Bayesian Framework-Based Adaptive Hybrid Filtering for Speckle Noise Reduction in Ultrasound Images Via Lion Plus FireFly Algorithm

Abstract

The existence of speckle noise in ultrasound (US) image processing distorts the image quality and also hinders the development of systematic approaches for US images. Numerous de-speckling schemes were established to date that concern speckle reduction; however, the models suffer from demerits like computational time, computational complexity, etc., that are to be rectified as soon as possible. This compulsion takes to the introduction of a new de-speckling model via an adaptive hybrid filter model that includes four filters like guided filter (GF), speckle-reducing bilateral filter (SRBF), rotation invariant bilateral nonlocal means filter (RIBNLM), and median filter (MF) respectively. Moreover, the novelty goes under the selection of optimal filter coefficients that make the process effective. Bayesian-based neural network is used to predict the appropriate filter coefficients, where the training library is constructed with the optimal coefficients. Along with this, the selection of optimal filter coefficients is done under the defined objective function using a new hybrid algorithm termed as Randomized FireFly (FF) update in Lion Algorithm (RFU-LA) that hybrids the concept of both LA and FF, respectively. Finally, the performance of the proposed de-speckling model is compared over that of other conventional models with respect to different performance measures. Accordingly, from the analysis, the mean MAPE of the proposed method are 39.13% and 49.28% higher than those of the wavelet filtering and hybrid filtering schemes for a noise variance of 0.1.

Introduction

US [1–4] imaging is comparatively cheap and quick, and is the easiest imaging process for medical analysis. The US equipment exists as the best-selling medical equipment followed by MRI. US imaging [5, 6] is a more effective scheme that views the internal parts of body. It is convenient, non-radioactive, non-invasive, and proficient when compared with other analysis schemes. Therefore, US imaging [7, 8] has turned out to be a significant application in the medical field, particularly in monitoring the development of the fetus in pregnant women and in the treatment of lesions in the abdominal regions. On the other hand, like all consistent imaging schemes, US imaging [9, 10, 20] also includes a major drawback, i.e., it is affected by speckle noise.

Speckle noise [11, 12] is defined as “the noise, which is generated by the interaction of the reflected waves from various independent scatters within a cell resolution.” The presence of speckle noise has corrupted the quality of US images significantly and it has also limited the progression of automatic analytic schemes. Speckle noise [13, 14] could be eliminated by post-processing approaches or by using compounding schemes. In the compounding technique, images are attained from diverse transducers that lessen the impact of noise. However, the compounding technique [15] is found to be more expensive. Several techniques [34–37] were introduced to minimize the speckle noise effect in US images that mainly involve local statistics-oriented schemes like Kaun, Frost, Lee, median [16], and so on. The median filter [17–19] is a constructive type of filter for reducing the impulsive noise when compared with the other types of filters. Moreover, it is exploited for minimizing the speckle noise from US images. In-kind of filter, the central pixel value of its nearby window is replaced with its median value by arranging it either in decreasing or increasing order.

During diagnosis, speckle noise occurs, which results in an inaccurate diagnosis, thus causing a great challenge for clinicians [35–38]. Thus, from the viewpoint of medical application, removal of this speckle noise, known as de-speckling [21–23, 30], has turned out to be a significant process, which has to be done before analysis and processing of US images. The process of de-speckling also provides a technological motivation for the doctors to attain more precise diagnosis, thus minimizing the possibilities of misdiagnosis.

The major contributions of this paper are depicted below.

1. As a novelty, this work establishes a novel de-speckling model using hybrid filters that hybrid GF, SRBF, RIBNLM, and MF, respectively.

2. To enhance the performance, the filter coefficients are optimally tuned, which is considered a major contribution to this work.

3. The Bayesian-based NN is exploited for predicting the appropriate filter coefficients, where the training library is constructed with the optimal coefficients.

4. Moreover, the optimal selection of filter coefficients is done using RFU-LA that hybrids the concept of both LA and FF correspondingly.

The arrangement of the paper is given as follows: "Literature Review" analyzes the literature work. "Proposed De-speckling Process Using an Adopted Hybrid Filter with Bayesian Regularized NN" describes the proposed de-speckling process using an adopted hybrid filter with Bayesian regularized NN and "Hybrid Filter with Optimized Filter Coefficient Prediction via Bayesian Regularized NN" portrays the hybrid filter with optimized filter coefficient prediction via BR-based NN. Furthermore, "A Hybrid Algorithm for Optimal or Fine-tuned Filter Coefficient Selection" illustrates the hybrid algorithm for optimal or fine-tuned filter coefficient selection outcomes. "Results and Discussion" portrays the results and the paper is concluded in "Conclusion".

Literature Review

Related Works

In 2019, Leal and Henrique [1] have established a novel wavelet-based method that was modeled to minimize the speckle noise in US images. A group of 110 images was exploited for designing the filters and for validating the outcomes. The adopted method has offered enhanced outcomes when evaluated over the conventional ones with respect to both edge conservation and noise minimization. In addition, the outcomes have offered better results over the traditional de-noising methods.

In 2017, Singh et al. [2] have presented a de-noising scheme that combined both the nonlocal and local information in a proficient way. The established approach includes three phases, where at the initial phase, the usage of local statistics was exploited for minimizing the speckle noise. Subsequently, an enhanced SRBF was presented for minimizing the speckle noise from medical images. At last, the post-processing method was adopted, which included both the advantages of NLM and BF for the reduction of speckle noise in an effective manner.

In 2016, Ju et al. [3] have established an approach that focused on a novel de-noising technique depending on an enhanced guided filter and wavelet filter. In addition, an enhanced threshold function depending on universal wavelet was introduced as per the features of US medical images. Moreover, a Bayesian maximum was deployed that attained a novel wavelet shrinkage model. Furthermore, the filtered image was attained by deploying the IWT. Finally, the experimentations have proved the better de-speckling capability of the adopted scheme over traditional methods.

