Tab. 1 displays the abbreviation list Image moment was firstly introduced by Hu [21], who used geometric moments to generate a set of invariants. Hu’s moments have been widely used in knee osteoarthritis classification [22], brain tumor classification [23], etc. However, geometric moments are sensitive to noise. Thus, Teague [24] introduced Zernike moments (ZMs) based on orthogonal Zernike polynomials. The orthogonal moments have been proven to be more robust in noisy conditions, and they can achieve a near-zero value of redundancy measure [25].

Later, pseudo Zernike moment (PZM) is derived from Zernike moment. PZMs have been proven to give better performances than other moment functions such as Hu moments, Zernike moments, etc. For example, for an order p, there are (p+1)2 linearly independent pseudo-Zernike polynomials of orders ≤p, while there are only 12(p+1)(p+2) Zernike polynomials. Hence, PZM is more expressive and offers more feature vectors than ZM.

The kernel of PZMs is a set of orthogonal pseudo-Zernike polynomials defined over the polar coordinate inside a unit circle (UC). The 2D PZM of order p with repetition q of an image g(r,θ) is defined as [26]

(1) Zpq=p+1π∫−ππ∫01Wpq∗(r,θ)g(r,θ)rdrdθ,

where the pseudo-Zernike polynomials Wpq(r,θ) of order p are defined as

(2) Wpq(r,θ)=Rpq(r)ejqθ,j=−1

(3) Rpq(r)=∑k=0p−|q|(−1)k(2p+1−k)!k!(p+|q|+1−k)!(p−|q|−k)!rp−k

where 0≤|q|≤p. In practice, pseudo Zernike functions (https://www.mathworks.com/matlabcentral/ fileexchange/33644-pseudo-zernike-functions) are used for simplicity and fast calculation. Fig. 3 displays pseudo Zernike functions of orders p≤5.

Note that PZM are defined in terms of polar coordinates (r,θ) with |r|≤1. Therefore, the computation of PZM requires a linear transformation of the image plane (IP) coordinates (w,h),1≤w≤W,1≤h≤H to the UC domain (x,y)∈R2. There are two commonly used transformations as shown in Fig. 4: (i) IP over UC; and (ii) IP inside UC. In this study, we use the former (IP over UC), because the lesions will not occur within the four corners of the CCT image.

Traditionally, p-order PZMs are sent into shallow classifiers, such as multi-layer perceptron [27], adaptive differential evolution wavelet neural network (Ada-DEWNN) [28], linear regression classifier (LRC) [29], kernel support vector machine (KSVM) [30]. In this study, we introduced a customized deep-stacked sparse autoencoder (DSSAE). DSSAE is a type of deep neural network technologies, and we expect DSSAE to achieve better performances than shallow models.

The fundamental element of DSSAE in the autoencoder (AE), which is a typical shallow neural network that learns to map its input X to output Y. There is an internal code output IN That represents the input X. The whole AE can be divided into two parts: An encoder part (AX,BX) that maps the input X to the code IN, and a decoder part (AY,BY) that maps the code to a reconstructed data Y.

The structure of AE is displayed in Fig. 5, where the encoder part is with weight AX and bias BX, and the decoder part is with weights AY and bias BY. We have

(4) IN=zLS(AXX+BX),

(5) Y=zLS(AYIN+BY),

where the output Y is an estimate of input X, and zLS is the log sigmoid function

(6) zLS(x)=11+exp⁡(−x).

The sparse autoencoder (SAE) is a variant of AE. SAE encourages sparsity into AE. SAE only allows a small fraction of the hidden neurons to be active at the same time. To minimize the error between the input vector X and the output Y, the raw loss function Jb of AE is deduced as:

(7) Jb(AX,AY,BX,BY)=1NS‖Y−X‖2,

where NS means the number of training samples. From Eqs. (4) and (5), we find the output Y can be expressed in the way of

(8) Y=zAE(X∣AX,AY,BX,BY),

where zAE is the abstract of AE function [31]. Hence, Eq. (7) can be revised as

(9) Jb(AX,AY,BX,BY)=1NS‖zAE(X∣AX,AY,BX,BY)−X‖2.

