Indian Cross Corpus Speech Emotion Recognition Using Multiple Spectral-Temporal-Voice Quality Acoustic Features and Deep Convolution Neural Network

Affective computing seeks to facilitate people's natural interaction with computers. One of the main goals is to enable computers to understand people's emotional states so that customized answers may be provided in response [1, 2]. Recent years have seen an increase in interest in SER, which is often done on the premise that spoken sounds in training and testing datasets are generated under the same circumstances. However, as voice data are often gathered from many devices or places, this assumption does not hold true in practice. Due to the disparity between the training and testing datasets, SER suffers from Class imbalance problem [3, 4].

Emotions reflect the psychological state of the human being. Various physiological and psychological signals such as speech, facial expressions, and electrocardiograms (ECG), electroencephalograms (EEG) are utilized for the manifestation of emotional reflection. Speech is the natural and easiest way of interaction that comprises huge emotional content and context. SER is the most straightforward way of human-machine interaction (HMI). Generalized SER systems use the same corpus for training as well as testing purpose, which may cause poor outcome for the new corpus [5-7]. SER is very challenging due to many factors such as age, health status, gender, linguistic variability, cultural variability, recording environments, and languages with distinct corpus. The speech attributes show high variance for different corpus which leads to poor recognition rate for the SER systems designed for single corpus. Now a days, various cross-corpus SER systems have been implemented that use one dataset for training and another for testing [8, 9].

In past decades, most of the SER techniques uses same corpus for the training as well as training and researchers have achieved noteworthy success for the SER under controlled experimental boundaries [10-12]. Earlier SER uses traditional machine learning (ML) techniques such as Gaussian Mixture Model (GMM) [13], Hidden Markov Model (HMM) [14], Support Vector Machine (SVM) [15], K-Nearest Neighbor (KNN) [16], Random Forest Classifier (RF) [17], Artificial Neural Network (ANN) [18], etc along with handcrafted feature extraction and pre-processing schemes. In recent years, deep learning (DL) has engrossed the extensive attention of investigators for the SER because of robustness, high feature depiction capability, ability to work for larger dataset, higher recognition rate, etc. various deep learning techniques has been presented successfully for the SER such as Auto encoder (AE) [19], Convolution Neural Network (CNN) [20], Deep Belief Network (DBN) [21], Long-Short Term Memory (LSTM) [22], Recurrent Neural Network (RNN) [23], etc. Though the traditional ML and DL based SER has achieved extraordinary progress under the controlled experimental environment, the generalization of SER system is key challenge that is key to endorse the SER systems for real time applications. Thus there is need to increase the generalization capability of the cross corpus SER to enhance the outcome of SER for different corpuses [24, 25]. Most of the SER techniques presented in the past used English and European languages for the training which limits the outcome for the other languages due to cultural, regional, and linguistic variations. Very less focus has been given on SER for Indian languages though there are vast variations in the Indian corpus and regions. Therefore, there is need to present the SER system for Indian languages that can help to model the linguistic variations in Indo-Aryan and Dravidian language family.

This paper presents cross corpus SER using multiple acoustic features and Deep Convolution Neural Network (DCNN) for four Indian Languages such as Hindi, Urdu, Telugu and Kannada. The chief influences of the work are summarized as follow:

• Robust feature representation of speech signal using multiple acoustic features consisting of spectral, temporal, and voice quality features.
• Salient feature selection with higher inter-class and lower intra-class variance using Fire Hawk based optimization scheme.
• Design of lightweight one-dimensional DCNN for improving the feature distinctiveness for CCSER.
• Analysis of the proposed CCSER scheme for single corpus and multi-corpus training and testing for four Indian languages such as Hindi, Urdu, Telugu and Kannada.

The outcomes of the anticipated SER scheme are validated using accuracy, precision, recall, and F1-score.

The remaining of the paper is arranged as follow: Section 2 describes the information regarding various techniques used for SER and CCSER in recent years. Section 3 gives exhaustive depiction of the dataset, acoustic features and DCNN model. Section 4 depicts the analysis and discussions of simulation results suggested CCSER scheme. Lastly, Section 5 concludes the work and offers the future scope for possible boost in the proposed CCSER scheme.

