*2.3. High-Resolution Parabolic Radon Transform with Lq*<sup>1</sup> − *Lq*<sup>2</sup> *Mixed Regularization*

To suppress multiple *<sup>m</sup>*<sup>2</sup> from *<sup>d</sup>*, we use *Lq*<sup>1</sup> − *Lq*<sup>2</sup> mixed regularization with 0 < *<sup>q</sup>*1, *<sup>q</sup>*<sup>2</sup> < 1 for sparsity promotion and use the following formulation:

$$\min\_{m\_1 m\_2} \left\{ \frac{1}{\beta} \|A\_1 m\_1 + A\_2 m\_2 - d\|\_2^2 + \mu \|m\_1\|\_{q\_1}^{q\_1} + \|m\_2\|\_{q\_2}^{q\_2} \right\} \tag{12}$$

where *μ* is a positive parameter, *β* > 0 is a penalty parameter, *A* = *A*<sup>1</sup> = *A*<sup>2</sup> is the Radon operator, *m*<sup>1</sup> is primary and *m*<sup>2</sup> is multiple. To determine *q*<sup>1</sup> and *q*2, we explore the *Lq*(0 < *q* < 1) regularization. Xu et al. [25] introduced *Lq*(0 < *q* < 1) regularization and proved that the *Lq* regularization can obtain much sparser solutions than *L*<sup>1</sup> regularization. Meanwhile, they proposed *L*1/2 regularization, and it is the most representative regularization method for *Lq*(0 < *q* < 1) regularization. The study shows that the *L*1/2 regularization is the sparsest and the most robust among the *Lq*(1/2 ≤ *q* < 1) regularization, and when 0 < *q* < 1/2, the *Lq* regularization has similar properties to the *L*1/2 regularization. Therefore, we select *q*<sup>1</sup> = *q*<sup>2</sup> = 1/2 in this paper.

*Lq*<sup>1</sup> − *Lq*<sup>2</sup> mixed regularization is the nonconvex problem, and it is a very complicated process to solve. In this paper, we use an improved alternative direction method of multipliers algorithm (ADMM) to approximate the solution [34]. The ADMM algorithm can be used to solve many high-dimensional problems and uses a decomposition-coordination procedure to decouple the variables [34–36,40]. This algorithm enables the difficult global problem to be properly solved. The Problem (12) can be formulated as

$$\min\_{m\_1 m\_2} \left\{ \|A\_1 m\_1 + A\_2 m\_2 - d\|\_2^2 + \beta \mu \|\|\mathbf{z}\_1\|\|\_{q\_1}^{q\_1} + \beta \|\|\mathbf{z}\_2\|\|\_{q\_2}^{q\_2} \right\} \tag{13}$$

where the auxiliary variables are *<sup>z</sup>*<sup>1</sup> = *<sup>m</sup>*1, *<sup>z</sup>*<sup>2</sup> = *<sup>m</sup>*2.

The augmented Lagrangian function is

*L*(*m*1, *m*2, *z*1, *z*2, *w*1, *w*2)

$$\begin{aligned} &=||A\_1\mathfrak{m}\_1 + A\_2\mathfrak{m}\_2 - d||\_2^2 + \beta\mu \|z\_1\|\_{q\_1}^{q\_1} + \beta\|z\_2\|\_{q\_2}^{q\_2} + \langle \mathfrak{w}\_1, \mathfrak{m}\_1 - z\_1 \rangle + \langle \mathfrak{w}\_2, \mathfrak{m}\_2 - z\_2 \rangle \\ &+ \frac{\rho\_1}{2} ||\mathfrak{m}\_1 - z\_1||\_2^2 + \frac{\rho\_2}{2} ||\mathfrak{m}\_2 - z\_2||\_2^2 \end{aligned}$$

2 where *ρ*<sup>1</sup> and *ρ*<sup>2</sup> are positive penalty parameters and *w*<sup>1</sup> and *w*<sup>2</sup> are the dual variables. The dual variables and *z*1, *z*2, *m*1, *m*<sup>2</sup> are alternatively updated as follows:

$$z\_1^{k+1} = \arg\min\_{z\_1} (\beta \mu ||z\_1||\_{q\_1}^{q\_1} + \frac{\rho\_1}{2} \left\| m\_1^k - z\_1 + \frac{w\_1^k}{\rho\_1} \right\|\_2^2) \tag{14}$$

$$\mathbf{z}\_{2}^{k+1} = \arg\min\_{\mathbf{z}\_{2}} (\beta \|\mathbf{z}\_{2}\|\_{q\_{2}}^{q\_{2}} + \frac{\rho\_{2}}{2} \left\|\mathbf{w}\_{2}^{k} - \mathbf{z}\_{2} + \frac{\mathbf{w}\_{2}^{k}}{\rho\_{2}}\right\|\_{2}^{2}) \tag{15}$$

$$m\_1^{k+1} = \arg\min\_{\mathbf{w}\_1} \left( \left\| A\_1 \mathbf{w}\_1 + A\_2 \mathbf{w}\_2^k - d \right\|\_2^2 + \frac{\rho\_1}{2} \left\| \mathbf{w}\_1 - \mathbf{z}\_1^{k+1} + \frac{\mathbf{w}\_1^k}{\rho\_1} \right\|\_2^2 \right) \tag{16}$$

