Algorithm 1.

Fluorescence imaging reconstruction

Require: $I_{f,\mathcal{l}}\left(\mathbf{r}\right)$, ${\mathbf{r}}_{\mathcal{l}}$, $\mathcal{l}=1,\dots ,N_{\text{img}}$
1.	initialize $o_f^{\left(1,0\right)}\left(\mathbf{r}\right)={\displaystyle \sum }_{\mathcal{l}}{I}_{f,\mathcal{l}}\left(\mathbf{r}\right)/N_{\text{img}}$
2.	initialize $p_f^{\left(1,0\right)}\left(\mathbf{r}\right)$ with all one values
3.	initialize ${\tilde{h}}_f\left(\mathbf{u}\right)$ with the non-aberrated incoherent OTF
4.	initialize ${\mathbf{r}}_{\mathcal{l}}^{\left(1\right)}$ with the scanning position from the registration step
5.	for $k=1:K_f$ do
6.	Sequential gradient descent
7.	for $\mathcal{l}=1:N_{\text{img}}$ do
8.	$o_f^{(k,\mathcal{l})}(\mathbf{r})=o_f^{(k,\mathcal{l}-1)}(\mathbf{r})-{\nabla }_{o_f}{f}_{f,\mathcal{l}}(o_f^{(k,\mathcal{l}-1)},p_f^{(k,\mathcal{l}-1)},{\mathbf{r}}_{\mathcal{l}}^{(k)})/\max {(p_f^{(k,\mathcal{l}-1)}(\mathbf{r}))}^2$
9.	$p_f^{(k,\mathcal{l})}(\mathbf{r})=p_f^{(k,\mathcal{l}-1)}(\mathbf{r})-{\nabla }_{p_f}{f}_{f,\mathcal{l}}(o_f^{(k,\mathcal{l}-1)},p_f^{(k,\mathcal{l}-1)},{\mathbf{r}}_{\mathcal{l}}^{(k)})/\max {(o_f^{(k,\mathcal{l}-1)}(\mathbf{r}))}^2$
10.	$\xi (\mathbf{u})=\mathcal{F}\{o_f^{(k,\mathcal{l}-1)}(\mathbf{r})\cdot \mathcal{C}\{p_f^{(k,\mathcal{l}-1)}(\mathbf{r}-{\mathbf{r}}_{\mathcal{l}})\}\}$
11.	${\tilde{h}}_f^{(k,\mathcal{l})}(\mathbf{u})={\tilde{h}}_f^{(k,\mathcal{l}-1)}(\mathbf{u})-{\nabla }_{{\tilde{h}}_f}{f}_{f,\mathcal{l}}(o_f^{(k,\mathcal{l}-1)},p_f^{(k,\mathcal{l}-1)},h_f^{(k,\mathcal{l}-1)},{\mathbf{r}}_{\mathcal{l}}^{(k)})\cdot \left|\xi (\mathbf{u})\right|/12[\max (|\xi (\mathbf{u})|)\cdot ({|\xi (\mathbf{u})|}^2+\delta )]$, where δ is chosen to be small
12.
13.	Scanning position refinement
14.	$x_{\mathcal{l}}^{(k+1)}$ = $x_{\mathcal{l}}^{(k)}-\alpha {\nabla }_{x_{\mathcal{l}}}{f}_{f,\mathcal{l}}(o_f^{(k,\mathcal{l}-1)},p_f^{(k,\mathcal{l}-1)},{\mathbf{r}}_{\mathcal{l}}^{(k)})$
15.	$y_{\mathcal{l}}^{(k+1)}$ = $y_{\mathcal{l}}^{(k)}-\alpha {\nabla }_{y_{\mathcal{l}}}{f}_{f,\mathcal{l}}(o_f^{(k,\mathcal{l}-1)},p_f^{(k,\mathcal{l}-1)},{\mathbf{r}}_{\mathcal{l}}^{(k)})$
16.	end for
17.	Nesterov’s acceleration
18.	if k = 1 then
19.	$t_1=1$
20.	$o_f^{(k+1,0)}(\mathbf{r})=o_f^{(k,N_{\text{img}})}(\mathbf{r})$
21.	$p_f^{(k+1,0)}(\mathbf{r})=p_f^{(k,N_{\text{img}})}(\mathbf{r})$
22.	else
23.	$t_{k+1}=\frac{1+\sqrt{1+4t_k^2}}{2}$
24.	$o_f^{(k+1,0)}(\mathbf{r})=o_f^{(k,N_{\text{img}})}(\mathbf{r})+\frac{t_k-1}{t_{k+1}}\left[o_f^{(k,N_{\text{img}})}(\mathbf{r})-o_f^{(k-1,N_{\text{img}})}(\mathbf{r})\right]$
25.	$p_f^{(k+1,0)}(\mathbf{r})=p_f^{(k,N_{\text{img}})}(\mathbf{r})+\frac{t_k-1}{t_{k+1}}\left[p_f^{(k,N_{\text{img}})}(\mathbf{r})-p_f^{(k-1,N_{\text{img}})}(\mathbf{r})\right]$
26.	end if
27.	end for