3.1 Dataset

The outcome of the proposed SER scheme is estimated on four languages from two Indian language families. It uses Hindi and Urdu corpus from the Indo-Aryan language family and Telugu and Kannada corpus from the Dravidian Language family. Four common emotions such as angry, happy, neutral and sadness are selected for the CCSER. Out of total data 70% and 30% data is considered for the training and testing purpose respectively. The summary of various datasets used is given in Table 2.

3.2 Methodology

The framework of the proposed SER scheme is given in Figure 1 that encompasses the preprocessing, multiple acoustic feature extraction and DCNN for CCSER. The multiple acoustic features are used to capture the temporal variations in the emotion signal using time domain features, spectral properties of the signal using various spectral features and variation in amplitude and frequency of the voice signal using voice quality features. The DCNN improves the connectivity and correlation of the different local and global acoustic features for effective CCSER.

Table 2. Details about dataset used for CCSER

1.png

Figure 1. Flow diagram of proposed system

3.3 Multiple acoustic features

Wide range of acoustic features are extracted for the signal to characterize the changes occurs in the voice due to emotion. The proposed feature set consists of time-domain features, spectral features and voice quality features as given in Table 3. The feature vector is further given to FH for prominent feature selection and these salient features are further given to DCNN architecture to improve the feature representation.

A. Multi-taper MFCC

Only one hamming window with a higher variance is used in generalized MFCC, which is unable to acquire the disparities over the frame of speech signal. The speech is filtered using moving average filter to lessen the amount of noise it contains during the pre-emphasis stage. As a part of framing process, the entire signal is divided into the frames of 40 ms each. This is necessary for multi-taper windowing to accumulate the adjoining spectral components together. The DFT is used to transform signals from the time to the spectral domain. The linearly scaled signal is then converted to Mel frequency, which can be perceived by human hearing. Discrete Cosine Transform (DCT) is used to transform signal back to time domain to reduce the signal's redundancies. 13 cepstral values are chosen as the features after log filter-bank energy has been calculated over the frames. Figure 2 depicts the MT-process MFCC's flow. In contrast, the windowing of the speech signal in the MT-MFCC uses various tappers with diverse variations, which aids in increasing frequency resolution.

2.png

Figure 2. MFCC coefficient extraction [7]

Table 3. Details regarding features

The sign weighted ceptrum estimator (SWCE) provides low error compared with traditional MFCC as given in Eq. (1) [29, 30].

${{w}_{P}}\left( j \right)=\sqrt{\frac{2}{N+1}}\sin \left( \frac{\pi p\left( j+1 \right)}{N+1} \right),~j=0,1,\ldots ,N-1.$             (1)

where, N denotes total frames, $w_p$ depicts tapper window, and p=1,2,3, …., M.

The weights of SWCEfor each taper are estimated using Eq. (2) [31].

$\lambda \left( p \right)=\frac{\cos \left( \frac{2\pi \left( p-1 \right)}{M/2} \right)+1}{\mathop{\sum }_{p=1}^{M}\left( \cos \left( \frac{2\pi \left( p-1 \right)}{M/2} \right)+1 \right)},p=1,,2,\ldots ,M.$         (2)

where, M represent total tapers, $\lambda(\mathrm{p})$ denotes the weight of ${{p}^{th}}$ taper, and p =1,2,3, ……., M. The power spectral density (PSD) for different taper windows for speech signal is computed using Eq. (3) [29, 30].

${{\hat{S}}_{MT}}\left( m,k \right)=\underset{p=1}{\overset{M}{\mathop \sum }}\,\lambda \left( p \right){{\left| \underset{j=0}{\overset{N-1}{\mathop \sum }}\,{{w}_{P}}\left( j \right)s\left( m,j \right){{e}^{\frac{2\pi ik}{N}}} \right|}^{2}},$            (3)

After multi-taper windowing the speech signal is converted in to spectral domain signal using fast Fourier transform (FFT). The Mel frequency spectrum is further changed in to time domain using DCT and to minimize the redundancy in the signal, Total 39 MTMFCC coefficients are considered for the representation of voice signal such as 13 MFCC coefficients, 13 $\Delta$-MTMFCC coefficients and 13 $\Delta$$\Delta$-MTMFCC coefficients. Figure 3 depicts the conception of the phases of MT-MFCC.

B. LPCC

The linear predictive analysis's spectral feature known as the LPCC is used to reflect the speech signal's emotion-specific phonological interpretation. The LPCC does a fantastic job of describing aspects of the human vocal tract that assist to specifically identify the emotional content of speech. In linear predictive analysis, the knowledge of prior p samples may be used to estimate the nth samples, as shown in Eq. (4).

