Tab. 1 enumerates all abbreviations and their associated meanings. The advantage of wavelet transform (WT) is that it holds both time/spatial and frequency information of the given signal/image. Nevertheless, the discrete wavelet transform (DWT) is chosen to convert the raw signal r(t) into the wavelet coefficient domain [15] in reality. Suppose the signal r(t) is one-dimension, first, we define the continuous wavelet transform (CWT) Eγ(sa,st) of r(t) as:(5) Eγ(sa,st)=∫−∞∞⁡r(t)×γ(t|sa,st)dt, in which E stands for the wavelet coefficient, γ the mother wavelet. γ(t|sa,st) is defined as:(6) γ(t|sa,st)=1saγ(t−stsa),sa>0,st>0, where the sa signifies the scale factor (SF) and st the translation factor (TF).

Now, we deduct the definition of DWT from CWT. The Eq. (5) is discretized by substituting sa and st with two discrete variables (DVs) c and v,

(7) {sa=2cst=v×2c

where c signifies the DV of the SF sa , and v the DV of the TF st [16]. Moreover, the original signal r(t) is a DV to r(q) , of which q signifies the DV of t. Like this, two subbands (SBs) can be calculated. The approximation SB EA(q|c,v) is determined as:(8) EA(q|c,v)=SD[∑o⁡r(q)×fA∗(q−2cv2c)], where fA(q) signifies the low-pass filter. SD is the down-sampling operation. The detail SB ED(q|c,v) is determined as:(9) ED(q|c,v)=SD[∑o⁡r(q)×fD∗(q−2cv2c)]. where fD(q) signifies the high-pass filter.

Suppose we handle a two-dimensional (2d) image Q; the 2d-DWT [17] is worked out by processing row-wise and column-wise 1d-DWT in succession [15]. Initially, the 2d-DWT operates on the original image Q. Later, four SBs (Z1,O1,F1,A1) are generated, where the subscript i means i -th level decomposition. Tab. 2 itemizes the description of four SBs. Note here MDL means the maximum decomposition level.

Assuming h2d−DWT signifies a 2D-DWT decomposition operation, we deduce(10) (A1Z1O1F1)=h2d−DWT(Q).

The subsequent decompositions run as:(11) (AmZmOmFm)=h2d−DWT(Am−1),m=2,…M, where M is the MDL and m the current decomposition level [18].

The subband A1 is further decomposed into four SBs (A2,Z2,O2,F2) at the 2nd level. The SB A2 is later decomposed to (A3,Z3,O3,F3) , and then SB A3 is decomposed accordingly. Fig. 3 portrays a diagram of 5-level 2d-DWT, whose pseudocode is represented in Algorithm 1. This study chooses a M -level decomposition. The optimal value of M is found via trial-and-error approach [19] and related in Section 4.1.

The (3M+1) SBs (AM,ZM,OM,FM,ZM−1,OM−1,FM−1,…,Z1,O1,F1) may contain redundant features. Here we use the db4 wavelet. To decrease the number of features, we employ two-dimensional wavelet entropy (2d-WE) layer. The pseudocode of 2d-WE is illustrated in Algorithm 2. For each SB s in the generated (3M+1) SBs, we imagine s to be a random DV S with H quantization values (s1,s2,…,sh,…,sH) . In the beginning, we gauge the matching probability mass function (PMF) p(s)={ph(s)} .(12) ph(s)=hPr(S==sh),h=1,2,⋯H, where hPr signifies the probability function.

Second, the entropy of the PMF p(s) is calculated as fe(s) :(13) fe(s)=−∑h=1H⁡ph(s)×log⁡ph(s), where fe is the entropy function.

Lastly, the entropy values of the whole SBs are concatenated to grow a feature vector I.(14) I=[fe(AM)fe(ZM)fe(OM)fe(FM)fe(ZM−1)fe(OM−1)fe(FM−1)………fe(Z1)fe(O1)fe(F1)], where the number of the features in I is NI=(3M+1) , which equals the number of SBs.

The NI features are thrown into a feed-forward neural network (FNN)—in which its inner connections do not make a loop. One-hidden-layer FNN (OFNN), represented in Fig. 4, is established due to the universal approximation theory. Assume (x,t) stands for a training case as: x=[x1,x2,…,xi,…,xNI]T signifies the input feature vector with NI -dimension, i denotes the neuron index at the input layer, t is the corresponding target label t=[t1,t2,…,tk,…,tNO]T, where NO signifies the number of prediction categories and k the node index at the output layer. Assuming n is the case index and N the number of entire training cases, this study symbolizes the training case (x,t) as {x(n),t(n)|n=1,,…,N} . The training of the weights/biases (WBs) of OFNN is considered an optimization problem that minimizes the loss between the target t and the real output y. This study chooses the loss as the sum of the mean-squared error (MSE) E :(15) E=∑n=1N⁡∑k=1NO⁡[yk(n)−tk(n)]2.

Assume β2 is the activation function (AF) in the output layer, and (B,S) are the WBs of neurons that connect the hidden layer (HL) to the output layer. B={b(j,k)},j=1,…,NL,k=1,…NO, and S={s(k)},k=1,…,NO. It is easy to reckon the output yk as

(16) yk(n)=β2[∑j=1NL⁡b(j,k)zj(n)+s(k)],

where zj(n),j=1,…,NH signifies the output of j -th neuron in the HL. The description of zj(n) is(17) zj(n)=β1[∑i=1NI⁡a(i,j)xi(n)+r(j)]. where A={a(i,j)},i=1,…,NI,j=1,…,NL and R={r(j)},j=1,…,NL are the WBs of the neurons that connect the input layer with the HL, and β1 the AF linked to the HL.

