A New Approach of Image Denoising Based on Adaptive Multi-Resolution Technique

: Medical imaging and diagnostic techniques have become popular over the last two decades with the advancement of data science, data analysis, data storage, and the internet. The impact of this evolution can be seen in the fields of telemedicine and medical sciences, which allow more effective detection and treatment of various diseases. Like any other form of imaging technique, medical images are sensitive to noise and artifacts. The images become unclear with the presence of noise, and the diseases cannot be identified properly. Therefore, image denoising plays a vital role in the field of biomedical image processing. As a result, work must be done to minimize noise without sacrificing image quality. Various methods for reducing noise have already been proposed in the literature. Each method has its own set of benefits and drawbacks. In this paper, we introduce a bi-dimensional empirical mode decomposition (BEMD)-based image de-noising approach. The principal purpose of this research is to decompose noisy images depending on frequency and create a hybrid algorithm that incorporates existing de-noising approaches. The proposed algorithm is an image-dependent technique that decomposes the noisy image into several IMFs with residue, then considering the individual attributes of the IMFs, they are separately filtered. Furthermore equalization is applied to the residue for preserving the edge information. A comprehensive study is conducted over the experimental results of the benchmark test images using different performance measure matrices to quantify the effectiveness of the presented approach. In terms of subjective and objective evaluation, the reconstructed image is found to be more accurate and visually pleasing. It also outperforms the state-of-the-art image-denoising methods, especially in terms of PSNR, RMSE, correlation, and structural similarity.


I. INTRODUCTION
One of the fundamental challenges in the field of medical imaging is the radiation-sensitive property. Due to this property, various noises are encountered during the image acquisition process. The emergence of noise is random in nature and is intimately linked to image quality assessment. In the presence of these undesirable elements, image processing operations are similarly hampered. As denoising plays an important role in the application areas of image processing, such as video tracking, image analysis, restoration, registration, segmentation, and classification, where visually pleasing images are essential, a special focus is required on it (Gonzalez and Woods 2007).
Removing noise from the noisy image is still a challenging problem for researchers. Over the previous few decades, a number of authors have presented many algorithms, each with its own set of benefits and drawbacks. Some of the publications focused on the classification of noises based on their behaviour. The noises discussed in the preceding articles are either additive or multiplicative. Classical filters like as mean filters, median filters, Gaussian filters, and others are employed for spatial domain denoising (Mallat, 2008). But these classical filters not only smooth the image. However, these classical filters not only smooth the image but also blur the edges of the information. In due course, to overcome the above limitations, transformation-based filters were introduced. The Fourier Transform (FT) is one of the transformation and decomposition methods used in image processing. Later on, wavelet transformation became popular as it has a low resolution and provides simultaneous localization in the time and frequency domains.
The Wavelet transform (WT) (Gupta and Ahmad 2018;Ellinas et al. 2004) has shown its efficiency in various signal processing applications. The beauty of this method is that the decomposed signal contains the different space-frequency components. At this stage, many authors have applied some mathematical operation such as thresholding to suppress the noise (Ellinas et al. 2004;Zhang 2016;Fedak and Nakonechnyy 2015;Kimlyk and Umnyashkin 2018). Then the denoised image is reconstructed by reversing the wavelet coefficients into the spatial domain. The whole process is known as the wavelet-based denoising technique (Fedak and Nakonechnyy 2015; Kimlyk and Umnyashkin 2018;Bnou et al. 2020;Sagheer and George 2020). As per image quality, this denoising method gives a better result in terms of PSNR. Moreover, in maximum denoising cases, wavelet thresholds are applied to remove the Gaussian noise. Popular thresholding techniques used in wavelets are soft thresholding and hard thresholding (Donoho, 1995;Kumar, 2013). In soft thresholding, over smoothing affects the reconstructed image, whereas, in the case of hard thresholding, many coefficients of wavelet become zero, which causes blur and artifacts. Therefore, even though threshold based image denoising methods present favourable results, the artefacts are still noticeable (Fan, Zhang et al, 2019;Srivastava et al, 2016;Madadi et al, 2013). In addition, the wavelet transform has lower singularity and directional effect issues. From an operational point of view, DWT decomposes an image into a set of mutually orthogonal wavelet basis, for which a constant set of filters are used and these filters are not image dependent. Moreover, the inverse DWT increases the computational complexity (Chang et al, 2000;Adamo et al, 2013).
This motivated us towards an image-dependent decomposition method such as bi-dimensional empirical mode decomposition (BEMD) (Yan et al. 2013). The BEMD method is a time-domain analysis suitably used for the analysis of non-linear and non-stationary signals. By applying EMD, the image is decomposed adaptively into integral oscillatory components, and these separate components are named Intrinsic Mode Functions (IMF). The major challenges considered in this paper are the smoothing of flat areas, the protection of edge information without blurring, the preservation of internal texture, and the new artefact suppression.
To outline the paper's objective, section II demonstrates the detailed methodology in algorithm form. Section III describes the experimental results as well as comparisons with other state-of-art methods with proper evidence. The conclusion and future work are given in section IV.

