3.1.2. CNN Module Architecture

To achieve the most suitable sparse representation of **h**(*k*) and enhance the performance of the SMD-Net, we exploited a CNN module to automatically learn the sparse transform instead of the traditional hand-crafted presets one. The CNN module is shown in Figure 3, and it contains the sparse transform ℵ(·), soft(·) function, and inverse transform ℵ−1(·). The front end of ℵ(·) is a local feature extraction module. In order to combine the global phase information, a GC block [44] is connected to the back end of ℵ(·) to extract the global feature of the phase. ℵ(·) can be represented by the following expression.

$$\mathcal{N}\left(\mathbf{h}^{(k)}\right) = \mathbf{G}\mathbf{C}\left(\delta \times \Gamma\_1 \mathbf{h}^{(k)} + (1-\delta) \times \Gamma\_2 \text{ReLU}\left(\Gamma\_1 \mathbf{h}^{(k)}\right)\right) \tag{19}$$

where Γ1 represents a convolution operator with *Mf* filters of the size *Ms* × *Ms*; Γ2 is another convolution operator corresponding to *Mf* filters of the size *Ms* × *Ms* × *Mf*; *δ* denotes the weight; ReLU (*x*) = max (0, *x*); and *GC*(·) represents a GC block.

Moreover, the inverse transform function ℵ−1(·) is the mirror-symmetrical architecture of ℵ(·). The constraint ℵ−<sup>1</sup>(·) × <sup>ℵ</sup>(·) = I is required to obtain the filtered phase in the spatial domain. The two convolution operations in ℵ−1(·) are the same as in ℵ(·), but the order is switched.

In the *k*th block of the SMD-Net, Equation (15) can be written as

$$\mathbf{p}\_c^{(k)} = \underset{\mathbf{p}\_c}{\operatorname{argmin}} \frac{1}{2} \left\| \mathbf{p}\_c - \mathbf{h}^{(k)} \right\|\_2^2 + \lambda \left\| \aleph(\mathbf{p}\_c) \right\|\_1 \tag{20}$$

Finally, according to ℵ(·) and ℵ−1(·), Equation (15) can be expressed in the following form.

$$\mathbf{p}\_c^{(k)} = \aleph^{-1}\left(\text{soft}\left(\aleph\left(\mathbf{h}^{(k)}\right), \lambda\right)\right) \tag{21}$$

The CNN module is exploited to automatically learn the appropriate sparse basis and parameters, which not only bridges the regre<sup>t</sup> of the hand-crafted setting in the conventional ISTA, but also enhances the performance of the SMD-Net for InSAR phase filtering.
