Complex-valued deconvolution DOA estimation method based on block-sparse BCS deep unfolding

Abstract

The complex-valued deconvolution DOA estimation method based on block sparse BCS deep unfolding includes: constructing an array receiving signal model under a single snapshot according to a sparse signal and a super-complete steering vector through a compressed sensing technology; constructing a complex-valued beam model according to the array receiving signal model under the single snapshot by using a traditional beam forming algorithm; constructing a coupled hierarchical Gaussian prior Bayesian compressed sensing model according to a complex number output of the complex-valued beam model by using a Bayesian compressed sensing technology; constructing a block sparse Bayesian compressed sensing complex-valued deconvolution model to realize accurate recovery of a block sparse signal; constructing a training data set, including a real source distribution function as a label and a complex-valued beam forming output as a sample corresponding to the label; a training loss function of a deep unfolding network based on a block sparse Bayesian compressed sensing; inputting the training data set into the deep unfolding network for model training; and inputting test data into the trained deep unfolding network to generate a direction of arrival estimation.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of signal processing, especially the technical field of array signal processing and direction of arrival estimation, and particularly relates to a complex single-snapshot deconvolution beamforming direction finding method using Bayesian compressed sensing. BACKGROUND

[0002] Array signal processing is a core and key technology in the fields of radar, mobile communication and satellite navigation, etc., and the direction of arrival (DOA) estimation is of great significance to target positioning and tracking. The traditional beamforming (CBF) method is limited by the physical aperture, has the Rayleigh limit constraint, and is difficult to distinguish close sources, and has high sidelobes, which seriously affects its use effect in dense target environment. The CBF process can be modeled as the convolution result of the real source distribution and the point spread function (PSF), so in order to solve the limitations of the CBF algorithm, the existing DOA estimation method uses deconvolution technology to invert from the beam output to obtain the real source distribution to realize super-resolution DOA estimation. However, most deconvolution-based DOA estimation algorithms are constructed in the real domain, while actual applications mostly need to be modeled and solved in the complex domain, and most deconvolution algorithms based on beam intensity are limited in performance when processing single-snapshot signals, which makes the existing framework not suitable for the actual application environment. In addition, in actual applications, the non-zero coefficients of signals often form a block sparse structure in a clustered distribution, which makes the random sparse assumption in the traditional DOA estimation method for block sparse signals have limitations such as hypothesis mismatch with reality.

[0003] At the same time, most of the existing deconvolution DOA estimation methods are traditional iterative algorithms, which have high computational complexity, low operation efficiency, and the problems of sensitivity of azimuth estimation to parameter setting and the influence of the weight matrix segment algorithm of the point spread function. SUMMARY

[0004] In view of the shortcomings of the prior art, the application provides a complex deconvolution DOA estimation method based on block sparse BCS deep unfolding, which improves the utilization rate of parameters, and enables the sparse mode of adjacent coefficients to be adaptively associated, so that the block sparse structure can be more flexibly and reliably enhanced, while the computational complexity is significantly reduced and the operation efficiency is improved, which is more suitable for real-time requirements.

[0005] To achieve the above purpose, the technical scheme adopted by the application is:

[0006] The complex deconvolution DOA estimation method based on block sparse BCS deep unfolding comprises:

[0007] Step 1, according to the sparse signal and the super-complete steering vector, a single-snapshot array received signal model is constructed through compressed sensing technology;

[0008] According to a single-shot array receiving signal model, a complex-valued beam model is constructed by using a traditional beam forming algorithm;

[0009] According to a complex number output of the complex-valued beam model, a coupled hierarchical Gaussian prior Bayesian compressed sensing model is constructed by using a Bayesian compressed sensing technology;

[0010] A block sparse Bayesian compressed sensing complex-valued deconvolution model is constructed, and a block sparse signal is accurately recovered;

[0011] Step 2, a training data set is constructed, including a real source distribution function as a label and a complex-valued beam forming output as a sample corresponding to the label;

[0012] Step 3, a block sparse Bayesian compressed sensing deep unfolding network training loss function is constructed;

[0013] Step 4, the training data set is input into the deep unfolding network for model training;

[0014] Step 5, test data is input into the trained deep unfolding network to generate a direction of arrival estimation.