In 2016, Jian et al. [4] have presented an effective framework that intended to reduce the speckle noise in the US image by means of the intensity distribution. Here, the NLM filter was used for filtering the noises by exploiting the redundancy information in noisy images. Here, this work had integrated the NLM filter along with local statistics for minimizing the speckle in US images. In addition, the weights of pixels were determined based on the similarity metrics among the patch intensities. In the end, a quantitative analysis has shown that the adopted scheme outperformed the traditional NLM and several other existing schemes.

In 2016, Iman and Ali [5] have presented a novel scheme that minimized the speckle noise in US images. The introduced scheme was formulated based on the modified Bayes shrink. Initially, TV regularization was carried out on the US image. Consequently, DWT was performed on the coefficients, which were attained from the entire deviations. Later, soft thresholding and BS were carried out on the coefficients of the detail sub-bands. Finally, the suggested scheme has outperformed other schemes by removing the speckle noise in a significant manner.

In 2011, Guo et al. [6] have suggested a new technique known as NLM, which operated well with the image de-noising and has also improved the edge information. The NLM filter was initially modeled for removing the Gaussian noise and it could not be applied directly to the US image for minimizing the speckle noise. Here, a MNLM filter was adopted for reducing the speckle noise. Furthermore, experimentations were performed on synthetic images for finding the best parameters. MNLM filter was compared with other filters and the outcomes have demonstrated that the adopted technique was capable of efficiently minimizing the speckle noise in US images.

In 2016, Iman et al. [7] have suggested a scheme that aims at reducing the speckle noise in breast cancer US images. Depending on filtering and wavelet analysis, MBS and homogeneity filtering were employed for eliminating the noise when maintaining the sharpness of the features. Initially, the threshold value of MBS was exploited for distinguishing the homogenous areas and speckle noise-affected areas. The presented scheme was thus termed as HMBS. In addition, the analysis was done with other techniques and the outcomes have demonstrated the supremacy of the adopted model method over existing models.

In 2015, Murat and Irfan [8] have introduced a scheme that intends to minimize the speckle noise in US images by enhancing the speckle-minimizing capability of the edge-sensitive filter. Here, a homogeneity map was produced, which exploited the valuable features of the filter with enhanced smoothing capability for increasing the quality of image. In addition, this work was simulated by deploying a cyst image and the outcomes have proved the better de-noising capability of the presented technique over the other existing schemes.

Review

Table 1 shows the reviews on the speckle noise reduction in US images. At first, the DWT model was introduced in [1], which offers better preservation of edge and also offers less execution time. However, it has to focus more on the combination of wavelet filter model. The Bayesian approach was exploited in [2] that provides reduced complexity and also offers improved de-noising capacity, but it requires consideration on contrast of the images. In addition, Gaussian distribution was deployed in [3] that preserves the details of the image and also offers improved PSNR. Anyhow, it needs consideration on time complexity. Likewise, NLM filter was exploited in [4], which offers reduced speckles in images and also provides high SNR. However, it requires knowledge about the location of pixels. Also, DWT method was employed in [5], which offers reduced RMSE and also enhances the PSNR; however, it needs consideration on individual interpretation. NLM filter was exploited in [6] that minimizes the speckle noise and also detects the boundaries; anyhow, it consumes more time. BS model was implemented in [7], which offers reduced RMSE and quicker simulation, but edge image has to be concentrated more. At last, the homogeneity map scheme was suggested in [8] that offers better speckle minimization and there are no edge losses.

Table 1.

Features and challenges of speckle-noise removal in US images using various techniques

Author [citation]	Adopted methodology	Features	Challenges
Leal and Henrique [1]	DWT	✥ Preservation of edge ✥ Less execution time	✥ Have to focus more on the combination of wavelet filter model
Singh et al. [2]	Bayesian approach	✥ Reduced complexity ✥ Improved de-noising capacity	✥ Requires consideration on the contrast of the images
Ju et al. [3]	Gaussian distribution	✥ Preserves the details of the image ✥ Improved PSNR	✥ Needs consideration on time complexity
Jian et al. [4]	NLM filter	✥ Reduce the speckles in images ✥ High SNR	✥ Requires knowledge about the location of pixels
Iman and Ali [5]	DWT	✥ Reduced RMSE ✥ Better PSNR	✥ Have to focus more on the individual interpretation
Guo et al. [6]	NLM filter	✥ Minimizes the speckle noise ✥ Detects the boundaries	✥ Consumes more time
Iman et al. [7]	BS model	✥ Reduced RMSE ✥ Quicker simulation	✥ Edge image has to be concentrated more
Murat and Irfan [8]	Homogeneity map scheme	✥ No edge losses ✥ Better speckle minimization	✥ Have to focus on edge filters

Several techniques were employed in speckle-noise removal in US images. However, the presented method offered some limitations and challenges such as more computational complexity, needing more focusing on the combination of wavelet filter model, contrast of the images, knowledge about the location of pixels, and requiring more focusing on edge filters [1–4]. Thus, this work presents a novel de-speckling model using hybrid filters. Moreover, for predicting the appropriate filter coefficients, the Bayesian-based NN is exploited. Furthermore, the optimal selection of filter coefficients is carried out by means of RFU-LA, which hybrids the concept of both LA and FF correspondingly.