II. MATERIALS AND METHODS
In this section, some of the fundamental issues related to image denoising with different types of noises having zero mean and finite variance are considered and their characteristics are elaborately discussed.

A. BEMD-based Denoising 1) Noise model
In the spatial domain, noise is broadly categorised as additive or multiplicative. The best additive noise used in maximum research work is Gaussian noise. The additive noise model is represented as: where I (x, y) is the noise-contaminated image function, M (x, y) is the original image, and n (x, y) represents the signalindependent additive Gaussian random noise with zero variance (Gonzalez and Woods, 2007). In some cases, noise arises due to environmental conditions such as voltage spikes in the circuits or random changes in the physical properties of materials. This kind of noise is categorised as multiplicative noise and is also known as "speckle" noise. The multiplicative noise model can be depicted as Eq. (2) (Gonzalez and Woods, 2007).
where 0≤e ≤ 1, with a probability p, I (t) is the noisy image at a particular time (t), M (t) is the original signal, and N (t) is the speckle noise introduced during image capture, transmission, or other processing.

2) Image decomposition
Image decomposition is an image processing technique where the image is segregated into multiple images based on its features and frequency. In this paper, we have used frequency-based decomposition using BEMD (Yan et al, 2013). The EMD can decompose the image into n levels based on the frequency of the input signal. In this model, we have used a four-level decomposition as illustrated in Figure 1.
3) Image denoising model Figure 2 demonstrates the proposed denoising model. In step 1, the data acquisition system is modelled such that the original signal is corrupted by the external noise. The noisy image (I) is decomposed into four parts using BEMD based on their frequency. Where the decomposed images from high frequency to low frequency are depicted as IMF1, IMF2, IMF3, and residue. The objective of this algorithm is to clean up the noise available in the homogeneous areas and preserve the structures like edges and corners. Owing to this, the conservation of valuable hidden information can be achieved by separately considering the decomposed images (Yan et al, 1998). Noise is often high-frequency in nature and the highfrequency.
Moreover, noise is often high-frequency in nature. Therefore, the high-frequency components of digital images are filtered to suppress the noise. As the low-frequency component holds the details of hidden structures as their pixel values change slowly over space, the residue of the image is made unfiltered. In addition, the residue is equalised to improve the brightness.

B. Bidimensional Empirical Mode Decomposition (BEMD)
The noisy medical image can be decomposed into a finite number of unique frequency components, which are known as intrinsic mode functions (IMF) (Satapathy et al, 2018;Dong et al, 2014). These IMFs are extracted by applying a sifting process that repeats the steps until less than 2 maxima points occur.
The uniqueness of the BEMD is similar to that of the EMD, which is used for one-dimensional signals. If I (x,y) is defined as the image which is to be decomposed into a series of BIMFs and a residue Eqn. (3).
where the IMFi (x, y) is the i th IMF component. The frequency of IMF1 is higher than the other IMFs. The detailed process is demonstrated in Figure 3.

C. Noise
Noise is treated as external energy that corrupts the signal and changes its characteristics. Salt-and-pepper noise is also known as impulse noise, which is a form of white and black pixel that can sometimes be seen on images. The probability density function (PDF) "S" of salt and pepper noise with variable "u" is formulated as follows: for u = k (0 < k < 2 n -1)

C) Gaussian Noise
This is statistical noise that is identically distributed at any two points in time. Sensor noise, which is caused by temperature and poor lighting, is the primary source of Gaussian noise. The probability density function G of a Gaussian random variable is given by: where Z represents the intensity of pixel the parameters µ represent the mean and standard deviation.

D) Speckle Noise
This is modelled as a multiplicative noise that arises due to the effect of environmental conditions. The probability density function F of speckle noise follows the gamma distribution and can be represented as follows in Eqn. 6.
where g is the grey level intensity, a and b are positive integers. The mean and variance of this density are b/a and b/a 2 . For an 8-bit imaging system, the a and b lies between 0 and 255.