[0015] Preferably, the coupled hierarchical Gaussian prior Bayesian compressed sensing model includes adding an inverse compressed sensing problem into a Bayesian compressed sensing framework for solving, and improving sparse signal coefficients by a spatial domain coupling strength parameter.

[0016] Preferably, in step 1, a first layer distribution of the coupled hierarchical Gaussian prior Bayesian compressed sensing model is:

[0017]

[0018] wherein, represents a complex Gaussian distribution, , , represents the spatial domain coupling strength parameter, used to express the correlation between the coefficient and a neighboring coefficient set ; is a relevant hyperparameter; represents the total number of elements; if , the sparsity of the coefficient is controlled by the hyperparameter and the relevant hyperparameter , and the sparsity patterns of the relevant coefficients are coupled by the hyperparameters shared by them.

[0019] As a preference, in the coupled hierarchical Gaussian prior Bayesian compressive sensing model, the coupled hierarchical Gaussian prior second layer distribution assumes that each signal precision imposes a Gamma prior distribution:

[0020]

[0021] wherein denotes a Gamma distribution, , is a predefined model parameter, and

[0022] As a preference, step 1 further comprises adopting different scale parameter values and the said spatial coupling strength parameter values for each coefficient in the sparse signal to improve the coupled hierarchical Gaussian prior distribution of the coupled hierarchical Gaussian prior Bayesian compressive sensing model.

[0023] As a preference, the improved coupled hierarchical Gaussian prior first layer distribution is:

[0024]

[0025] wherein ,

[0026] The sparse pattern of the related coefficients is forced to have correlation by the shared hyperparameters coupling the adjacent elements, ;

[0027] The second layer distribution is:

[0028]

[0029] As a preference, step 2 comprises:

[0030] Randomly generating sparse signals received by the array and random noise under a single signal-to-noise ratio and a single snapshot;

[0031] Randomly selecting incident angles from an angle set to form a set of incident angles , and forming corresponding real source distribution functions , and randomly selecting times to form a corresponding label set , wherein denotes the sparse signal formed by the set of incident angles; and the output of the complex-valued beam model ​As a sample corresponding to the label; the last available common The training data set composed of the label and its corresponding sample .

[0032] Compared with the prior art, the beneficial effects of the present application are embodied in:

[0033] 1. In the method of the present application, different scale parameter values and coupling strength parameter values are used to characterize the non-uniform correlation within the block, which improves the utilization of parameters compared to the method of using scale parameters and coupling parameters as shared parameters, and enables adaptive correlation of sparse patterns of adjacent coefficients, and more flexible and reliable enhancement of block sparse structure.

[0034] 2. The method of the present application expands the iterative algorithm into an interpretable deep iterative neural network through deep learning technology, learns and trains parameters and and the weight matrix of the point spread function , obtains more optimal parameters , and the point spread function matrix , effectively reduces or even eliminates the uncertainty of manually set parameters and , and obtains a more optimal point spread function matrix through the learned weight matrix, improving the direction of arrival estimation accuracy. At the same time, the use of shallow network design significantly reduces the computational complexity and improves the operation efficiency, and is more suitable for real-time requirements. BRIEF DESCRIPTION OF DRAWINGS

[0035] Figure 1 is a method flowchart of embodiment 1 of the present application;

[0036] Figure 2 is a general structure diagram of the deep iterative network of embodiment 1 of the present application;

[0037] Figure 3 is a single iteration network structure diagram of embodiment 1 of the present application;

[0038] Figure 4 is a bearing spectrum comparison diagram of embodiment 2 of the present application;

[0039] Figure 5 is a direction of arrival estimation result diagram of embodiment 3 of the present application under different signal-to-noise ratios;

[0040] Figure 6 is a direction of arrival estimation result diagram of embodiment 4 of the present application under different angle intervals. Detailed Implementation

[0041] To make the technical means, inventive features, objectives, and effects of the invention readily understandable, the invention is further described below with reference to specific illustrations. However, the invention is not limited to the embodiments described below.

[0042] It should be noted that the structures, proportions, sizes, etc., illustrated in the accompanying drawings of this specification are only used to complement the content disclosed in the specification for those skilled in the art to understand and read, and are not intended to limit the conditions under which the present invention can be implemented. Therefore, they have no substantial technical significance. Any modifications to the structure, changes in the proportions, or adjustments to the size, without affecting the effects and objectives that the present invention can produce, should still fall within the scope of the technical content disclosed in the present invention.