Proposed De-speckling Process Using an Adopted Hybrid Filter with Bayesian Regularized NN

The pictorial demonstration of the presented scheme is shown in Fig. 1. The presented approach for speckle-noise reduction in US image comprises two contributions, namely (i) proposed filtering design and (ii) optimal filter coefficient prediction. Here, the de-speckling is handled by a hybrid filter model that includes four filters, namely GF, SRBF, RIBNLM filter, and MF, respectively. Here, the innovation lies in the optimal selection of filter coefficients, by which the process can be made more effective. Here, the considered filter coefficients for GF are epsilon and window size; for SRBF at two stages, the filter coefficients are domain parameter for spatial kernel, range parameter for intensity level; for RIBNLM, the filter coefficients are radio of search window and radio of similarity window respectively. Consider the filter coefficients, FC₁, FC₂, FC₃……. FC_N, where N specifies the total number of filter coefficients. Consequently, Bayesian-based NN is exploited in this work for predicting the appropriate filter coefficients, where the training library is constructed with the optimal coefficients. Together with this, the optimal selection of filter coefficients is performed under the defined objective function by means of a novel hybrid algorithm known as RFU-LA, which hybridizes the concepts of both LA and FF respectively.

Fig. 1.

Overall demonstration of the presented model

Hybrid Filter with Optimized Filter Coefficient Prediction via Bayesian Regularized NN

Proposed Hybrid Filters with NN-Based Optimal Filter Coefficient Prediction

At the initial phase [2], the GF is deployed for suppressing the speckle noise effect. The GF has included local statistics, by which the impact of the speckle noise could be reduced from the major regions of medical images. In the 2nd phase, the SRBF filter is deployed for improved restoration of fine structural information of the previously filtered; and thus, the quality of the de-noised image could be enhanced. In addition, to improve de-noising and for effectual exploitation of differently oriented “self-similar structures” available in the image, RIBNLM is proposed at the final phase. The steps involved in hybrid filters with NN-based optimal filter coefficient prediction are as follows:

• Step 1:

  Deploy GF [2] for suppressing the speckle noise effect.

• Step 2:

  Deploy SRBF [2] on pre-filtered GF image by means of the weights depending on square chord distance.

• Step 3:

  At last, for retaining the edges and or attaining the details of fine structure, RIBNLM filter [2] is deployed with weights evaluated with the rotation invariant similarity measure.

The adopted hybrid model does not include the iterative process, which diffuses the fine sharp details and weak edges. The exploitation of 2nd-order statistics at the 2nd phase assists in restoring the required fine edge details and the weight computation improves the NLM filter’s performance. The adopted scheme has reduced computational needs and the usage of diverse phases in a single hybrid model is found to be more appropriate for dealing with speckle noise–interrupted US images [2]. Generally, the de-noising performance of the presented hybrid model is considerably better over that of the individual filter. In addition, a thorough analysis indicates that the specified sequence of these 3 phases offers the optimal outcomes and any variation in the order results in “sub-optimal de-noising” performance.

GF [2]

“It is a local linear transformation that generates the outputs with respect to some guidance information.” It is an enhancement technique that performs image smoothening at a reduced computational cost. Therefore, GF is effectively deployed in numerous image development tasks.

MF [32]

It is an extensively exploited technique for smoothing and de-noising the areas of an image. This method has attained much consideration from analysts of image forensics.

SRBF [31]

It is established as a method for preserving the edges and for smoothing the images. It was effectively deployed in various computer graphics and image processing applications. Moreover, when compared over other filters, this type of filters is found to be non-iterative and simple and it also consumes less time for de-noising applications.

RIBNLM [2]

It was established for removing the noises in MR images. It exploits the ODCT3D filter for reducing the noise from MR images and, furthermore, RIBNLM was deployed for improving the de-noising performance.

Optimal Filter Coefficient Prediction by the BR-Based NN model

The major contribution relies on increasing the performance of the proposed de-speckling model; therefore, it is planned to predict the optimal or fine-tuned filter coefficients using the Bayesian-trained NN model since the training process generates a training library with optimal filter coefficients. In fact, the selection of optimal or fine-tuned filter coefficients is a critical issue. For this, the optimization concept is influenced in this work by introducing a new hybrid algorithm introduced in this paper, which is explained clearly in the subsequent section. The proposed BP-Bayesian regularization–based NN training model is shown in Fig. 2.

Fig. 2.

Proposed BP-Bayesian regularization–based NN training model

The input given to the model is the SSIM of each speckle image, and it is defined in Eq. (1), where nu indicates the total count of features.

$$\mathrm{Im}=\left\{{\mathrm{Im}}_1,,,{\mathrm{Im}}_2,,,\cdots ,\cdots ,{\mathrm{Im}}_{\mathit{nu}}\right\}$$ 1

Since the NN [28] includes an input layer, an output layer, and a hidden layer, it is a prerequisite to find out the output of hidden layer before computing the overall network output. The hidden layer output $e^{\left(H\right)}$ is determined as per Eq. (2); nf indicates the activation function, $\stackrel{⌢}{i}$ denotes the hidden neuron, j denotes the input neurons, $W_{\left(B,\stackrel{⌢}{i}\right)}^{\left(H\right)}$ denotes bias weight to ${\stackrel{⌢}{i}}^{\mathit{th}}$ hidden neuron, $n_{\stackrel{⌢}{i}}$ specifies count of input neurons, $W_{\left(j,\stackrel{⌢}{i}\right)}^{\left(H\right)}$ denotes the weight from $j^{\mathit{th}}$ input neuron to ${\stackrel{⌢}{i}}^{\mathit{th}}$ hidden neuron, and $\mathrm{Im}$ refers to the SSIM. Moreover, the general network output $G_o$ is determined from the output layer, which is shown in Eq. (3); $\widehat{o}$ refers to the output neurons, $n_h$ indicates the number of hidden neurons, $W_{\left(B,\widehat{o}\right)}^{\left(G\right)}$ denotes bias weight to output of ${\widehat{o}}^{\mathit{th}}$ neuron, and $W_{\left(\stackrel{⌢}{i},\widehat{o}\right)}^{\left(G\right)}$ indicates weight from ${\stackrel{⌢}{i}}^{\mathit{th}}$ hidden neuron to output of ${\widehat{o}}^{\mathit{th}}$ neuron. In addition, the error among the predicted and actual values is calculated as depicted in Eq. (4) that has to be reduced. In Eq. (4), $n_G$ indicates the number of output neurons, $G_{\widehat{o}}$ denotes the actual output, and ${\widehat{G}}_{\widehat{o}}$ denotes the predicted output.

