Image Fusion of Noisy Images Based on Simultaneous Empirical Wavelet Transform

2.1 2D simultaneous empirical wavelet transform

Since in the real world, two-dimensional signals need to be processed, in the one-dimensional SEWT method, the shape of the signals was changed, and it was written to the approximate layer and the minor layers to the two-dimensional space. In such processes, some vital location information may be lost among adjacent pixels. That's why the Littlewoods-Paley two-dimensional wave converter was used [13].

I_i(p)(i=1,…….,N_i) Processed images (two-dimensional signals) with p where the pixel locations are displayed and $\mathfrak{F}_{\mathrm{P}}\left(\mathrm{I}_{\mathrm{i}}\right)(\theta,|\omega|)$ The Fourier Transform shows the polarity that "θ" refers to the Fourier Transform angle. Before identifying the February boundary for each image, it is necessary to calculate the mean spectrum with respect to "θ" such as relation (1):

$\mathscr{F}\left(I_{i}\right)(|\omega|)=\frac{1}{N_{\theta}} \sum_{k=1}^{N_{\theta}} \mathfrak{F}_{p}\left(I_{i}\right)(\theta,|\omega|)$            (1)

In relation (1), the Fourier spectrum indirectly affects the experimental Littlewoods-Paley waves. So, all the images that need to be decomposed are calculated. In relation (2), the average of the spectrum is calculated twice, according to i:

$\mathrm{F}(|\omega|)=\frac{1}{\mathrm{~N}_{\mathrm{i}}} \sum_{\mathrm{i}=1}^{\mathrm{N}_{\mathrm{i}}} \mathrm{F}\left(\mathrm{I}_{\mathrm{i}}\right)(|\omega|)=\frac{1}{\mathrm{~N}_{\mathrm{i}} \mathrm{N}_{\theta}} \sum_{\mathrm{i}=1}^{\mathrm{N}_{\mathrm{i}}} \sum_{\mathrm{i}=1}^{\mathrm{N}_{\theta}} \mathfrak{F}_{\mathrm{p}}\left(\mathrm{I}_{\mathrm{i}}\right)\left(\theta_{1},|\omega|\right)$           (2)

The mean range then identifies the Fourier boundary as in relation (2). Therefore, the diagnostic results are stored in the set, $\omega_{i}^{m}(m=1 \ldots M)$. So the set of two-dimensional simultaneous experimental wavelet transforms as that $B=\left\{\phi_{I}(p),\left\{\varphi_{m}(p)\right\}_{m=1}^{M-1}\right\},$ and $\phi_{I}(\omega)$ refers to the function of the experimental scale, and the set $\left.\left\{\varphi_{m}(t)\right\}_{m=1}^{M-1}\right\}$ shows the experimental wave transform function. Finally, the approximation layer of $\mathrm{I}_{i}(\mathrm{p})$ is defined as the relation (3):

$\mathrm{W}\left(\mathrm{I}_{\mathrm{i}}\right)(0, \mathrm{p})=\mathfrak{F}_{\omega}^{-1}\left(\mathfrak{F}_{\mathrm{p}}\left(\mathrm{f}_{\mathrm{i}}\right)(\omega) \overline{\mathscr{F}_{\mathrm{P}}\left(\phi_{1}\right)(\omega)}\right)$         (3)

The detail layers of image I_i(p) are defined as relation (4):

$\mathrm{W}\left(\mathrm{I}_{\mathrm{i}}\right)(\mathrm{m}, \mathrm{p})=\mathfrak{F}_{\mathrm{p}}^{-1}\left(\mathscr{F}_{\mathrm{p}}\left(\mathrm{I}_{\mathrm{i}}\right)(\omega) \overline{\left.\mathscr{F}_{\mathrm{P}}\left(\varphi_{\mathrm{m}}\right)(\omega)\right)}\right.$              (4)

Finally, the inverse of the two-dimensional simultaneous Empirical Wavelet Transform is defined by Eq. (5):