To avoid over-complete mapping or learn a trivial mapping, we define one L2 regularization term ΓA of the weights (AX,AY) and one regularization term Γs of the sparsity constraint. Therefore, the loss function Jl of SAE is derived as:

(10) Jl(AX,AY,BX,BY)=1NS‖zAE(X∣AX,AY,BX,BY)−X‖2+as×Γs+aA×ΓA,

where as stands for the sparsity regulation factor, and aA the weight regulation factor. The sparsity regularization term Γs is defined as:

(11) Γs=∑m=1|IN|zKL(ρ,ρ^m)=∑m=1|IN|ρlog⁡ρρ^m+(1−ρ)log⁡1−ρ1−ρ^m,

where zKL stands for the Kullback–Leibler divergence [32] function, |IN| is the number of elements of internal code output IN, ρ^m is the m-th neuron’s average activation value over all NS training samples, and ρ is its desired value, viz., sparsity proportion factor. The weight regularization term ΓA is defined as

(12) ΓA=12×‖AXAY‖22.

The training procedure is set to scaled conjugate gradient descent (SCGD) method.

We use SAE as the building block and establish the final deep-stacked sparse autoencoder (DSSAE) classifier by following three operations: (i) We include input layer, preprocessing layer, PZM layer; (ii) We stack four SAEs; (iii) We append softmax layer at the bottom of our AI model. The details of this proposed PZM-DSSAE model are listed in Tab. 2 and illustrated in Fig. 6. After processing, all the CCT images are normalized to fixed grayscaled images with the size of W×H. Then, PZM is applied to obtain feature vector with size of (p+1)2×1. In the classification stage, four SAE blocks with number of neurons of (M1,M2,M3,M4) are employed. Finally, a softmax layer with neurons of Mc is appended, where Mc means the number of categories to be identified.

The small size of training images causes overfitting, one solution to data augmentation (DA) that creates fake training images. Multiple-way DA (MDA) is an enhanced method of DA. Wang [33] proposed a 14-way data augmentation, in which they employed seven different DA techniques on k-th training image g(k) and its mirrored image g(m)(k).

In this study, we add two new DA techniques, speckle noise (SN) [34] and salt-and-pepper noise (SAPN). SN altered image is defined as

(13) gSN(k)=g(k)+NSN∗g(k),

where NSN is uniformly distributed random noise. The mean and variance of NSN is set to mSN and vSN, respectively.

For the k-th training image g(k), the SAPN altered image [35] is defined as gSAPN(k) with its values are set as

(14) {P(gSAPN=g)=1−γdsapn,P(gSAPN=gmin)=γdsapn2,P(gSAPN=gmax)=γdsapn2,

where γdsapn stands for noise density, and P the probability function. gmin and gmax correspond to black and white colors, respectively. The definitions of gmin and gmax can be found in Algorithm 1.

First, QD different DA methods as shown in Fig. 7 are applied to g(k). Let Hm,m=1,…,QD denote each DA operation, we have the augmented dataset on raw image g(k) as:

(15) Hm[g(k)],m=1,…,QD.

Suppose QN stands for the size of generated new images for each DA method, we have

(16) |Hm[g(k)]|=QN.

Second, horizontal mirrored image is generated as:

(17) g(m)(k)=zb[g(k)]

where zb stands for horizontal mirror function.

Third, all the QD different DA methods are performed on the mirror image pc(k), and generate QD different dataset.

(18) {Hm[g(m)(k)],m=1,…,QD|Hm[g(m)(k)]|=QN,m=1,…,QD

Fourth, the raw image g(k), the mirrored image g(m)(k), all the above QD-way results of raw image Hm[g(k)], and QD-way DA results of horizontal mirrored image Hm[g(m)(k)], are combined together. The final generated dataset from g(k) is defined as F(k):

(19) g(k)↦F(k)=za{g(k)g(m)(k)H1[g(k)]⏟QNH1[g(m)(k)]⏟QN⋯⋯HQD[g(k)]⏟QNHQD[g(m)(k)]⏟QN}

where za stands for the concatenation function. Suppose augmentation factor is QA, which stands for the number of images in F(k), we have

(20) QA=|F(k)||g(k)|=(1+QD×QN)×21=2×QD×QN+2

Algorithm 2 summarizes the pseudocode of proposed 18-way DA method.