E) Filters
Nowadays, filters are used for the suppression of highfrequency components of an image. As a result, the image is smoothed and the edge is preserved. Compared with the frequency-domain, in the spatial domain, noise removal is easier because it requires much less processing time. The filters are broadly divided into two categories: (i) linear filters and (ii) non-linear. The linear filter has the advantage of faster processing but fails to preserve the edge where a nonlinear filter can preserve the edge with the compromise of processing speed.

F) Median Filter
This is a non-linear filter having the ability to remove salt and pepper type noise by using a pre-defined window size. During the filtering process, the median filter replaces the pixel values with the median value of neighbouring pixels. Since edge information is important for an image, the median filter is useful for protecting edges during smoothing.

G) Gaussian filtering
This is a linear filter. During the filter process, it usually blurs theedges andreduces the details. The standard deviation used in the Gaussian function is playing a vital role in its behavioral feature.In the digital image domain,a twodimensional Gaussian function is used.
where p and q are the horizontal and vertical distances of the pixel from the origin. σ is the standard deviation. A Gaussian filter reduces the contrast and preserves the brightness of the filtered image. As per its characteristics, it is designated as the ideal time-domain filter.

H) Wiener Filter
This is a stationary linear filter used for inverse filtering and noise smoothing. In inverse filtering, the filter works as a high-pass filter by using de-convolution. In compression mode, it functions as a low-pass filter to remove noise and minimise the overall mean square error. This technique gives a better result in the case of additive white Gaussian noise (AWGN). The limitation of Wiener filtering is that it requires knowledge of the power spectra of the noise and the original image. The Wiener filter can be mathematically expressed as follows: where, ( 1 , 2 ) is the power spectral of the original image, S ( 1 , 2 ) is the power spectral of additive Gaussian noise, ( 1 , 2 )is the blurring filter (Vaseghi, 2001).

I) Equalization
Histogram Equalization is an image processing technique used to improve image contrast. It achieves this by effectively spreading out the most common intensity values of pixels across the screen, stretching out the intensity range.

J) Weighted Average
Over-enhancement is a common issue with image filtering. Here, a weighted average is applied to the filtered and equalised images to reduce the excessive enhancement.
Both A and B have the same size, p × q. For optimal brightness preservation the range of γ is fixed between [0, 5] and for this observation, γ is set to 1.5.The maximum and minimum grey values of image A are represented by max(A) and min(A). The function used in Eq. (11) is described as: 2) Peak signal-to-noise ratio The peak signal-to-noise ratio (PSNR) is the ratio of the signal power of the processed image to the referral image. The Higher value of PSNR represents a better quality of performance. PSNR is represented as: = 10 10 ( 2 ) (14) =20 10 (MAX) − 10 10 ( ) MAX is the maximum possible pixel value of the original image which is 255 in 8-bit image systems.

3) Structural similarity index
The SSIM is a perceptual metric used for quantifying the image quality which was degraded by the process of data compression, data transmission, and data acquisition. It is a full reference metric comparison method that requires a minimum of two images as the reference image and a processed image. The range of the SSIM is between -1 to1 to indicate the similarity. The closer to the value of 1 is more similar in structure. If α = β = γ = 1 and C3= C2/2 then the above index is simplifying to:

4) Absolute Average Difference (AAD)
The absolute average difference provides the average amount of change between the processed and reference image (Jagalingam and Hegde, 2015). AAD can be expressed as follows:

5) Maximum Difference (MD)
The maximum value of the absolute error between the processed and reference image is one of the important factor in quality assessment and is represented as follows:

6) Mean Absolute Error (MAE)
MAE is defined as the maximum absolute value, the difference between the original image and the reconstructed image (Naidu and Raol 2008). As the name suggests, the mean

Figure 8 (a) Input image (b) Image with Gaussian noise and variance 0.01 (c) median filtered image (d) wiener filtered image (e) Applying Gaussian filtered image (f) Proposed BEMD with Gaussian filter method with Gaussian filter method.
absolute error is an average of the absolute errors. MAE values range from 0 to 255, and the lower MAE value, the better demosaicing quality.

7) Normalize Absolute Error (NAE)
This quality measure can be expressed as follows: A higher NAE value shows that image is of poor quality.

8) Correlation (CORR)
The Pearson correlation coefficient computes the similarity features of the reference image and fused image. The benchmark correlation value is one when the fused and reference are exactly alike. A higher value of SC (Structural Content) shows that image is of poor quality.