[0043] Example 1:

[0044] like Figure 1 The complex-valued deconvolution DOA estimation method based on block sparse BCS depth unrolling shown includes the following steps:

[0045] Step 1: Complex-valued beamforming output modeling:

[0046] 1.1 By using compressed sensing (CS) technology, sparse signals are input and overcomplete steering vectors are used to obtain array output in the form of a single snapshot, so as to reduce the dependence on the number of snapshots and improve the resolution.

[0047] First, a horizontal linear array is deployed to construct a single-shot array signal reception model:

[0048] Let the number of array elements be M, and divide the target space into N angular grids. And assume that the set of actual azimuth angles of K narrowband far-field signals incident on the array. Constructing an array at a certain moment The received single-shot array output is:

[0049]

[0050] in, Indicates in Complex signal at time, This represents the complex Gaussian white noise added to the array. This represents noise power. Denotes the array manifold matrix, where This indicates that the array is for the first... The steering vector of the received signal, where This indicates the transpose operation. Indicates the spacing between array elements. denotes signal wavelength, j denotes imaginary unit.

[0051] Based on the DOA (Direction of Arrival) estimation theory of the compressed sensing framework, the above single snapshot array output is converted into:

[0052]

[0053] Since the present application is performed under the condition that the Point Spared Function (PSF) is shift-invariant, at this time , is the array output of dimension, is the super-complete steering vector of dimension , is the sparse signal of dimension (the vector has only non-zero elements on the grid points closest to the real incident direction, and the values correspond to the complex amplitudes of the signal, and the other elements are all zero), is the Gaussian white noise of dimension.

[0054] 1.2, using the traditional beamforming (CBF) algorithm, constructing the complex-valued beam model according to the array signal model under single snapshot

[0055] The normalized weighting vector of beamforming in the observation angle is , and the complex-valued beam response model of traditional beamforming in output is:

[0056]

[0057] Among them: denotes the conjugate transpose operation, and the Sinc function .

[0058] 1.3, using Bayesian compressive sensing (BCS) technology, constructing a coupled hierarchical Gaussian prior Bayesian compressive sensing model according to the complex output of the complex-valued beam model

[0059] To address the ill-posedness of the deconvolution problem (which makes it sensitive to noise), and considering that most beam intensity-based deconvolution methods significantly degrade performance when processing coherent sources, this paper extends the deconvolution technique from the real domain to the complex domain for modeling. Furthermore, taking into account the spatial sparsity of sources in real-world scenarios, based on the CS theoretical framework and combined with complex-valued deconvolution techniques, the following complex-valued inverse compressed sensing model is obtained:

[0060]

[0061] in, , They represent respectively by The complex numerical beam vector and point spread function matrix composed of observation directions ( (This is the weight matrix of the point spread function). Represents the source distribution function in the complex domain. Let be a complex noise vector, whose real and imaginary parts follow an independent and identically distributed zero-mean Gaussian distribution.

[0062] The source distribution function (SDF) is the mathematical representation of a sparse signal. The coefficients of a sparse signal typically exhibit a special structure, which is used as known information and incorporated into the sparse pattern. Furthermore, the correlation between coefficients is integrated; this is model-based compressed sensing. Block structure, as a sparse signal structure developed based on the CS model, has its prior information applied to model-based CS. However, in reality, the position and size of the block structure are usually unknown, and each coefficient does not exist in isolation. To overcome this difficulty, this method incorporates the aforementioned inverse compressed sensing problem into the Bayesian compressed sensing framework for solution. It abandons the assumption of independent modeling of source coefficients in traditional sparse Bayesian learning and designs an improved coupled hierarchical Gaussian prior for each coefficient. The first layer is:

[0063]

[0064] in, Indicates a complex Gaussian distribution. , , This represents the spatial coupling strength parameter, used to describe the coefficients. With adjacent coefficient set The correlation between the coefficients; , express Total number of elements. When When the above equation is expressed, it represents the prior form in traditional Bayesian Compressed Sensing (BCS). Conversely, if ,but The sparsity is not only determined by hyperparameters Control is also determined by related hyperparameters. Control. Assumption As it approaches infinity, then its corresponding coefficient It will become zero, and the correlation coefficient It is highly likely that it will also be zero. Similarly, the relevant hyperparameters... Approaching infinity can also easily lead to Take a value of zero. In summary, the sparsity pattern of the correlation coefficient is determined by... The shared hyperparameters are coupled with each other, which forces the activity of adjacent elements to be correlated, thus "preferring" block sparse structure (clustering of non-zero elements) rather than random sparsity (isolation of non-zero elements) in the mathematical model.