$$e^{\left(H\right)}=nf\left(W_{\left(B,i\right)}^{\left(H\right)},+,\sum _{j=1}^{n_i},W_{\left(j,i\right)}^{\left(H\right)},\mathrm{Im}\right)$$ 2

$${\widehat{G}}_{\widehat{o}}=nf\left(W_{\left(B,\widehat{o}\right)}^{\left(G\right)},+,\sum _{i=1}^{n_h},W_{\left(i,\widehat{o}\right)}^{\left(G\right)},e^{\left(H\right)}\right)$$ 3

$$\begin{array}{c}{E}^{\ast }=\mathrm{argmin}\sum ^{n_G}\left|G_{\widehat{o}},-,{\widehat{G}}_{\widehat{o}}\right|\\ \left\{W_{\left(B,\stackrel{⌢}{i}\right)}^{\left(H\right)},,,W_{\left(j,\stackrel{⌢}{i}\right)}^{\left(H\right)},,,W_{\left(B,\widehat{o}\right)}^{\left(G\right)},,,W_{\left(\stackrel{⌢}{i},\widehat{o}\right)}^{\left(G\right)}\right\}=1\end{array}$$ 4

Bayesian Regularization Model [27]

Several approaches are exploited for training NN. Here, a BR framework is deployed to the training data for computing the weights among the hidden and input layers and among the output and hidden layers. The transfer functions of the output layer and hidden layer are set to the log–sigmoid function as shown in Eq. (5), in which $x$ denotes the input vector.

$$f\left(x\right)=1/\left(1,-,e^x\right)$$ 5

The BRBPNN exploits the regularization technique for enhancing its ability of generalization. The objective function of training denoted by D is specified by Eq. (6), in which $E_r$ denotes the squared sum of network weights, $E_H$ indicates the squared sum of residuals among objective values and network response values, and $\zeta$ and $\vartheta$ signify the hyperfunction constraints or objective constraints.

$$D=\zeta E_r+\vartheta E_H$$ 6

The weights of the network are regarded as the arbitrary constraints in the Bayesian approach. Initially, the functions are assigned with certain previous distributions. While the data is detected, the posterior distribution of the weights could be updated by means of Bayes’ rule as shown in Eq. (7), in which $R$ denotes the NN framework, $P\left(r,|,\zeta ,,,R\right)$ signifies the prior density, $r$ indicates the network weight vector, $P\left(M,|,\zeta ,,,\vartheta ,,,R\right)$ symbolizes the normalization factor [26], and $P\left(M,|,r,,,\vartheta ,,,R\right)$ specifies the likelihood function. Therefore, Eq. (17) can be represented as in Eq. (8).

$$P\left(r,|,M,,,\zeta ,,,\vartheta ,,,R\right)=\frac{P\left(M,|,r,,,\vartheta ,,,R\right)P\left(r,|,\zeta ,,,R\right)}{P\left(M,|,\zeta ,,,\vartheta ,,,R\right)}$$ 7

$$Posterior=\frac{Likelihood.\mathrm{Pr}ior}{\mathit{Evidence}}$$ 8

The likelihood function can be written as in Eq. (9) on considering the data probability distribution and weights as Gaussian, in which $Y_H\left(\vartheta \right)$ denotes the normalization factor as given by Eq. (10).

$$P\left(M,|,r,,,\vartheta ,,,R\right)=\frac{\exp \left(-,\vartheta ,E_H\right)}{Y_H\left(\vartheta \right)}$$ 9

$$Y_H\left(\vartheta \right)={\left(\pi ,/,\vartheta \right)}^{{}^n{/}_2}$$ 10

Likewise, Eq. (11) shows the prior probability, where $Y_H\left(\zeta \right)$ indicates the normalization factor as specified by Eq. (12).

$$P\left(r,|,\zeta ,,,R\right)=\frac{\exp \left(-,\zeta ,E_r\right)}{Y_H\left(\zeta \right)}$$ 11

$$Y_H\left(\zeta \right)={\left(\pi ,/,\zeta \right)}^{{}^k{/}_2}$$ 12

At last, the posterior probability is given by Eq. (13).

$$P\left(r,|,M,,,\zeta ,,,\vartheta ,,,R\right)=\frac{\exp \left(-,D,\left(w\right)\right)}{Y_D\left(\zeta ,,,\vartheta \right)}$$ 13

Here, Bayes’ rule is exploited for optimizing the objective function constraints such as $\zeta$ and $\vartheta$. Therefore, Eq. (14) can be attained, in which $P\left(\zeta ,,,\vartheta ,|,R\right)$ denotes the prior probability for regularization constraints $\zeta$ and $\vartheta$, as well as $P\left(M,|,\zeta ,,,\vartheta ,,,R\right)$ which indicates the likelihood function that is said to be the proof for $\zeta$ and $\vartheta$ [26].

$$P\left(\zeta ,,,\vartheta ,|,M,,,R\right)=\frac{P\left(M,|,\zeta ,,,\vartheta ,,,R\right)P\left(\zeta ,,,\vartheta ,|,R\right)}{P\left(M,|,R\right)}$$ 14

The optimal values for $\zeta$ and $\vartheta$ could be considered as in Eq. (15), in which $\gamma$ denotes the effectual constraint [28], n is the number of sample sets, m indicates the entire count of network constraints, and Hes specifies the Hessian matrix of the objective function $D\left(r\right)$.

$$\zeta =\gamma |2E_r;\vartheta =\left(n,-,\gamma \right)/2E_H;\gamma =\sum _{i=1}^{m}m-\zeta .trac{e}^{-1}\left(H,e,s\right)$$ 15

As per [29], the iteration process is mentioned below.