$x\left( n \right)={{a}_{1}}x\left( n-1 \right)+{{a}_{2}}x\left( n-2 \right)+{{a}_{3}}x\left( n-3 \right)+~\ldots \ldots \ldots \ldots ~+~{{a}_{p}}x\left( n-p \right),$        (4)

where, $a_1, a_2, \ldots \ldots a_p$ are the constants over the speech frames. The speech sample is predicted by these linear predictor coefficients. The suggested method takes into account a total of 13 LPCC coefficients [13] as characteristics.

C. Spectral Kurtosis(SK)

This denotes the sequence of transients together with their spectral domain locations. It describes the non-Gaussianity or smoothness of the speech frequency spectrum around its centroid, which demonstrates the impact of varying levels of arousal and emotional valence on the speech spectrum. The spectral kurtosis of the voice is calculated using Eq. (5).

$SK=\frac{\mathop{\sum }_{k=b1}^{b2}{{\left( {{f}_{k}}-{{\mu }_{1}} \right)}^{4}}{{s}_{k}}}{{{\left( {{\mu }_{2}} \right)}^{4}}\mathop{\sum }_{k=b1}^{b2}{{s}_{k}}}$         (5)

Here, $\mu_1$ depicts the spectral centroid, $\mu_2$ denotes spectral spread respectively, ${{s}_{k}}$ is spectral value over $k$ bins, ${{b}_{1}}$ and ${{b}_{2}}$ are the lowest and highest bound of the bins where SK of voice is computed.

D. Formants

Formants are energetically intense frequency peaks in the spectrum. They are predominantly noticeable in vowels. Every formant has an associated vocal tract resonance that characterizes the influence of the emotion on the speech signal. Here, 3 formants, mean and standard deviation of formants are considered to characterize the spectral changes due to emotion in the voice signal.

E. Pitch Frequency

To illustrate the vocal component of communication, pitch $({{f}_{0}})$ is important. By calculating the disparity between the peaks obtained by the speech signal's autocorrelation, the pitch of the speech may be determined.

F. ZCR

ZCR offers the signal's passage through the zero line, which represents the degree of noise in the voice signal. ZCR is computed in the time domain via Equation 6. Over a time period, the sign function returns a value of ‘1’ for the positive amplitude and ‘0’ for negative amplitude (t) of speech.

$ZC{{R}_{t}}=\frac{1}{2}\left( \underset{n=1}{\overset{N}{\mathop \sum }}\,(sign\left( x\left[ n \right] \right)-sign(x\left[ n-1 \right] \right)$           (6)

G. Jitter and Shimmer

The fluctuations in frequency and amplitude of the speech induced by aperiodic vocal fold vibrations are known as jitter and shimmer, respectively. The breathiness, hoarseness and roughness of the emotional voice are portrayed by the jitter and shimmer. The mean jitter is given by Eq. (7).

$Jitter=\frac{1}{N-1}\underset{i=1}{\overset{N-1}{\mathop \sum }}\,\left| {{T}_{i}}-{{T}_{i+1}} \right|$         (7)

where,$~{{T}_{i}}$ denotes the time period in sec and $N$ provides total periods. Eq. (8) represents mean shimmer.

$Shimmer=\frac{\frac{1}{N-1}\mathop{\sum }_{i=1}^{N-1}\left| {{A}_{i}}-{{A}_{i+1}} \right|}{\frac{1}{N}\mathop{\sum }_{i=1}^{N}{{A}_{i}}},$            (8)

where, A_i is peak-to-peak amplitude of speech signal and N is the number of periods.

Eq. (9) represents the feature representation $\left( Feat \right)~$provided to FHO for selecting prominent feature.