$\mathrm{I}_{\mathrm{i}}(\mathrm{p})=\mathrm{W}\left(\mathrm{I}_{\mathrm{i}}\right)(0, \mathrm{p}) \diamond\left(\phi_{1}\right)(\mathrm{p})+\sum_{\mathrm{m}=1}^{\mathrm{M}-1} \mathrm{~W}\left(\mathrm{I}_{\mathrm{i}}\right)(\mathrm{m}, \mathrm{p}) \diamond \varphi_{\mathrm{m}}(\mathrm{p})$             (5)

2.2 Image fusion algorithm

I_A and I_B show the input images with the size of M×N, and I_F shows the output or fusion image [13]. The process of fusion images is done in three steps:

(1) According to the two input images I_A and I_B, the Simultaneous Empirical Wavelets is simultaneously created as $\mathrm{B}=\left\{\phi_{1}(\mathrm{p}),\left\{\varphi_{i}(\mathrm{p})\right\} \begin{array}{l}L-1 \\ i=1\end{array}\right\}$. These two images are then decomposed into sub-bands by using Eq. (3) and Eq. (4), $W_{A}=\left\{W_{A}^{0}, W_{A}^{1}, \ldots, W_{A}^{L}\right\} \quad$ and $\quad W_{B}=\left\{W_{B}^{0}, W_{B}^{1}, \ldots, W_{B}^{L}\right\}$ respectively.

$\mathrm{W}_{*}^{0}$ represents the approximation layer and $\mathrm{W}_{*}^{1}, \ldots, \mathrm{W}_{*}^{\mathrm{L}}$ represents the details layers.

(2) According to the designed rules, the approximation layer and details layers are combined to obtain the fused layers $\mathrm{W}_{F}=\left\{\mathrm{W}_{F}^{1}, \mathrm{~W}_{\mathrm{F}}^{1}, \ldots, \mathrm{W}_{F}^{\mathrm{L}}\right\}$. Also, $\mathrm{r}=\left\{r_{0}\left\{r_{i}\right\}_{i=1}^{L}\right\}$ shows the fusion map set, in which $r_{0}$ is fusion map for approximation layer, and $\left\{r_{i}\right\}_{i=1}^{L}$ are fusion map for detail layers.

$W_{F}^{0}=r_{0} W_{A}^{0}+\left(1-r_{0}\right) W_{R}^{0}$       (6)

and

$W_{F}^{i}=r_{i} W_{A}^{i}+\left(1-r_{i}\right) W_{B}^{i} \quad(\mathrm{i}=1, \ldots \mathrm{L})$       (7)

(3) Conducts inverse 2D SEWT on $\mathrm{W}_{F}$. This is done to obtain the merged image using relation (5) [9].

2.3 Fusion rules

For each input image in the approximation layer, relation (8) is used [13].

$\mathcal{S}\left(\mathrm{W}_{*}^{0}(\mathrm{p})\right)=\frac{1}{\mathrm{M} \times \mathrm{N}} \sum_{\mathrm{w}_{(\mathrm{q}) \square \mathrm{W}_{*}^{0}}} \mathrm{D}\left(\mathrm{W}_{*}^{0}(\mathrm{p}), \mathrm{W}_{*}^{0}(\mathrm{q})\right)$          (8)

where, $\mathrm{D}\left(\mathrm{W}_{*}^{0}(\mathrm{p}), \mathrm{W}_{*}^{0}(\mathrm{q})\right)$ indicates the distance between the intensities of the coefficients P and Q, $\mathrm{D}\left(\mathrm{W}_{*}^{0}(\mathrm{p}), \mathrm{W}_{*}^{0}(\mathrm{q})\right)=\left|\mathrm{W}_{*}^{0}(\mathrm{p})-\mathrm{W}_{*}^{0}(\mathrm{q})\right|$. According to the sigmoid function, approximation layer rule is stated:

$\mathrm{r}_{0}=\frac{1}{1+\mathrm{e}^{-\mathrm{a}\left(\mathcal{S}\left(\mathrm{w}_{\mathrm{A}}^{0}\right)-\mathcal{S}\left(\mathrm{w}_{\mathrm{B}}^{0}\right)\right)}}$          (9)

In which a is set to 0.5. The fusion rule is defined for detail layers by Eq. (10) as,

$r_{i}=\left\{\begin{array}{l}1\left|W_{A}^{i}\right|>\left|W_{A}^{i}\right| \\ 0 \text { otherwise }\end{array} \quad(i=1, \ldots . L)\right.$       (10)

In the final step, the approximation layer and the partial layers are obtained using Eq. (6) and Eq. (7) [13].