[0065] The second-level distribution assumes that a Gamma prior distribution is applied to each signal precision:

[0066]

[0067] in Represents the Gamma distribution. , , are predefined model parameters that represent the shape and scale parameters of the Gamma distribution, respectively, and belong to the prior parameters.

[0068] 1.4 Construct a block-sparse Bayesian compressed sensing complex-valued deconvolution model to accurately recover block-sparse signals.

[0069] The scale parameter mentioned in the bilayer distribution in step 1.3 Spatial coupling strength parameters These parameters are shared by all coefficients, resulting in low parameter utilization and an inability to characterize the non-uniform correlation between coefficients within a block. Therefore, this method further improves upon the above to obtain an improved coupled hierarchical Gaussian prior to more accurately represent sparsity, i.e., by using different scale parameter values ​​for each coefficient in the sparse signal. Spatial coupling strength parameter value Improved Coupled Hierarchical Gaussian Prior: First Layer Distribution:

[0070]

[0071] in, ,

[0072] The sparsity pattern of the correlation coefficient is determined by... The shared hyperparameters are coupled with each other, which forces the activity of adjacent elements to be correlated, thus "preferring" block sparse structure (clustering of non-zero elements) rather than random sparsity (isolation of non-zero elements) in the mathematical model. Second layer distribution:

[0073]

[0074] In the BCS framework, noise satisfies the following hierarchical prior distribution:

[0075] First layer distribution

[0076]

[0077] in, Indicates the noise variance. Indicates noise accuracy. The dimension is The identity matrix, This represents a complex Gaussian distribution. The second-layer distribution affects noise accuracy. Apply Gamma prior:

[0078]

[0079] Under the aforementioned assumed prior probability conditions, sparse signals can be obtained. The posterior distribution:

[0080]

[0081] Meanwhile, the complex Gaussian likelihood function of the beam output in the BCS framework is:

[0082]

[0083] Therefore, the mean and variance of the posterior probability density function are calculated to be...

[0084]

[0085] With the placement of the hyperprior, learning the hyperparameters becomes searching for their posterior patterns, i.e., maximizing the expected value of the joint distribution function (posterior probability density function) with respect to the posterior:

[0086]

[0087] Obtained through the EM algorithm and The estimated values ​​are respectively

[0088]

[0089] ,

[0090] in , To adjust the parameters.

[0091] Step 2: Generation of Deep Learning Training Dataset: This addresses the challenges of array space constraints, single-shot signal processing, and point spread function weights in traditional complex-valued deconvolution DOA estimation, which are present in practical applications. To address the challenges of optimization difficulties, sensitivity to prior parameters, and high computational complexity, this invention combines deep learning technology to transform the parameters and weight matrices that require manual tuning in traditional methods into learnable parameter arrays and weight matrices. Furthermore, it employs a shallow network design, significantly reducing computational complexity. The specific dataset generation method is as follows:

[0092] In the case of a single signal-to-noise ratio (SNR) and a single snapshot, the sparse signal received by the randomly generated array is... and random noise Because the report's study used a uniform linear array, and the PSF of a uniform linear array is related to... It has shift invariance, and at this point there is an overcomplete guided vector dictionary. At the same time, it can be Random selection A set of incident angles is formed by several incident angles. And form the corresponding real source distribution function. , conducted The corresponding tag set is formed by random selection. ,in Indicates the first The sparse signal is formed by the incident angles. Based on the beamforming formula above, the complex-valued beamforming output is obtained. As samples corresponding to the labels, the total number can be obtained. The training dataset consists of labels and their corresponding samples. .

