**2. Theory and Method**

### *2.1. Shearlet Transform*

Shearlet transform is an asymptotically optimal sparse representation of multidimensional functions constructed by a special form of affine systems with synthetic expansion, which is composed of the Laplace pyramid transform and a set of directional filters.

When the dimension is two-dimensional, the scale matrix and the shear matrix are:

$$A = \begin{bmatrix} j & 0 \\ 0 & \sqrt{j} \end{bmatrix}, B = \begin{bmatrix} 1 & -I \\ 0 & 1 \end{bmatrix} \tag{1}$$

In Equation (1), *A* is the scale matrix, which can be divided into scales; *j* > 0, *j* is the scale parameter; *B* is the shear matrix, which can be dissected into directions; and *l* is the shear parameter.

The affine system with synthetic expansion is defined as:

$$\begin{aligned} \psi\_{AB}(\boldsymbol{\varrho}) &= \left\{ \boldsymbol{\varrho}\_{j,l,k}(\boldsymbol{x}) = \left| \det A \right|\_{j/2} \right. \\ \left. \left. \boldsymbol{\varrho} \left( \boldsymbol{B}^{l} \boldsymbol{A}^{j} \boldsymbol{x} - \boldsymbol{k} \right) : j\_{\prime} \boldsymbol{l} \in \mathbb{R} \; \boldsymbol{k} \in \mathbb{R}^{2} \right\} \end{aligned} \tag{2}$$

In Equation (2), *ϕ* ∈ *L*<sup>2</sup> - *R*2 , *ϕ* is the shear wave function, *L* represents the productable space, *x* is the data to be processed, *det* (·) is the determinant, and *k* is the translation parameter.

If *ψAB*(*ϕ*) satisfies the Parseval framework (the tight framework), that is, for any function, it satisfies the following equation:

$$\mathcal{C}\_{j,l,k} = \mathcal{S}\{f\} = \left\langle f, \varphi\_{j,l,k} \right\rangle \tag{3}$$

In Equation (3), *Cj,l,k* is the Shearlet domain coefficient, *S* is the Shearlet forward transform, and · is the inner product. *ψAB*(*ϕ*) is called the composite wavelet. Parameters *j*, *l*, *k* can control the division of the direction and the scale. By taking the inner product of the function and the basis function *f* and the basis function *ϕj*,*l*,*<sup>k</sup>* of different scales, directions, and positions, the Shearlet domain coefficients in the corresponding scale, direction, and position can be obtained. This process essentially uses the basis function after translation, expansion, and rotation to approximate the signal.

The frequency domain subdivision of the shear wave and its support interval are shown in Figure 1. As can be seen from Figure 1, the support interval of any shear wave element *ϕj*,*l*,*<sup>k</sup>* in the frequency domain is a trapezoidal region symmetric about the origin; the direction is distributed along a straight line with the slope of *l*2−*<sup>j</sup>* . The size of the trapezoidal region is about 22*<sup>j</sup>* × 2*<sup>j</sup>* .

**Figure 1.** Frequency domain subdivision and its support interval.