1. Assign values for weights and $\zeta$ and $\vartheta$.

2. For reducing $D\left(r\right)$, deploy a step of the Levenberg–Marquardt (LM) model.

3. Evaluate $\gamma$ by means of the “Gauss–Newton approximation” to Hessian matrix in LM training framework.

4. Evaluate novel values for $\zeta$ and $\vartheta$.

5. Steps 2 to 4 are iterated until convergence.

In the testing phase, the Bayesian-trained NN model predicts the filter coefficients under the reference of the constructed training library (explained in the subsequent section). Furthermore, the predicted filter coefficients are once again given as the solution to the proposed hybrid algorithm, from which the final updated filter coefficients are obtained.

Training Library Construction on Fine-tuned Filter Coefficients: Offline Process via BR Framework

Training library construction seems to be the most important phase in the proposed work. Once the eight optimal filter coefficients are determined, the data library is constructed. The library includes the details regarding the speckle-noised image, and its SSIM, along with its corresponding optimal filter coefficients, where $N^i$. Figure 3 shows the pictorial demonstration of the constructed data library with the defined columns.

Fig. 3.

Data library construction

A Hybrid Algorithm for Optimal or Fine-tuned Filter Coefficient Selection

Solution Encoding and Objective Function

The filter coefficients of the hybrid filters are provided as a solution to the proposed RFU-LA, where they are tuned optimally. Eight filter coefficients are selected, N = 8, in Fig. 4. Furthermore, the optimal selection directly dealt with the defined objective function in Eq. (16), which says the maximization of PSNR.

$$Objectivefunction=Max\left(P,S,N,R\right)$$ 16

Fig. 4.

Solution encoding

Conventional Lion Optimization

The LA [25] was established by B.R Raja Kumar that exhibits the living nature of the lion species. Different from other cat species, lions continue to exist with a fascinating social system/behavior, known as pride, to make stronger their own species at all generations [39–41]. Usually, a pride includes one to three lion pairs, where the resident females attract males to give birth to young ones. The residing region of these lions is known as territory. The dominating lion of a territory governs a particular area by combating against other animals together with nomadic lions. The territorial defense persists until sexual maturity is achieved by the cubs, i.e., 2–4 years. LA approach includes four major steps, namely, mating, generation of pride, territorial takeover, and territorial defense. Here, s denotes the solution vector and is addressed as $s=\left[s_1,,,s_2,,,\cdots ,.,s_{\widehat{m}}\right]$.

Pride Generation

The pride initialization is done using a nomadic lion $s^{\mathit{nd}}$ and a lioness $s^{\mathit{fem}}$ and a territorial lion $s^{\mathit{mal}}$. The vector elements of $s^{\mathit{nd}}$, $s^{\mathit{fem}}$, and $s^{\mathit{mal}}$, that is $s_{\mathit{len}}^{\mathit{nd}}$, $s_{\mathit{len}}^{\mathit{fem}}$, and $s_{\mathit{len}}^{\mathit{mal}}$, are as random numbers, which rely on the minimum and maximum limits while $\widehat{m}>1$, where $len=1,2\cdots ..,Len$. Here, $\mathit{Len}$ denotes the lion’s length as shown by Eq. (17), where $\widehat{n}$ and $\widehat{m}$ are variables. When $\widehat{m}=1$, the parameters specified in Eqs. (18) and (19) should be satisfied and $V\left(f,l,e,n\right)$ is specified in Eq. (20).

$$Len=\left\{\begin{array}{c}\widehat{m};\widehat{m}=1\\ \widehat{n};otherwise\end{array}\right)$$ 17

$$V\left(s_{\mathit{len}}\right)\in \left(s_{\mathit{len}}^{\min },,,s_{\mathit{len}}^{\max }\right)$$ 18

$$\widehat{n}\%2=0$$ 19

$$V\left(s_{\mathit{len}}\right)=\sum _{len=1}^{\mathit{Len}}{s}_{\mathit{len}}{2}^{\left(\frac{\mathit{Len}}{2},-,l,e,n\right)}$$ 20

Fertility Evaluation

If $S^{\mathit{fem}}$ and $S^{\mathit{mal}}$ get saturated, then they could have reached local or global optimum, by which they could not attain the best solution. In this approach, the rate of laggardness is indicated as $L{a}_r$ and laggard is considered $S^{\mathit{mal}}$, while $h\left(S^{\mathit{mal}}\right)$ is beyond $h^r$ and denotes the reference fitness. The fertility of $S^{\mathit{fem}}$ is denoted by sterility rate $S{t}_r$ after crossover and is increased by 1. While $S{t}_r$ is greater than tolerance $S{t}_r^{\max }$, $S^{\mathit{fem}}$ update is performed according to Eq. (21). When updated female $S^{fem+}$ is regarded as $S^{\mathit{fem}}$, the mating process can be performed. In Eq. (21) and Eq. (22), $s_{\mathit{len}}^{fem+}$ and $s_d^{\mathit{fem}}$ are concerned as $le{n}^{\mathit{th}}$ and $d^{\mathit{th}}$ vector elements of $S^{fem+}$.