[0093] Step 3: Design of the training loss function for the block sparse Bayesian compressed sensing deep unfolding network: To achieve a balance between allowable approximation error and improved convergence speed, the network in this algorithm adjusts the learnable parameters by performing end-to-end task-driven training. That is, to minimize the label (i.e., the true signal) with low computational cost. With network output The approximation error between the two values. The following formula is used as the loss function Loss to quantitatively measure the approximation error:

[0094]

[0095] in The proposed deep unfolded network is for samples The estimated result of the reconstructed true signal distribution. A smaller loss indicates better reconstruction quality and higher estimation accuracy. During training, it can learn to approximate the true signal distribution as closely as possible. The optimal solution.

[0096] Step 4: Neural Network Architecture and Training: The neural network has L layers. The training dataset is input into the dual-drive deep unfolded network of this data model to obtain the final network output. And save the trained model (including the parameters obtained during training). The neural network is trained using the forward propagation method.

[0097] Step 5, Direction of Arrival Estimation: Input the test data into the trained model to obtain the network output. When plotted as an azimuth map, the angles corresponding to the largest k or q spectral peaks are the angles of arrival when there are k incident signal sources or q clusters of sparse signals.

[0098] To verify the performance of this invention, simulation examples are used below, and the verification results are compared with dCv-PC-CBCS, dCv-CBCS, dCv-RBCS and dCv-CISTA. The Cramer-Rao lower bound (CRB) is introduced as a theoretical performance upper limit reference, and the CBF algorithm is used as the basic comparison.

[0099] The array was configured as a 13-element uniform linear array, using single snapshots, and the spatial domain was divided into 181 grids. In step 4 of constructing the training dataset, the signal-to-noise ratio (SNR) was 15, and the number of signal sources was 2. A total of 60,000 training data points were generated using the aforementioned method. The deep iterative network was configured as follows... Figure 2 As shown: The deep learning network has L=10 layers, using sigmoid and softplus activation functions, Adam optimizer, a maximum number of epochs of 400, a batch size of 256, and an initial learning rate of 0.001. The results of a single layer are shown below. Figure 3 As shown, the simulation results of this method are compared with those of the dCv-PC-CBCS algorithm, the dCv-CBCS algebraic algorithm, the dCv-RBCS algorithm, and the convex optimization algorithm dCv-CISTA. The comparison metrics are 3dB beamwidth, maximum sidelobe intensity, computation time, and root mean square error (RMSE).

[0100] Example 2:

[0101] With a fixed signal-to-noise ratio of 15dB, assume two narrowband far-field signal clusters are incident on a uniform linear array from approximately -5° and 5° respectively. Each signal cluster consists of 5 signal sources, which are closely distributed at 0.1° intervals in the angular dimension. The specific angles are as follows: and .

[0102] The comparison results are as follows Figure 4 As shown in Figure 1 and Table 1, the present invention and the dCv-PC-CBCS algorithm identify the five closely connected sources within each cluster as a continuous energy diffuser, rather than five independent sources. Furthermore, the present invention's method significantly sharpens the main lobe and reduces sidelobe intensity. Compared to the dCv-CISTA and dCv-RBCS algorithms, it exhibits more significant sidelobe suppression and a narrower main lobe. Although the main lobe width is slightly wider than that of the dCv-CBCS algorithm, its ability to suppress sidelobes is stronger. Simultaneously, the iteration time of this algorithm is significantly reduced, making it more suitable for testing environments with high real-time requirements.

[0103] Table 1: Comparison of Performance Parameters of Various Algorithms

[0104]

[0105] Example 3:

[0106] Two narrowband far-field signals were incident on a uniform linear array from -5° and 5° respectively. The angles corresponding to the two largest peaks in the model's output layer were used as the angle estimation results. Through 300 Monte Carlo simulations, the RMSE of four algorithms (dCv-PC-CBCS, dCv-CBCS, dCv-RBCS, and dCv-CISTA) and the method of this invention were compared under 300 iterations, showing changes in signal-to-noise ratio from 0dB in 5dB increments to 40dB. The Cramer-Rao lower bound (CRB) was introduced as a theoretical performance upper limit reference, with the CBF algorithm used as a baseline comparison. The comparison results are as follows: Figure 5 As shown in the figure, compared with other algorithms, this algorithm has the lowest root mean square error and has excellent estimation performance.