F-fold cross-validation was used in this study. The whole dataset is divided into F folds. At f-th trial, 1≤f≤F, the f-th fold is selected as the test, and the rest F−1 folds [36]: [1,…,f−1,f+1,…,F] are selected as training set (Fig. 8). In this study, suppose F=10, then each fold will contain 32 COVID-19 images and 32 HC images.

To avoid randomness, we run the whole above procedure NR times with different initial random seeds and different cross-validation partitions. The ideal confusion matrix (CM) Rideal is defined as

(21) Rideal={rideal(m,n)}=NR×[C100C2],

Note here the off-diagonal entries of Rideal are all zero, viz., rideal(m,n)=0,∀m≠n. C1 and C2 are the number of samples of each category, which can be found in Algorithm 1. Seven measures are defined based on realistic CM [37] defined as:

(22) Rreal={rreal(m,n)}=[TPFNFPTN],

The first four measures are sensitivity, specificity, precision and accuracy, common in most pattern recognition papers. The last three measures are F1 score, Matthews correlation coefficient (MCC) [38], and Fowlkes–Mallows index (FMI) [39]. They are defined as:

(23) {F1=2TP2TP+FP+FNMCC=TP×TN−FP×FN(TP+FP)×(TP+FN)×(TN+FP)×(TN+FN),FMI=TPTP+FP×TPTP+FN

Besides, the receiver operating characteristic (ROC) curve [40] is used to provide a graphical plot of our model. ROC curve is created by plotting the true positive rate against the false-positive rate at various threshold settings. The area under the curve (AUC) is also calculated.

Tab. 3 displays the parameter setting of this study. The number of samples of each class is 320. The minimum and maximum grayscale values are set to (0,255). For the crop operation, 200 pixels are removed from all four sides. The preprocessed image is with size of 256×256. The max order of PZM is set to 19, so we have (19+1)2=400 PZM features. The weight regularization factor aA=0.001, the sparsity regulation factor as=1.1, and the sparsity proportion factor is ρ=0.05. The neurons of four SAEs are 300, 200, 100, and 50, respectively. The number of classes to be classified is set to 2. The number of folds in cross-validation is set to 10. The mean and variance of uniformly distributed random noise in SN are set to 0 and 0.05, respectively. The noise density of SAPN is set to 0.05. The number of different DA methods is set to 9, and the number of the newly generated image is set to 30. The augmentation factor is obtained as QA=542 (See Algorithm 2). The number of runs is set to 10.

Fig. 9 shows the QD-way DA to the raw image. Due to the page limit, the mirrored image and its corresponding DA results are not displayed. As can be observed in Fig. 9, the multiple-way DA can increase our training images’ diversity.

Tab. 4 gives the 10 runs of 10-fold cross-validation, where we can see our method achieves a sensitivity of 92.06% ± 1.54%, a specificity of 92.56% ± 1.06%, a precision of 92.53% ± 1.03%, and an accuracy of 92.31% ± 1.08%. Its F1 score, MCC, and FMI arrive at 92.29% ± 1.10%, 84.64% ± 2.15%, and 92.29% ± 1.10%, respectively. The AUC is 0.9576.

In addition, we compared the two transformation settings: IP over UC against IP inside UC (See Fig. 4). The IP inside the UC setting achieves a sensitivity of 91.84% ± 2.18%, a specificity of 92.44% ± 1.31%, and an accuracy of 92.14% ± 1.12%, which are worse than IP over UC setting. This comparison result demonstrates the reason why we choose IP over UC in this study. Particularly, the receiver operating characteristics (ROC) curves of both settings are displayed in Fig. 10.

This proposed PZM-DSSAE method is compared with 8 state-of-the-art methods. The comparison results are carried out on the same dataset via 10 runs of 10-fold cross-validation, and the results are displayed in Tab. 5. Fig. 11 displays the error bar of the proposed method against 8 state-of-the-art methods. We can see that the proposed PZM-DSSAE gives the best performance among all the methods. The reason is three folds: (i) We try to use PZM as the feature descriptors, (ii) DSSAE is used as the classifier, (iii) 18-way DA is employed to solve the overfitting problem.