$$s_{\mathit{len}}^{fem+}=\left\{\begin{array}{c}{s}_d^{fem+iflen=d}\\ s_{\mathit{len}}^{\mathit{fem}}otherwise\end{array}\right)$$ 21

$$s_d^{fem+}=\min \left[s_d^{\max },,,\max ,\left(s_d^{\min },,,{\mathrm{\nabla }}_d\right)\right]$$ 22

$${\mathrm{\nabla }}_d=⌊s_d^{\mathit{fem}},+,\left(0.1,{\stackrel{⌣}{r}}_2,-,0.05\right),\left(s_d^{\mathit{mal}},-,{\stackrel{⌣}{r}}_1,s_d^{\mathit{fem}}\right)⌋$$ 23

The variable d is generated that lies among [1, Len], $\mathrm{\nabla }$ indicates the female update process, and ${\stackrel{⌣}{r}}_1$ and ${\stackrel{⌣}{r}}_2$ specify the random constraints, which lie among [0, 1].

Mating

The initial mating process is the crossover and mutation, and subsequently, gender-based clustering is the subsequent phase. By carrying out mutation and crossover, the cubs are created as $S^{\mathit{cubs}}$ and $S^{\mathit{new}}$, where $S^{\mathit{cubs}}$ denotes the cub produced from crossover and $S^{\mathit{new}}$ denotes the cubs produced from mutation. Accordingly, four cubs are formed during the pregnancy of a lioness and hence four cubs are formed by crossover process. Furthermore, the mutation process is carried out using these four cubs to generate four new cubs.

Lion Operators

The territorial defense is represented by the survival fight that generates nomad coalition and coalition updates. $S^{e-nd}$ is selected if Eq. (24) to Eq. (26) are met with

$$h\left(S^{e-nd}\right)<h\left(S^{\mathit{mal}}\right)$$ 24

$$h\left(S^{e-nd}\right)<h\left(S^{mal\_cub}\right)$$ 25

$$h\left(S^{e-nd}\right)<h\left(S^{fem\_cub}\right)$$ 26

The pride update occurs after the failure of $S^{\mathit{mal}}$, while the nomad coalition update occurs only after the failure of $S^{\mathit{nd}}$. Territorial takeover updates $S^{\mathit{mal}}$ and $S^{\mathit{fem}}$ if $S^{mal\_cub}$ and $S^{fem\_cub}$ get matured, i.e., when the cub’s age is greater than the maximum age for cub maturity, $A^{\max }$.

Termination

The algorithm gets terminated only if Eq. (27) or Eq. (28) is satisfied. In Eq. (27), the generation count is denoted by $\mathit{it}$ that is initially set as 0 and raised by 1 when the territorial takeover is carried out.

$$it>{\mathit{it}}_{\max }$$ 27

$$h\left(S^{\mathit{mal}}\right)-h\left(S^{\mathit{opt}}\right)\le e{r}_{\mathit{th}}$$ 28

The maximum generations and threshold for error are denoted as it_max and $e{r}_{\mathit{th}}$, respectively.

Conventional FF Algorithm

Fireflies are said to be one of the charismatic insects of all [24]. The flashing light of fireflies is their major aspect that comprises two basic characteristics such as (a) attraction of mating partners and (b) warning of predators [42–44]. The flashing light obeys several rules. Consequently, distance w gets increased when the light intensity I is minimized as specified by Eq. (29).

$$I\alpha \frac{1}{w^2}$$ 29

Two vital issues are taken into account for modeling a precise FF scheme. They are attraction formulation and the variations in _{light intensity}. In the FF method, I representing the solution S is proportional to the fitness value. Here, the intensity of light $I\left(w\right)$ is changed as per Eq. (30), in which $I_0$ indicates the intensity of light attained from the source, and the light absorption is computed by the deployment of light absorption coefficient $\gamma$.

$$I\left(w\right)={I_{0^e}}^{-\gamma w^2}$$ 30

The similarity at $w=0$ in $I/w^2$ is avoided by involving the inverse square law effects and estimation of absorption in the type of Gaussian. The attractiveness is denoted by $\beta$ that is proportional to $I\left(w\right)$. Therefore, Eq. (31) could be used for portraying the attractiveness, $\beta$, in which ${\beta }_0$ signifies the attraction level at $w=0$. In addition, the light intensity and $\beta$ are almost the same.

$$\beta ={{\beta }_{0^e}}^{-\gamma w^2}$$ 31

The distance among two fireflies $S_i$ and $S_j$ is signified by Eq. (32), in which n refers to the dimensionality issues.

$$w_{\mathit{ij}}=\Vert S_i-S_j\Vert =\sqrt{\sum _{k=1}^{k=n}{\left(S_{\mathit{ik}},-,S_{\mathit{jk}}\right)}^2}$$ 32

Here, the specified equation could be exploited as shown in Eq. (33), where ${\epsilon }_i$ is a random number, achieved from Gaussian distribution, and $\alpha$ indicates a randomization factor. The pseudocode of the conventional FF model is specified by Algorithm 1.

$$S_b^{\ast }=S_i+{{\beta }_{0^e}}^{-\gamma {w_{\mathit{ij}}}^2}\left(S_j,-,S_i\right)+\alpha {\epsilon }_i$$ 33

Proposed Algorithm

In order to make the de-speckling model more appropriate, this paper aims to choose the optimal filter coefficients using a new hybrid algorithm for overcoming the drawbacks of the existing LA and FF algorithms. In fact, even though the conventional DA algorithm poses various advantages, including accurate approximations, it also poses certain disadvantages such as reduced internal memory and slow convergence. On the other hand, the Firefly algorithm offers fast convergence and efficiently converges the global maximum point. Hence, it is planned to mingle both the concepts in a certain way that obviously solves the defined optimization problems with better convergence. The hybridization concept of the proposed model is as follows: Here, as per the proposed model, a random value ra is initialized that lies among 0 to 1 (decides the update evaluation). If $ra<0.5$, FF gets updated as per Eq. (33), else female lion gets updated as per Eq. (21). As the modification is done based on the random value, the presented model is termed as the RFU-LA model. Algorithm 2 depicts pseudo-code of the proposed RFU-LA algorithm and the flow chart of the proposed algorithm is given in Fig. 5.