2.4 Denoising proposed

In digital image processing, de-noising technique is used to reduce, or if possible remove, noise without losing detail. There are various techniques for de-noising digital images. Gaussian noise can be reduced using a spatial filter. While smoothing an image, the blurring of fine-scaled image edges and details are possible for corresponding to the high frequencies blocked by the process [14, 15]. Among the spatial filtering techniques, the most prevalent noise removal techniques are comprised of mean (convolution) filtering, median filtering and Gaussian smoothing. Most of the standard de-noising algorithms are dealt with an individual filtering process [16]. The result decreases the noise, but the image is blurred or smoothed at lines and edges due to high frequency losses [17, 18].

In order to effectively reduce the noise together with minimum adverse outcome, this research is developed to examine and evaluate various filters and wavelet transform for analyzing a digital image in useful sub-bands. Wavelet transform and filtering, as well as a median filter are individually used for image de-noising in various domains [14]. In this work, we explore some combinations, which have the ability to produce better outcomes for digital image de-noising.

Application of empirical Wavelet Transform (DWT) approach is highlighted for digital image de-noising in the following six steps.

(1) Apply EWT on a noisy image to Divide into four sub-bands (A, H, V, D) by using wavelet filter families [14].

(2) Choose one of the suitable threshold rule methods to compute a threshold value for one of the detail bands H, V, D.

(3) Implement a comparison between pixels in the sub-bands (H, V, D) and specific threshold value for that band. Set the pixel to zero if its value is less than the threshold of the band. Otherwise, test the next pixel value. This process is repeated for all of the pixels in the selected sub-band.

(4) The inverse EWT (IEWT) is used for selected substrates to obtain a noise-free image.

(5) Apply median filter within WT alone. It is done either before or after threshold as needed in each case. The median filter is applied to the entire image using a 3x3 mask filter. Sort all of the pixel values from surrounding neighborhood in an ascending manner. Exchange the pixel to the centeral pixel value [14].

(6) implement a comparison between the de-noised and original image. It is accomplished by computing image quality for the denoising technique.

Thresholding is a simple and effective method for image segmentation, which is used for generating digital images from a grayscale image. As such, threshold technique is utilized for digital image de-noising in a beneficial manner. In this study, the threshold technique is used in digital wavelet domain. Because the conversion of the experimental waveform decomposes the images into details layers and the approximation layer, the details layers with different scales and resolutions are examined. The details layers have a higher frequency, and noise can be eliminated by different methods of thresholding.

With developing this method, the noise, from the image of the loss of vital information, should be significantly reduced, which is of interest to medical [14]. Therefore, a good threshold value would lead to less image distortion. Threshold value is computed by applying Eq. (11) as follows:

$\mathrm{Th}=\frac{\sigma^{2}}{\sigma_{x}}$       (11)

where, $\sigma^{2}$ is related to noise value of noisy image. It is computed by Robust Median Estimator, which is estimated from HH sub-band. It is defined by Eq. (12).

$\sigma^{2}=\left[\frac{\operatorname{median} \mid\left\{\mathrm{x}_{i, j}: \mathrm{i}, \mathrm{j} \in \mathrm{HH}_{1}\right\}}{0.6745}\right]^{2}$        (12)

and $\sigma_{x}$ refers to standard deviation (STD) of the sub-bands. It is computed by Eq. (13).

$\sigma=\sqrt{\frac{\sum_{i=0}^{n-1}\left(x_{i}-m\right)^{2}}{n}}$       (13)

where:

σ: standard deviation (STD),

m: mean,

n: Indicates the number of pixels in each sub-band,

x_i: Indicates each sub-band.

1.png

Figure 1. Denoising proposed

In the proposed denoising technique, EWT is applied on a noisy image to obtain detail sub-bands. Then, median filter is applied on each sub-band independently using Eq. (11), (12) and (13). Then, IEWT is considered for the sub-bands to obtain a de-noisy (de-noised) image. Figure 1 shows the block diagram for the third case and Median filter before threshold.