$$\mathbf{m}\_{2}^{k+1} = \arg\min\_{\mathbf{m}\_{2}} (\left\| \mathbf{A}\_{1}\mathbf{m}\_{1}^{k+1} + \mathbf{A}\_{2}\mathbf{m}\_{2} - d \right\|\_{2}^{2} + \frac{\rho\_{1}}{2} \left\| \mathbf{m}\_{2} - \mathbf{z}\_{2}^{k+1} + \frac{\mathbf{w}\_{2}^{k}}{\rho\_{2}} \right\|\_{2}^{2}) \tag{17}$$

$$\mathfrak{w}\_1^{k+1} = \mathfrak{w}\_1^k + \rho\_1(\mathfrak{w}\_1^{k+1} - z\_1^{k+1}) \tag{18}$$

$$\mathfrak{w}\_2^{k+1} = \mathfrak{w}\_2^k + \rho\_2(\mathfrak{w}\_2^{k+1} - \mathfrak{z}\_2^{k+1}) \tag{19}$$

Due to the proximity operator of *Lq* regularization, *proxq*,*η*(*t*) is defined as

$$\operatorname{prox}\_{q,\eta}(\mathbf{t}) = \operatorname\*{arg\,min}\_{\mathbf{m}} \left\{ \|\mathbf{m}\|\_{q}^{q} + \frac{\eta}{2} \|\mathbf{m} - \mathbf{t}\|\_{2}^{2} \right\} \tag{20}$$

where *η* > 0 is a penalty parameter.

When 0 < *q* < 1, it can be updated as [41]

$$prox\_{q,\eta}(t)\_i = \begin{cases} 0, & |t\_i| < \tau \\ \{0, sign(t\_i)\beta\}, & |t\_i| = \tau \\ sign(t\_i)z\_i, & |t\_i| > \tau \end{cases} \tag{21}$$

for *i* = 1, ··· , *n*, where *β* = [2(1 − *q*)/*η*] 1/(2−*q*) and *<sup>τ</sup>* <sup>=</sup> *<sup>β</sup>* <sup>+</sup> *<sup>q</sup>βq*<sup>−</sup>1/*η*,*zi* is the result of *h*(*z*) = *qzq*−<sup>1</sup> + *ηz* − *η*|*ti*| = 0. The exact solutions of *m*<sup>1</sup> and *m*<sup>2</sup> denote

$$\boldsymbol{\sigma} \boldsymbol{\mathfrak{w}}\_{1}^{k+1} = \left(2\boldsymbol{A}\_{1}^{T}\boldsymbol{A}\_{1} + \rho\_{1}\boldsymbol{I}\right)^{-1} \left[2\boldsymbol{A}\_{1}^{T}\left(\boldsymbol{d} - \boldsymbol{A}\_{2}\boldsymbol{\mathfrak{w}}\_{2}^{k}\right) + \rho\_{1}\boldsymbol{z}\_{1}^{k+1} - \boldsymbol{w}\_{1}^{k}\right] \tag{22}$$

$$\boldsymbol{m}\_{2}^{k+1} = \left(2\boldsymbol{A}\_{2}^{T}\boldsymbol{A}\_{2} + \rho\_{2}\boldsymbol{I}\right)^{-1} \left[2\boldsymbol{A}\_{2}^{T}\left(\boldsymbol{d} - \boldsymbol{A}\_{1}\boldsymbol{m}\_{1}^{k+1}\right) + \rho\_{2}\boldsymbol{z}\_{2}^{k+1} - \boldsymbol{w}\_{2}^{k}\right] \tag{23}$$

The standard ADMM algorithm often fails to converge and converges under some conditions. Therefore, the sufficient condition for the ADMM convergence is [34]

$$
\rho\_1 > \frac{16\lambda\_1^2}{\rho\_1} + \frac{16\lambda\_1\lambda\_2}{\rho\_2} - 2\rho\_1 \tag{24}
$$

$$
\rho\_2 > \frac{16\lambda\_2^2}{\rho\_2} + \frac{16\lambda\_1\lambda\_2}{\rho\_1} - 2\rho\_2\tag{25}
$$

where *λ<sup>i</sup>* = *λmax*(*A<sup>T</sup> <sup>i</sup> Ai*) and *<sup>ϕ</sup><sup>i</sup>* = *<sup>λ</sup>min*(*A<sup>T</sup> <sup>i</sup> Ai*), i = 1,2. The convergence condition of (24) and (25) causes the sequence (*z<sup>k</sup>* <sup>1</sup>, *<sup>z</sup><sup>k</sup>* <sup>2</sup>, *<sup>m</sup><sup>k</sup>* <sup>1</sup>, *<sup>m</sup><sup>k</sup>* <sup>2</sup>, *<sup>w</sup><sup>k</sup>* <sup>1</sup>, *<sup>w</sup><sup>k</sup>* 2) generated by (14)–(19) to converge to a critical point of the Problem (13).