Fig. 5.

Flowchart of the proposed model

Results and Discussion

Simulation Procedure

The proposed RFU-LU model was implemented in MATLAB and the corresponding results were achieved. Here, the performance of the presented technique was compared over the other conventional schemes like wavelet filtering [1] and hybrid filtering schemes [2] by varying the noise variances from 0.1, 0.2, 0.3, 0.4, and 0.5 with respect to MAPE, PSNR, SDME, and SSIM, respectively. The sample image of the proposed and conventional schemes is given in Fig. 6 and the sample image is given in Fig. 7.

Fig. 6.

Sample image of the proposed vs. conventional schemes by varying the nosie variances like (a) 0.1, (b) 0.2, (c) 0.3, (d) 0.4, and (e) 0.5

Fig. 7.

Sample image of the adopted filtering schemes by varying the nosie variances like (a) 0.1, (b) 0.2, (c) 0.3, (d) 0.4, and (e) 0.5

Dataset Description

The dataset was downloaded from [33]. The database includes images of the CCA of ten volunteers with diverse weights. The images were captured using a Sonix OP ultrasound scanner with various depth, gain, TGC curve, and linear array transducer settings. In the longitudinal region of the image database, there are 84 B-mode ultrasound images of CCA. The resolution of images is around 390 × 330px. The ultrasound scanner’s configuration determines the actual resolution. Two different linear array transducers were employed, each with a different frequency (10 MHz and 14 MHz). These frequencies were chosen because they are good for imaging superficial organs. Patients were in the supine position with their neck turned to the left side while the right CCA was evaluated, as per standard protocol.

Performance Analysis

The performance analysis of the adopted RFU-LA for effective de-speckling is given in Fig. 8 for different noise variances. The simulation outcomes have revealed that the MAPE measure attained by the presented scheme is low when compared over the traditional schemes for the five noise variances. More particularly, from Fig. 8a, for a noise variance of 0.1, the mean MAPE of the presented scheme are low, which are 39.13% and 49.28% superior to wavelet filtering and hybrid filtering schemes. Also, the mean MAPE of the presented scheme are 39.13% and 49.28% better than wavelet filtering and hybrid filtering schemes at a noise variance of 0.2.

Fig. 8.

Performance analysis of the proposed and conventional models in terms of (a) MAPE, (b) PSNR, (c) SDME, and (d) SSIM

Likewise, from Fig. 8b, the mean PSNR of the implemented RFU-LA is higher for all the five noise variances when evaluated over the other compared schemes. From the analysis, the mean PSNR of the presented scheme are 28.89% and 26.09% better than wavelet filtering and hybrid filtering schemes at a noise variance of 0.1. In the same way, for the noise variance of 0.3, the mean PSNR of the suggested model are 39.13% and 49.28% superior to the compared existing schemes.

The attained outcomes of mean SDME are given in Fig. 8c. From the results, the mean SDME of the proposed RFU-LA is better than wavelet filtering; however, the hybrid filtering model attains a higher SDME value than the proposed method. This variation can be considered negligible as the other measures attained by the presented scheme are optimal.

Moreover, the graphical outcomes of mean SSIM are demonstrated in Fig. 8d. From the results, better mean SSIM value is found to be attained by the adopted model over the conventional methods for the five noise variances. Specifically, the mean SSIM of the presented scheme at 0.1 noise variance are 20.97% and 29.31% superior to wavelet filtering and hybrid filtering schemes. In addition, the mean SSIM values of the adopted scheme are 20.97% and 29.31%, superior to the compared schemes at a noise variance of 0.4. Thus, the effectiveness of the proposed RFU-LA algorithm under the effective de-speckling model has been proved by the attained results.

Convergence Analysis

The convergence analysis of the presented RFU-LA scheme for effective de-speckling is given in Fig. 9. Here, convergence analysis of the proposed model is carried out for the PSNR measure, which is found to be higher for varied values of SSIM.

Fig. 9.

Convergence analysis of the proposed model: PSNR vs. SSIM

PSNR Analysis: With and Without Optimization

Figure 10 shows the PSNR analysis of the adopted RFU-LA approach for effective de-speckling. On observing the attained outcomes, improved performance is achieved by the RFU-LA approach with optimization when compared to RFU-LA approach without optimization. Accordingly, from the analysis, at a noise variance of 0.5, the RFU-LA approach with optimization is 7.02% superior to the RFU-LA approach without optimization. Thus, the effectiveness of the presented RFU-LA algorithm with optimization has been proved to form the attained results.

Fig. 10.

PSNR analysis of the proposed model: with and without optimization

PSNR and SSIM Analysis by Varying Noise

The performance analysis with respect to PSNR and SSIM on noise variance (0.1, 0.2, 0.3, 0.4, and 0.5) is given in Table 2 for 10 images. From the simulation results, better values are attained by the presented scheme for PSNR and SSIM measures for all the noise variances in the case of 10 images.

Table 2.