$Feat=\left\{ MTMFC{{C}_{1-39}},~LPC{{C}_{1-13}},~Formant{{s}_{1-3}},~MeanForman{{t}_{1}},~StdForman{{t}_{1}},~~PitchFre{{q}_{1}},~ZC{{R}_{1}},~Jitte{{r}_{1}},~Shimme{{r}_{1}} \right\}$           (9)

3-ab.png

3-cd.png

3-ef.png

Figure 3. Visualizations of MT-MFCC process a) Original speech Signal, b) Pre-emphasis effect, c) Multi-taper Windows (p=6), d) Mel filter bank, e) Mel log filterbank energy, f) Mel frequency cepstrum

3.4 FHO based feature selection

Finding significant features and reducing feature vector length both depend on feature selection. The SER system's output is enhanced by the less complex but still useful features. The FHO is a metaheuristic algorithm that mimics the fire hawk’s (FHs) fire-starting, fire-spreading, and prey-catching food foraging behaviour. To prevent local optimum entrapment, which improves global optimal solution, the FHO takes into account the average of the solution candidates in a certain region. Figure 4 shows the flow chart for effectively choosing characteristics from a collection of various acoustic features.

The biggest risks to animals and ecosystems come from wildfires that are caused by either a natural phenomenon or by humans. Many times, birds referred to as "fire hawks" such whistling kites, black kites, and brown falcons may purposely start a fire in order to catch prey to eat.

By holding the flaming sticks in its beak and dumping them over the area that hasn't yet caught fire, the fire hawk carefully spreads the flames in order to capture its victim. The fire hawks start little flames to frighten their prey, which includes snakes, rats, and other small animals. This causes the prey to make a hurried, panicked decision, which makes it easier for the FHs to capture the prey.

The number of viable solutions (A) shows where the FHs and their prey were first located. By taking into account the precise restrictions for each parameter as provided in (10) and (11), the population of FHs and preys is first initialized at random. Values of $\alpha, \beta$, and $K$ are taken into account by the populace. Here, N stands for the number of solutions, ${{A}_{ij}}$ signifies the ${{j}^{th}}$ decision variable of the ${{i}^{th}}$ solution, $\text{A}_{\text{i},\text{max}}^{\text{j}}$ and $\text{A}_{\text{i},\text{min}}^{\text{j}}$stands for the decision variable's upper and lower bounds, respectively.

$\begin{aligned} & A=\left[\begin{array}{c}A_1 \\ A_2 \\ A_3 \\ \vdots \\ A_N\end{array}\right]=\left[\begin{array}{ccc}A_{11} & A_{12} & A_{13} \\ A_{21} & A_{22} & A_{23} \\ A_{31} & A_{32} & A_{33} \\ \vdots & \vdots & \vdots \\ A_{N 1} & A_{N 2} & A_{N 3}\end{array}\right]=\left[\begin{array}{ccc}\alpha_{11} & \beta_{12} & K_{13} \\ \alpha_{21} & \beta_{22} & K_{23} \\ \alpha_{31} & \beta_{32} & K_{33} \\ \vdots & \vdots & \vdots \\ \alpha_{N 1} & \beta_{N 2} & K_{N 3}\end{array}\right]\end{aligned}$                (10)

4.png

Figure 4. FHO based feature selection strategy

$A_{i}^{j}\left( 0 \right)=A_{i,min}^{j}+rand.\left( A_{i,max}^{j}-A_{i,min}^{j} \right),\left\{ \begin{matrix} i=1,2,\ldots ,N.  \\ j=1,2,\ldots ,d.  \\ \end{matrix} \right.$           (11)

To assess the fitness of the solutions, the objective function based on the intra-class and inter-class variance is utilised. Other solutions are seen as prey while the ones with the best fitness are kept as FH. Global solutions are used to start fires since they are thought of as the original chief fires. To facilitate hunting, the chosen FHs are employed to start the fire in the unburned region. (12) and (13) include descriptions of the preys and FH. Here, $P{{R}_{m}}$ is ${{m}^{th}}$ prey in search space and $F{{H}_{n}}$ is ${{n}^{th}}$FH in search space.

$P R=\left[\begin{array}{c}P R_1 \\ P R_2 \\ P R_3 \\ \vdots \\ P R_m\end{array}\right]$                  (12)

$F H=\left[\begin{array}{c}F H_1 \\ F H_1 \\ F H_1 \\ \vdots \\ F H_n\end{array}\right]$              (13)

In next phase, the FH catches the nearby prey based on distance metrics given in 14. Here, $D_{k}^{l}$ shows distance between ${{l}^{th}}$FH and ${{k}^{th}}$ prey, $\left( {{x}_{1}},{{y}_{1}} \right)$ and $\left( {{x}_{2}},{{y}_{2}} \right)$indicates the coordinates of location of FH and prey.