9) Normalized Cross Correlation (NCC)
NCC (Normalized Cross Correlation) measure shows the comparison of the processed image and reference image (Tiwari and Singh 2004). NCC is expressed as follows: Furthermore, the Normalized Cross Correlation is confined in the range between -1 and 1 (Tiwari and Singh, 2004). NCC is expressed as follows:

10) Structural Content (SC)
This quality metric is expressed as follows:

B) Performance Analysis
In this section, the suggested model is tested according to the method discussed in Section II. For this experiment, a set of grey-level brain MRI images of 256x256 pixels is considered (Hamada 2020). The results are validated using a four-level empirical mode decomposition technique. Each test was repeated with ten different measuring parameters, and the average value of each performance measure is shown in Tables 1 through 3. The range of noise variance examined in each experiment is 0.001 to 0.1.
A few examples of MRI brain images are shown in Figures 7 to 9 for objective analysis of the proposed method. In figure 7, we have considered salt and pepper noise. Figure  7 (b) represents a noisy image with noise density 0.01. The sub Figure 7(c), 7(d) and 7(e) are showing the results of median, wiener and Gaussian filter processed image. These results are evidence that these filters are not only filtering the noise but also capable of preserving brightness. But when the Gaussian filter is considered with BEMD, the contrast of the image is also improved and the image is objectively looks better. Similar experiments were conducted for Gaussian noise and Speckle noise, and the evidence were recorded in Figures  8 and 9. In each case BEMD with Gaussian filtered images are looks better as compared to median filter, wiener filter and Gaussian filter processed image.   Figure 10 provides a comparative analysis of PSNR values in a bar chart. For a variance of 0.001, the graph shows that the bemd based filtering methods has a higher PSNR value than the median, Gaussian and wiener filter. As per literature the higher PSNR is the evidence of better noise reduction. Similarly the effectiveness of the method interms of MSE and SSIM can be seen in Fig. 11 and 12 respectively. Increasing SSIM and decreasing MSE validates the proposed method's effectiveness. The higher the SSIM value, the more likely the structure will be retained.
The average value of RMSE, PSNR, and SSIM for a set of images is calculated, and all results are listed in Table I-III.  Table I depicts the result of the images being affected by Gaussian noise. It was observed that when bemd-based filtering was compared to direct filtering, the PSNR values increased. The higher value of SSIM indicates the structural closeness between the original and the processed image in the proposed method. The lower value of RMSE is an indicator of brightness preservation. Table 1 also shows some of the error metrics, including absolute average difference, maximum difference, mean absolute error, normalised absolute error and structural content. The lower the value of the metrics, the more effective the method is. The greater Pearson correlation coefficient and normalised cross correlation value indicate that the two images have a very strong, significant, and positive relationship. Tables 2 and 3, metrics were calculated using speckle noise and salt and pepper noise. It was observed from the tables that in the presence of speckle noise and salt and pepper noise, the Gaussian filter and BEMD with Gaussian filter have nearly same PSNR value. However, in the case of salt pepper noise in Table 3, it was discovered that the median filter's PSNR value is higher than BEMD with median filter, which contradicts the effectiveness of the method. According to Wang and Bovik, the MSE and PSNR can yield contradictory findings in some circumstances, even if the outcome appears to be visually pleasing (Wang and Bovik, 2009). As per the findings presented in Tables 1 to 3, the median filter likewise outperforms the others for all noises.

IV. CONCLUSION
In this study, we have proposed a new method of algorithm for image denoising. The objective of this method is to exploit the advantages of empirical mode decomposition with a multi-resolution structure and also to demonstrate the model's resemblance to the human visual system as well as its remarkable spatial and frequency localization features. In this technique, the first IMF does not contain all the noise, so the standard filter is applied to all IMFs except the residuals. The suggested method was tested on a set of images contaminated by different types of noise. Experiments on benchmark images demonstrate that the proposed technique outperforms similar types of denoising algorithms, particularly in terms of PSNR, MSE, SSIM index, and visual effect. The obtained results do not suffer from over smoothing and loss of details. Future research will look into a soft computing-based threshold factor, which will be applied to the coefficients of the decomposed image to improve the denoising performance.

CONFLICT OF INTERESTS
The authors declare that they have no conflict of interests.

AVAILABILITY OF DATA AND MATERIALS
The datasets used and/or analyzed during the current study are available from the corresponding author on request.

AUTHOR CONTRIBUTIONS
Both authors contributed equally in this work. The authors read and approved the final manuscript.

FUNDING SOURCE
There is no source of funding for the research. .