[0107] Example 4:

[0108] With a signal-to-noise ratio of 15dB, the incident angles of the two signal sources are 0° and 1° respectively. It is assumed that angle 1 changes from 0° to 12° in 1° increments, while angle 2 remains constant at 0°. Different azimuth differences between the two signal sources are processed, with each azimuth estimation processing a single snapshot. After 300 Monte Carlo simulations, the RMSE of four algorithms (dCv-PC-CBCS, dCv-CBCS, dCv-RBCS, and dCv-CISTA) and the method of this invention are compared with the angle difference over 300 iterations. The Cramero lower bound (CRB) is introduced as a theoretical performance upper limit reference, and the CBF algorithm is used as the basis for comparison. The comparison results are as follows: Figure 6 As shown in the figure, the algorithm of this invention is not much different from dCv-CBCS, but its performance is better than other comparative algorithms.

[0109] Therefore, it can be seen that the direction finding performance of the method proposed in this invention is significantly better than that of the comparative method, and it has strong adaptability and superiority under different test conditions.

[0110] The above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A complex-valued deconvolution DOA estimation method based on block sparse BCS depth unrolling, characterized in that, include: Step 1: Based on the sparse signal and the overcomplete steering vector, construct a single-shot array signal receiving model using compressed sensing technology; Using traditional beamforming algorithms, a complex-valued beam model is constructed based on the single-shot array received signal model; Using Bayesian compressed sensing technology, a coupled hierarchical Gaussian prior Bayesian compressed sensing model is constructed based on the complex output of the complex-valued beam model, including: The inverse compressed sensing problem is incorporated into the Bayesian compressed sensing framework for solution, and the sparse signal coefficients are improved by the spatial coupling strength parameter. The first layer distribution of the coupled hierarchical Gaussian prior is: ； in, Indicates a complex Gaussian distribution. , , The spatial coupling strength parameter represents the coefficients. With adjacent coefficient set The correlation between the coefficients; , which are the relevant hyperparameters; express Total number of elements; if Then the coefficient The sparsity is determined by the hyperparameters and related hyperparameters The sparse patterns of the correlation coefficients are coupled to each other through their shared hyperparameters. Coupled hierarchical Gaussian prior second-layer distribution assumes that a Gamma prior distribution is applied for each signal precision: ； in Represents the Gamma distribution. , , are predefined model parameters, representing the shape and scale parameters of the Gamma distribution, respectively, and are considered prior parameters; Construct a block-sparse Bayesian compressed sensing complex-valued deconvolution model to accurately recover block-sparse signals; Step 2: Construct a training dataset, including the real source distribution function as the label and the complex-valued beamforming output as the sample corresponding to the label; Step 3: Train the loss function of the deep unfolded network based on block sparse Bayesian compressed sensing; Step 4: Input the training dataset into the deep unfolded network for model training; Step 5: Input the test data into the trained deep unfolded network to generate the direction of arrival (DOA) estimate.

2. The complex-valued deconvolution DOA estimation method based on block sparse BCS depth unrolling according to claim 1, characterized in that, Step 1 also includes applying different scale parameter values ​​and the spatial coupling strength parameter values ​​to each coefficient in the sparse signal to improve the coupled hierarchical Gaussian prior distribution of the coupled hierarchical Gaussian prior Bayesian compressed sensing model.

3. The complex-valued deconvolution DOA estimation method based on block sparse BCS depth unrolling according to claim 2, characterized in that, The improved coupled hierarchical Gaussian prior first-layer distribution is as follows: ； in, , The sparsity pattern of the correlation coefficient is determined by... Shared hyperparameters are coupled together, forcing the activity levels of adjacent elements to be correlated. ; The second layer distribution is as follows: 。 4. The complex-valued deconvolution DOA estimation method based on block sparse BCS depth unrolling according to claim 1, characterized in that, Step 2 includes: Under a single signal-to-noise ratio and a single snapshot, sparse signals received by a randomly generated array are generated. and random noise It has a highly complete guided vector dictionary. ; in the angle set Random selection A set of incident angles is formed by several incident angles. And form the corresponding real source distribution function. , conducted The corresponding tag set is formed by random selection. ,in Indicates the first Sparse signal formed by incident angles; output of complex-valued beam model As samples corresponding to the labels; finally, the total can be obtained. The training dataset consists of labels and their corresponding samples. .

Images