$D_k^l=\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2},\left\{\begin{array}{l}l=1,2, \ldots, n \\ k=1,2, \ldots, m\end{array}\right.$                   (14)

The FHs choose burning sticks and start fires in unburned areas of their territory during the next phase to trap their prey and make their escape fast and challenging. The FHs use (15) to update their location in order to defend their area and stop other fire hawks from grabbing burning sticks. In this case, the values of ${{r}_{1}}$ and ${{r}_{2}}$ are arbitrary numbers between 0 and 1 that control the direction of the FHs raids on the main fire (global solution).

$\text{FH}_{\text{l}}^{\text{new}}=\text{F}{{\text{H}}_{1}}+\left( {{\text{r}}_{1}}\times \text{GB}-{{\text{r}}_{2}}\times \text{F}{{\text{H}}_{\text{Near}}} \right),\text{ }\!\!~\!\!\text{ l}=1,2,\ldots ,\text{n}.$                 (15)

When the prey sees the flaming sticks laid out on the ground, it begins to flee, hide, or inadvertently run towards the hawks. Using (16), the prey's location is updated for this case. Here, ${{r}_{3}}$ and ${{r}_{5}}$ are random numbers between [0,1] which represent coefficients for the movement towards hawks and a safe spot, and $\text{PR}_{\text{q}}^{\text{new}}$ represents the prey encircled by lthfire hawks.

$\begin{aligned} & \mathrm{PR}_{\mathrm{q}}^{\text {new }}=\mathrm{PR}_{\mathrm{q}}+\left(\mathrm{r}_3 \times \mathrm{FH}_1\right. \left.-r_4 \times S P_1\right),\left\{\begin{array}{l}l=1,2, \ldots, n. \\ q=1,2, \ldots, r.\end{array}\right. \\ & \end{aligned}$                 (16)

The prey may sometimes relocate to another hawk's territory or to the safest location outside the area. The updated position is shown by the number (17). Here,$~{{r}_{5}}$and ${{r}_{6}}$stand for the random number between [0, 1] that designates the coefficients for the migration of prey towards other hawks and towards safe locations beyond the region.

$\begin{aligned} & \mathrm{PR}_{\mathrm{q}}^{\text {new }}=\mathrm{PR}_{\mathrm{q}}+\left(\mathrm{r}_5 \times \mathrm{FH}_{\text {Alter }}\right.\left.-r_6 \times \text{SP}\right),\left\{\begin{array}{l}l=1,2, \ldots, n. \\ q=1,2, \ldots, r.\end{array}\right. \\ & \end{aligned}$                    (17)

The safest region where all animals assemble together for shelter which is given by (18) and (19) where $\text{P}{{\text{R}}_{\text{q}}}$ is $q$th prey encircled by $l$th FH.

$\text{S}{{\text{P}}_{1}}=\frac{\mathop{\sum }_{\text{q}=1}^{\text{r}}\text{P}{{\text{R}}_{\text{q}}}}{\text{r}},\left\{ \begin{matrix}q=1,2,\ldots ,r.  \\ l=1,2,\ldots ,n.  \\ \end{matrix} \right.$                (18)

$\text{SP}=\frac{\mathop{\sum }_{\text{k}=1}^{\text{m}}\text{P}{{\text{R}}_{\text{k}}}}{\text{m}},\text{k}=1,2,\ldots ,\text{m}.$                (19)

The fitness of the solution is calculated using (20) which is based on the intra-class and inter-class variability of the speech features. The higher inter-class and lower intra-class feature variability helps to selects the hugely discriminative combination of feature set for SER.

$Fitnes{{s}_{F{{H}_{i}}}}=\frac{{{\text{ }\!\!\sigma\!\!\text{ }}_{\text{inter}-\text{class}}}}{{{\text{ }\!\!\sigma\!\!\text{ }}_{\text{intra}-\text{class}}}}$               (20)

