of structuring elements decide the final result of detected edges.Multiscale morphological edge detection is to use different sizes of structuring elements to extract edge characteristics of the image.Large scale structuring elements have good ability to remove the noise, but the detection result is rough.On the other hand, the ability to remove noise is weak for small scale structuring elements but can detect the edge details very well.
We use multiscale morphological edge detection to filter noise and keep image details simultaneously.Generally speaking, the 3 × 3, 5 × 5, and 7 × 7 windows are usually used, among which 3 × 3 window is the fastest, and its edge is the most exquisite [20].In this paper, we choose six representative structuring elements as follows: where   is 3 × 3 square window:  = 1, 2, . . ., 6.
Multiscale structuring elements are defined as  =  ⊕  ⊕ ⋅ ⋅ ⋅ ⊕ , where  is scale parameter (a positive integer) and  is the given structuring element.In addition,  indicates that the large scale structuring elements are obtained by the dilation of small scale elements.
The edge detection operators   (, ) ( = 1, 2, . . ., 6) of six representative structuring elements above can be obtained by (13).Because different structuring elements have different adaptability to detect image edge, in order to preserve more edge information and make the edge more smooth, we first give different weight coefficients   to   (, ).Because performance of antinoise of the large scale structuring elements is more obvious but that of small scale structuring elements is relatively weak, it is necessary to utilize this feature to distinguish the edge of image.So we introduce the adaptive coefficient   , which can better filter out the pseudoedge at large scale and take the advantages of multiscale edge detection and better detect the edge of the image.Then sum up   (, ); that is,   (, ) = ⋃ 6 =1     (, ), where  represents scale.Image edge can be detected using   (, ).By changing the scale  of the structuring elements, the image edge information is obtained.The new edge is as follows: where [ 1 ,  2 ] is the range of scale .
The weight coefficient   is calculated as follows: (1) Get the mean filtered image under different scales structuring elements: (2) Calculate the standard deviation under different scales: (3) Based on the fact that the weight coefficients   are inversely proportional to the standard deviation Δ  , that is to say, large scale weight coefficients are bigger and small scale weight coefficients are smaller, we have where   represents adjustable parameters related with   :  = 1, 2, . . ., 6.

Improved Threshold Denoising
Bayesian threshold is derived through the statistical theory of Bayes.For a given high frequency subband, the traditional threshold value is where  Noise and  Image are the estimation for the noise variance and the standard deviation of the image on the subband, respectively.The noise variance is estimated as follows: where  , is the wavelet coefficient of subband HH 1 of wavelet decomposition and median(⋅) is the median function.
The standard deviation of the image is estimated as follows: where  × presents a template with a size of  × ;  × ( , ) present coefficient values of the image in the template.However, the distribution characteristics of the noise during the wavelet decomposition are not taken into account in (18).
Here, we introduce an adaptive coefficient , which changes with wavelet coefficients adaptively.Therefore, the adaptive threshold is as follows: where  = 1/log( + 1);  is the number of decomposition levels.We improve the traditional threshold in (18) where  can be adjusted according to the wavelet coefficients (, ) and usually  = 20.Here,  is the percentage of the wavelet coefficients (, ) and is greater than 2.The improved threshold function  can automatically change with wavelet coefficients, which can effectively overcome the defects of hard threshold function and soft threshold function.

Improved Wavelet Domain Wiener Filtering
Wiener filter is a locally adaptive linear filter using observation window.Assuming that the size of the observation window is  × , usually take  = .The following steps are locally adaptive wiener filtering in wavelet domain and can obtain the denoised image.
(1) Calculate the variance of the wavelet coefficients of the noisy image with observation window units: where (, ) is the wavelet coefficient;  =  • .
(2) Calculate the variance of the wavelet coefficients of the original image without noise: where  Noise is the noise variance and can be calculated by (19).
In the following, we will improve wavelet domain wiener filtering through adding threshold processing before step (3).The steps added are as follows: (1) Select the appropriate wavelet "coif3" to decompose the noisy image and the decomposition level is  = 4.
(2) Estimate the noise variance  2 Noise from the high frequency subband HH 1 .
(3) Calculate parameter  of each scale in order to further improve the adaptability of threshold.
where   is the length of th subband,  is the length of the noisy image,  = 1, 2, . . ., , and  is the number of decomposition levels.It is obvious that the parameter  will change adaptively with .
(4) The wavelet coefficients are processed as follows: Parameter  in ( 26) is related to SNR (signal to noise ratio).The bigger SNR becomes, the bigger parameter  becomes, the more details are retained, and the clearer the image becomes.In addition, the improved wavelet domain wiener filtering implements the minimum variance estimation.What is more, this treatment is consistent with the visual characteristics of human and the denoised image has better visual effects.