The parameter training is an optimization problem that guides us to search for the optimal WB parametric vector θ=(A,B,R,S) . The length of θ is the number of parameters we need to optimize and is calculated as Nθ=NI×NL+NL×NO+NL+NO . The training algorithm we choose is adaptive chaotic PSO (ACP) [14].

Recap that two attributes (position x and velocity v ) are linked with each particle p in the standard PSO algorithm. Those two attributes are defined as the position of the particle (PoP) and the velocity of the particle (VoP). In each epoch, the fitness function E is re-calculated for the entire particles {p} in the swarm. The VoP v is re-evaluated by keeping track of the two best positions (BPs).

The first is the BP a particle p has traversed till now. It is dubbed pBest and symbolized as xpB . The second is the BP that any neighbor of p has traversed till now. It is a neighborhood best and is named nBest and symbolized as xnB .

If p takes the entire swarm as its neighborhood, the nBest turns to the global best and is for that reason named gBest. In standard PSO, the VoP v of particle p is updated as:(18) v←ωv+b1r1(xPB−x)+b2r2(xnB−x) where ω signifies the inertia weight (IW) controlling the influence of the preceding velocity of the particle on its present one. b1 and b2 stand for two positive constants named acceleration coefficients. r1 and r2 mean two random numbers, uniformly distributed in the range of [0,1]. r1 and r2 are re-calculated whenever they occur. The PoP x of the particle p is updated as:(19) x←x+vΔt where Δt is the assumed time step and always equals 1 for simplicity.

The ACP algorithm proposed an adaptive IW factor (AIWF) strategy. It uses ωAIWF to replace ω .(20) ωAIWF=ωmax−ωmax−ωminkmax×k

Here, ωmax signifies the maximum IW, ωmin the minimum IW, kmax the epoch once the IW goes to the final minimum IW, and k the present epoch.

Another improvement in ACP is upon the two random numbers (r1,r2) . In reality, (r1,r2) are created by pseudo-random number generators (RNG), which cannot guarantee the optimization’s ergodicity in solution space since they are pseudo-random. Rossler attractor (RA) is a good choice to calculate the random numbers (r1,r2) . RA equations are defined:

(21) {dxdt=−(y+z)dydt=x+δay    dzdt=δb+xz−δcz,

where δa , δb , and δc are inherent parameters of RA. We choose δa=0.2,δb=0.4,δc=5.7 via the trial-and-error method [20]. The corresponding curve is drawn in Fig. 5a. We agree r1=x(t) and r2=y(t) to implant the chaotic properties of RA into the two parameters (r1,r2) in standard PSO. The (x,y) plane of RA is displayed in Fig. 5b.

The parameters of this study are listed in Tab. 3. The sizes of the original images are 1024×1024 if we do not consider the number of color channels. The sizes of MTCed images are 624×624 , and the sizes of preprocessed images are 256×256 . The DCR is z1=48 , and the SSR is z2=97.92 . The MDL is M=4 . The number of features is NI=13 . The number of neurons in HL is NL=8 . The number of output neurons is NO=2 . The number of parameters to be optimized is Nθ=130 . The parameters in RA are δa=0.2,δb=0.4,δc=5.7 .

Fig. 6 shows the wavelet decomposition results with M=4 . The raw image is shown in Fig. 2a. The reason why we choose M=4 is the trial-and-error method. We test other values of M and find M=4 can obtain the best result.

Tab. 4 shows the ten runs of 10-fold CV via the parameters shown in Tab. 3, where pr=1,2.,..,10 means the run index. The final row in Tab. 4 presents the mean and standard deviation (MSD) of the results of 10 runs. WACPN attains a sensitivity of 91.87 ± 1.37%, a specificity of 90.70 ± 1.19%, a precision of 91.01 ± 1.12%, an accuracy of 91.29 ± 1.09%, an F1 score of 91.43 ± 1.09%, an MCC of 82.59 ± 2.19%, and an FMI of 91.44 ± 1.09%.

If we remove the AIWF from our WACPN model, the results using the same configuration are shown in Tab. 5. Similarly, the results of removing RA from our WACPN model are shown in Tab. 6. After comparing the results in Tab. 4 against the results in Tabs. 5 and 6, we can deduce that both strategies—AIWF and RA—are beneficial to our WACPN model.

Fig. 7 represents the ROC curves together with their upper and lower bounds of the proposed WACPN model and its two ablation studies (without AIWF and without RA). The AUC of WACPN model is 0.9577. The AUCs of the models removing AIWF or RA are only 0.9319 and 0.9456, respectively, demonstrating that both AIWF and RA help improve the standard PSO.

The proposed WACPN model is judged with six state-of-the-art models: GAN [4], CADe [5], SVM [6], IQNN [7], DT [8], and CSO [9]. The evaluation results on the same dataset via ten runs of 10-fold CV are listed in Tab. 7.

Error bar (EB) is an excellent tool for ease of visual evaluation. Fig. 8 presents the EB of model comparison, from which we can observe that the proposed WACPN model is superior to six state-of-the-art models. The causes are triple. First, the 2d-WE layer stands as a proficient way to designate CCT images. Second, ACP is efficient in training FNN. Third, we fine-tune and select the best parameters for the RA. In the future, our model may be applied to other fields [21,22].