Proposed model with respect to PSNR and SSIM: by varying noise

	Measures
	PSNR					SSIM
Noise variance	0.1	0.2	0.3	0.4	0.5	0.1	0.2	0.3	0.4	0.5
Image 1	31.891	32.028	30.032	31.891	32.462	0.85515	0.85807	0.81029	0.85515	0.87356
Image 2	29.935	30.035	30.075	29.935	30.422	0.84227	0.8477	0.84552	0.84228	0.85995
Image 3	28.971	28.971	28.971	28.971	29.288	0.79748	0.79735	0.79733	0.79748	0.81153
Image 4	29.103	29.104	29.104	29.103	29.414	0.7983	0.79821	0.7982	0.7983	0.81167
Image 5	30.412	30.552	30.552	30.412	30.98	0.81447	0.81833	0.81833	0.81447	0.83125
Image 6	29.442	29.442	29.492	29.442	29.839	0.7977	0.79773	0.79678	0.7977	0.81127
Image 7	28.779	28.834	28.779	28.779	29.17	0.78552	0.78451	0.78551	0.78552	0.79965
Image 8	29.136	29.264	29.15	29.136	29.54	0.7629	0.7667	0.76083	0.7629	0.77873
Image 9	29.715	29.715	29.715	29.715	30.139	0.7566	0.75662	0.7566	0.7566	0.77307
Image 10	27.583	27.479	27.582	27.583	27.951	0.71624	0.71064	0.71626	0.71624	0.73369

Obtained Filter Coefficients: with and without Optimization

As mentioned above, once the NN model predicts the respective filter coefficients, it again goes for the optimal tuning via the proposed hybrid algorithm. Hence, this section makes an analysis of obtained coefficients with and without optimization after prediction.

The eight filter coefficients acquired without optimization and with optimization for the five varied values of noise variances are mentioned in Tables 3 and 4, respectively (for each SSIM).

Table 3.

Obtained filter coefficients without optimization for five different noise variances

Noise variance	SSIM	Filter coefficient 1	Filter coefficient 2	Filter coefficient 3	Filter coefficient 4	Filter coefficient 5	Filter coefficient 6	Filter coefficient 7	Filter coefficient 8
0.1	0.65462	23,393	5.6887	30.377	1.0359	29.372	1.0192	5.664	2.7208
0.2	0.52171	23,496	5.9404	29.973	1.0225	30.49	1.0011	5.9683	1.9149
0.3	0.4446	23,254	6.2265	34.509	1.0415	28.671	1.0432	5.825	1.9229
0.4	0.39819	23,065	6.592	38.252	1.0539	27.756	1.0566	6.0522	1.1084
0.5	0.36763	23,065	6.5914	38.244	1.0538	27.759	1.0565	6.0513	1.1087

Table 4.

Obtained filter coefficients with optimization for five different noise variances

Noise variance	SSIM	Filter coefficient 1	Filter coefficient 2	Filter coefficient 3	Filter coefficient 4	Filter coefficient 5	Filter coefficient 6	Filter coefficient 7	Filter coefficient 8
0.1	0.73811	23,393	5.6886	30.374	1.0359	29.373	1.0192	5.6625	2.7208
0.2	0.74187	23,486	5.917	29.974	1.0227	30.467	1.0035	5.9101	1.9644
0.3	0.74258	23,300	6	31.5	1.0455	28.5	1.0476	5.8954	1.6686
0.4	0.73811	23,393	5.6886	30.374	1.0359	29.373	1.0192	5.6625	2.7208
0.5	0.80756	23,300	4	28.5	1	28.5	1	4	1

Conclusion

This paper has presented a novel de-speckling model using four filters, namely GF, SRBF, RIBNLM, and MF. Here, the filter coefficients were optimally selected, which was considered a major contribution in this work. Accordingly, the Bayesian-based NN was exploited for predicting the appropriate filter coefficients, where the training library was constructed with the optimal coefficients. Moreover, the optimal selection of filter coefficients was done using RFU-LA that hybrids the concept of both LA and FF correspondingly. At last, the performance of the adopted scheme was evaluated over other traditional schemes and the outcomes were attained. From the analysis, for noise variance of 0.1, the mean MAPE of the presented scheme were 39.13% and 49.28% superior to wavelet filtering and hybrid filtering schemes. Also, the mean PSNR of the presented scheme were 28.89% and 26.09% better than wavelet filtering and hybrid filtering schemes at a noise variance of 0.1. Hence, by optimally selecting the filter coefficients using RFU-LA, the proposed hybrid RFU-LA optimization model attains a fast convergence rate. Thus, the betterment of the presented scheme was proved. In the future, the proposed method could be studied to denoise other classes of medical images other than ultrasound.

Abbreviations

US

Ultrasound

GF

Guided filter

SRBF

Speckle-reducing bilateral filter

RIBNLM

Rotation invariant bilateral nonlocal means filter

MF

Median filter

FF

FireFly

RFU-LA

Randomized FF update in Lion Algorithm

MRI

Magnetic resonance imaging

DWT

Discrete wavelet transform

BF

Bilateral filter

RMSE

Root mean square error

IWT

Inverse wavelet transformation

MNLM

Modified NLM

MBS

Modified Bayes shrink

HMBS

Homogeneity MBS

PSNR

Peak signal to noise ratio

BR

Bayesian regularization

SSIM

Structural Similarity Index

MAPE

Mean absolute percentage error

SDME

Second-derivative–like measure of enhancement

CCA

Common carotid artery

TGC

Time gain compensation

Author Contribution

Shabana Sulthana S L conceived the presented idea and designed the analysis. Also, he carried out the experiment and wrote the manuscript with support from Sucharitha M. All the authors discussed the results and contributed to the final manuscript. All the authors read and approved the final manuscript.

Declarations

Ethics Approval

This paper does not contain any studies with human participants or animals performed by any of the authors.

Conflict of Interest

The authors declare no competing interests.