3.5 DCNN architecture

The DCNN provides the short term and long-term correlation and connectivity in the local and global features extracted using MAF. It characterizes the changes occurs on the valence and arousal on the voice due to emotion. It helps to represent the variations in intonation, prosody and timbre of the speech because of the emotions and language. It provides high level abstract features that give better distinctiveness for different emotions. The proposed lightweight DCNN consist of three layers of sequential CNN. Each layer of CNN consists of convolution layer (Conv), Rectified Linear Unit Layer (ReLU), and maximum pooling layer (MaxPool). The DCNN accepts the handcrafted feature vector that represents various spectral, temporal and voice quality features of the emotion signal. The first layer in proposed DCNN includes three layers {Conv1(KernelSize-1×3, NumFilter-64, Stride-1, ZeroPadding-Yes) →ReLU1 (Stride-1) →MaxPool1 (Stride-2)} which accepts multiple acoustic feature vector as an input with dimensions of (1×318) and produces the output feature map of (1×318×64). Zero padding maintains the original dimensions of the multiple acoustic features. The second layer is made up of {Conv2(KernelSize-1×3, NumFilter-128, Stride-1, ZeroPadding-Yes) →ReLU2(Stride-1) →MaxPool2(Stride-2)}. Third layer encompasses {Conv3(KernelSize-1×3, NumFilter-256, Stride-1, ZeroPadding-Yes) →ReLU3 (Stride-1) →MaxPool2(Stride-2)}.

The convolution layer feature map $y\left( n \right)$ of 1-D acoustic features A$\left( n \right)$ and L convolution filter $k\left( n \right)$using Eq. (21). Eq. (22) describes convolution feature map where $\text{y}_{\text{i}}^{\text{l}}$ stands for i^th feature map of ${{l}^{th}}$ layer, $\text{y}_{\text{j}}^{\text{l}-1}$ denotesjth feature of ${{\left( l-1 \right)}^{th}}$ layer, $\text{k}_{\text{ij}}^{\text{l}}$ representthe filter kernel of ${{l}^{th}}$ layer linked to $j$ feature, $\text{b}_{\text{i}}^{\text{l}}$ denotes for bias and $\text{ }\!\!\sigma\!\!\text{ }$ symbolizesReLU activation function. The ReLU activation function is faster and simple that replaces negative values by 0 to overcome the vanishing gradient issue using Eq. (23).

$y\left( n \right)=A\left( n \right)\times k\left( n \right)=\underset{m=0}{\overset{L-1}{\mathop \sum }}\,A\left( m \right).k\left( n-m \right)$             (21)

$y_{i}^{l}=\sigma \left( b_{i}^{l}+\underset{j}{\mathop \sum }\,y_{j}^{l-1}\times k_{ij}^{l} \right)$             (22)

$\sigma \left( y \right)=\max \left( 0,y \right)$                   (23)

Subsequent to three CNN layers, a FC layers is used having 4 hidden layers. Lastly, Softmax classifier offers the likelihood of the output class where utmost class label probability results in output class label using Eqs. (24)-(26). The Softmax classifier selected for the SER classification as it is simple probabilistic classifier which is simple, needs less computation compared with traditional classifiers and helps to minimize computational burden on the system.

${{z}_{i}}=\underset{j}{\mathop \sum }\,{{h}_{j}}{{w}_{ji}}$             (24)

${{p}_{i}}=\frac{\text{exp}\left( {{z}_{i}} \right)}{\mathop{\sum }_{j=1}^{n}\text{exp}\left( {{z}_{j}} \right)}$             (25)

$\hat{y}=arg\begin{matrix}   max  \\   i  \\\end{matrix}{{p}_{i}}$             (26)

Here, h_j denotes penultimate layer's weight, ${{\text{w}}_{\text{ji}}}$describes the weights joining Softmax and penultimate layer, ${{\text{z}}_{\text{i}}}$ representsSoftmax layer input, ${{\text{p}}_{\text{i}}}$stands for class label probability and $\hat{y}$ indicates label of predicted class. The outcome of proposed DCNN is estimated mini-batch gradient descent optimization (MBGD) learning algorithm. In MBGD algorithm n complete dataset is divided in to small batches b, then the model weights are reorganized utilizing model error given in Eq. (27).

${{E}_{t}}[f\left( w \right)=\frac{1}{b}\underset{k=\left( t-1 \right)b+1}{\overset{tb}{\mathop \sum }}\,f\left( w,{{T}_{i}} \right)$               (27)

The weights are updated using Eq. (28).

${{w}^{t+1}}={{w}^{t}}-\mu {{\nabla }_{w}}{{E}_{t}}\left[ f\left( {{w}^{t}} \right) \right]$             (28)

where, $E_t$ provides model error, $T_i$ stands for training samples, $W$ denotes weights of filter kernel, $F$ provides cost function, $\mu$ shows learning rate, $\nabla$ describes gradient of cost function and $t_b$ provides number of batches.