Denoising Method Proposed in This Paper
It is notable that many problems still exist in the threshold denoising.Worst of all is the fact that too much detail is eliminated, so that the image quality is severely reduced; even pseudo-Gibbs phenomenon is produced.
Combing improved Bayesian threshold with wavelet domain wiener filtering, a new method based on adaptive multiscale morphological edge detection is proposed to solve the threshold denoising problems above.Steps are as follows.
Step 1. Use the wavelet base "sym3" to decompose the noisy image into four layers.
Step 2. For each group of wavelet coefficients, apply adaptive multiscale morphological edge detection method to detect wavelet coefficients belonging to the edge of the image.
Step 3. Deal with wavelet coefficients belonging to the edge through improved wavelet domain wiener filtering, and window size of 5 × 5 is chosen for each group.
Step 4. Deal with wavelet coefficients not belonging to the edge with improved threshold denoising.
Step 5. Inversely process wavelet coefficients through the wavelet base "sym3" for each group to get the final denoised image.
The proposed method in this paper has the following characteristics.Firstly, this method based on adaptive multiscale morphological edge detection can not only detect edge details but also protect and highlight them.So this method has higher PSNR and SSIM.Secondly, the traditional Bayesian threshold is improved by adding adaptive coefficient  = 1/log( + 1), which can change with the number of decomposition levels  adaptively.To be more precise, when the decomposition level  increases, the threshold becomes smaller.This feature is in line with distribution characteristics of the noise decomposed by wavelet.By this way, the noise is removed to the maximum extent possible, but the wavelet coefficients of the original image are not affected.Finally, wavelet domain wiener filtering is improved by adding parameter  in order to further enhance the adaptability of threshold, which can retain edge features and increase clarity of the image.Thus, the proposed method has better visual effects and can provide better visual performance.

Image Denoising Evaluation
At present, there are two ways to evaluate the quality of the denoised image: subjective evaluation and objective evaluation.Subjective evaluation is to evaluate the denoising effect from the qualitative point of view.On the contrary, objective evaluation is to evaluate the denoising effect from the quantitative point of view.However, there are no uniform evaluation criteria to evaluate the denoising effect until now.PSNR and SSIM are adopted to evaluate the results of denoised image in [29,30].
Assuming that (, ) is the gray value of the original image without noise and (, ) is the gray value of the denoised image, the size of the image is  × .
PSNR is defined as follows: The smaller MSE value is, the bigger PSNR value is.It means that the difference between the original image and the denoised image is small, and the denoising effect is better.
SSIM is a kind of image quality evaluation index based on structural similarity.Assuming that the human eye is suitable for extracting the structural information from the perspective of the visual field, we have where  1 and  2 are the pixel means before and after the treatment, respectively.  is the pixel variance,  12 is the pixel covariance, and   is constant:  = 1, 2. In this paper, we take  1 = (0.01 × 255) 2 and  2 = (0.03 × 255) 2 .The range of SSIM is [0, 1] and its maximum value is 1.If SSIM is closer to 1, it indicates that the denoised image is more similar to the original image.This is to say that more edge details of the original image can be retained.

Simulation Experiments
In this section, three groups of experiments are provided to illustrate the effectiveness of the proposed method in this paper.Figure 1 is an example of remote sensing image.The denoised image is Aerial image with the size of 512 × 512 pixels.We compare the denoising results with articles [23][24][25][26][27][28] from the qualitative and quantitative aspects.The denoised images are shown in Figure 1 (the noise standard deviation   = 30), and the comparison results of denoised index PSNR and SSIM are shown in Table 1.
In Figure 1, the image processed by [23,24,27] or [28] is more blurred because it loses much edge information.The image processed by [25] or [26] includes much noise which destroys the clarity of the image.On the contrary, the denoising effect of the new method proposed in this paper is more significant and can preserve more edge information than the others.
From Table 1, we find that the new method has better performance than the others for Aerial image.The results of experiments show the high effectiveness of the proposed method in this paper for remote sensing image, no matter from the subjective or objective aspects.Figure 2 is an example of Lena, Barbara, and Boat image with the size of 512 × 512 pixels.We compare denoising results with articles [23][24][25][26][27][28] from qualitative and quantitative aspects.The denoised images are shown in Figure 2 (the noise standard deviation   = 30), and the comparison results of denoised index PSNR and SSIM are shown in Table 2.
In Figure 2, the new method highlights the features of the edge and retains most of the edge information.So the denoised edge details are more obvious and closer to the original image.Furthermore, we magnify the details to show the subtle differences between different denoised images.Because image denoising method based on adaptive multiscale morphological edge detection can detect the image edges with different direction structuring elements and each structuring element can detect an image edge, the ideal image edge with detailed character in Figure 2(k) can be obtained by combining different orientations.
In addition, from Table 2, we find that PSNR and SSIM of the reconstructed image obtained by the new method are highest.It is shown that the new method has strong denoising ability no matter in the low SNR or in the high SNR.In other words, the denoised images processed by the new method have higher peak signal to noise ratio and mean structural similarity.Figure 3 is an example of infrared image.The denoised image is infrared image of Cassiopeia A with the size of 640 × 480 pixels.We compare denoising results with articles [23][24][25][26][27][28] from qualitative and quantitative aspects.The denoised
(21)hat the improved Bayesian threshold in(21)can change with the number of decomposition levels  adaptively.When the decomposition level  increases, the threshold becomes smaller, which is in line with distribution characteristics of the noise decomposed by wavelet.By this way, the noise is removed to the maximum extent possible, but the wavelet coefficients of the original image are not affected.Next, we propose the improved threshold function as follows: (, )     >  0      (, )     ≤ ,