Algorithm 3 t-TNN based background completion [52].
Input: Original XCA data $\mathcal{D}\in {ℝ}^{n_1\times n_2\times n_3}$, non-vessel mask region Ω (acquired from M in Algorithm 2).
1:	Initialize: ρ0 > 0, η > 1, k = 0, $\mathcal{X}={\mathcal{P}}_{\Omega }(\mathcal{D})$, $\mathcal{Y}=\mathcal{W}=0$.
2:	while ${\Vert \mathcal{X}-\mathcal{Y}\Vert }_F/{\Vert \mathcal{X}\Vert }_F>tol$ and k < K do
3:	$\mathcal{X}$ sub-problem:
	${\mathcal{X}}^{k+1}={\text{argmin}}_{\mathcal{X}:{\mathcal{X}}_{\Omega }={\mathcal{D}}_{\Omega }^k}{\Vert \mathcal{X}-{\mathcal{Y}}^k+\frac{1}{{\rho }^k}{\mathcal{W}}^k\Vert }_F^2$
	solved by:
	${\mathcal{X}}^{k+1}={\mathcal{P}}_{\Omega }\left({\mathcal{D}}^k\right)+{\mathcal{P}}_{{\Omega }^{\perp }}\left({\mathcal{Y}}^k-\frac{1}{{\rho }^k}{\mathcal{W}}^k\right);$
4:	$\mathcal{Y}$ sub-problem:
	${\mathcal{Y}}^{k+1}={\text{argmin}}_{\mathcal{Y}}{\Vert \mathcal{Y}\Vert }_{\overrightarrow{⊛}}+\frac{{\rho }^k}{2}{\Vert {\mathcal{X}}^{k+1}-\mathcal{Y}+\frac{1}{{\rho }^k}{\mathcal{W}}^k\Vert }_F^2$
	solved by:
	$\tau =\frac{1}{{\rho }^k},{\tau }^{\prime }=\tau \lceil \frac{n_2+1}{2}\rceil ,\mathcal{Z}={\mathcal{X}}^{k+1}+\tau {\mathcal{W}}^k;$
	${\overrightarrow{\mathcal{Z}}}_f=fft(\overrightarrow{\mathcal{Z}},[],3);$
	for $j=1,\dots ,\lceil \frac{n_2+1}{2}\rceil$ do
	$\left[{\mathcal{U}}_f^{(j)},{\mathcal{S}}_f^{(j)},{\mathcal{V}}_f^{(j)}\right]=SVD\left({\overrightarrow{\mathcal{Z}}}_f^{(j)}\right);$
	${\mathcal{J}}_f^{(j)}=diag\left\{{\left(1-\frac{{\tau }^{\prime }}{{\mathcal{S}}_f^{(j)}(i,i)}\right)}_{+}\right\};$
	${\mathcal{S}}_{f,{\tau }^{\prime }}^{(j)}={\mathcal{S}}_f^{(j)}{\mathcal{J}}_f^{(j)};$
	${\mathcal{H}}_f^{(j)}={\mathcal{U}}_f^{(j)}{\mathcal{S}}_{f,{\tau }^{\prime }}^{(j)}{\mathcal{V}}_f^{{(j)}^T}$
	end for
	for $j=\lceil \frac{n_2+1}{2}\rceil +1,\dots ,n_2$ do
	${\mathcal{H}}_f^{(j)}=\text{conj}\left({\mathcal{H}}_f^{\left(n_2-j+2\right)}\right);$
	end for
	$\mathcal{H}=ifft\left({\mathcal{H}}_f,[],3\right),{\mathcal{Y}}^{k+1}=\overleftarrow{\mathcal{H}};$
5:	${\mathcal{W}}^{k+1}={\mathcal{W}}^k+{\rho }^k\left({\mathcal{X}}^{k+1}-{\mathcal{Y}}^{k+1}\right);$
6:	ρk+1 = ηρk, k = k + 1
7:	end while
8:	Vessel layer $\mathcal{V}=\mathcal{D}-\mathcal{X};$
Output: Background layer tensor $\mathcal{X}$, vessel layer tensor $\mathcal{V}$.