## 3 Basic Principles of the Proposed Method
### Discrete Nonseparable Shearlet Transform  
The wavelet analysis method has been applied in many fields due to its advantages of multi-scale decomposition and fast computation speed. However, the drawback of the wavelet is that the direction decomposition is insufficient. It can only decompose the horizontal, vertical, and diagonal directions. In order to compensate for the deficiency of directional decomposition, the multi-scale geometric analysis (MGA) method was proposed. Typical multi-scale geometric analysis methods include Ridgelet transform [20], Curvelet transform [21], Contourlet transform [22], Shearlet transform [23], and Bandelet transform [24]. The most widely used ones are Contourlet transform and Shearlet transform, while the other methods are limited by their slow computational speed. The advantage of Contourlet is fast computational speed, but its direction representation is limited. The computational speed of Shearlet transform is slower than that of Contourlet, but the direction representation is more flexible. The continuous casting slabs run very slowly on the track, with a speed less than 1 m/s, so the computational speed of the defect recognition algorithm is not required to be fast. The discrete nonseparable shearlet transform (DNSTS) [13; 14] is a new kind of shearlet transform. [PERSON] [13] proposed discrete nonseparable shearlet transform (DNST) based on the discrete frame. DNST is constructed from a 2D nonseparable fan filter (improved directional selectivity) and a separable compactly supported shearlet generator (excellent localization properties). It is a direction representation system that extends the wavelet frame. DNST exhibits the same advantages wavelet, namely a unified treatment of the continuum and digital situation. Two-dimensional discrete shearlet transform is usually obtained using a cone adapted discrete shearlet system defined by scaling functions \(\phi_{m}\) and shearlet functions \(\psi_{j,k,m}\), \(\vec{\psi}_{j,k,m}\) (by swapping the order of two variables of \(\psi_{j,k,m}\)). \[\left\{\phi_{m}:m\in\mathbb{Z}^{2}\right\}\cup\left\{\psi_{j,k,m},\vec{\psi}_{ j,k,m}:j\in Z,j\geq 0,|k|\leq 2^{\frac{j}{2}};m\in\mathbb{Z}^{2}\right\} \tag{1}\]  
where \(j\) is scaling parameter, \(m\) is translation parameter, and \(k\) is shear (direction) parameter. Non-separable generator \(\psi^{nom}\) is defined as follows:  
\[\vec{\psi}^{nom}(\xi){=}\ P(\xi_{1}/2,\xi_{2})\vec{\psi}(\xi) \tag{2}\]  
where the trigonometric polynomial \(P\) is a 2D fan filter, which can improve directional selectivity in the frequency domain at each scale. \(\psi\) is the 2D separable shearlet generator. which can provide excellent localization properties. \(\vec{\psi}\) is the Fourier transform of \(\psi\). The nonseparable shearlets \(\psi^{nom}_{j,k,m}(x)\) generated by \(\psi^{nom}\) by setting  
\[\psi^{nom}_{j,k,m}(x)=2^{\frac{3}{2}j}\psi^{nom}\Big{(}S_{k}A_{2j}x-M_{c_{j}}m \Big{)} \tag{3}\]  
where \(A_{2^{j}}\) is parabolic scaling matrix, \(S_{k}\) is shear matrix, and \(M_{c_{j}}\) is a sampling matrix given by \(M_{c_{j}}=\text{diag}(c_{1}^{j},c_{2}^{j})\). We only discuss the case of the shearlet coefficients associated with \(A_{2j}\) and \(S_{k}\); the same procedure can be applied to compute the shearlet coefficients associated with \(\widetilde{A}_{2j}\) and \(\widetilde{S}_{k}\). After faithfully digitizing \(\psi^{non}_{j,k,m}\), the digital formulation of the discrete nonseparable shearlet transform (DNST) is given by  
\[DNST_{j,k,m}\big{(}f_{j}\big{)} =\Big{(}f_{j}\ast\overline{\psi}^{d}_{j,k}\Big{)}\Big{(}2^{I}A_{ 2j}^{-1}M_{c_{j}}m\Big{)}\text{ for }j=0,\ \ldots,\ J-1 \tag{4}\]  
where \(f_{j}\) is the scaling coefficients. \(\psi^{d}_{j,k}\) is digital shearlet filters, \(\psi^{d}_{j,k}=S^{d}_{k2^{-j/2}}\big{(}p_{j}\ast W_{j}\big{)}\), \(S^{d}_{k2^{-j/2}}\) is the discrete shear operator, \(p_{j}\) is the Fourier coefficients of \(P\), and \(W_{j}=gl_{j-j}\otimes h_{j-j/2}\), \(gl_{j-j}\) and \(h_{j-j/2}\) are 1D filters. Please refer to reference [13] for details.