Algorithm 1.

TGV regularized MR-PET reconstruction

1:	function TGV-MR-PET(u0, v0, c0, λ, μ)
2:	M ← rescale_operator(M, ζM)
3:	P ← rescale_operator(P, ζP)
4:	u0 ← rescale_data(M*u0) · u0
5:	(v0, c) ← rescale_data(P*(v0 – c0)) · (v0, c0)
6:	u ← M*u0, v ← P*(v0 – c0), (ū, v̄) ← (u, v)
7:	w → 0, w̄ ← 0, p ← 0, q ← 0, r ← 0, s ← 0
8:	choose η > 0, σ > 0, τ = σ/η2
9:	repeat
10:	p ← projα1 (p + σ(∇(ū, v̄) – w̄))
11:	q ← projα0 (q + $\sigma \mathcal{E}\overline{w}$)
12:	r ← prox(${\left(D_{\text{MR}}^{\lambda }\right)}^{\ast }$),σ(r + σMū – σu0)
13:	s ← prox(${\left(D_{\text{PT}}^{\mu }\right)}^{\ast }$),σ(s + σPv̄ + σc0)
14:	(u+, v+) ← (u, v) – τ(−div p + (M*r, P*s))
15:	(u+, v+) ← projc((u+, v+))
16:	w+ ← w – τ(−p − div2 q)
17:	(ū, v̄, w̄) ← 2(u+, v+, w+) – (u, v, w)
18:	${\sigma }_{+}\leftarrow \eta \mathcal{S}\left(\sigma \tau ,\frac{\mid \mid (u_{+},v_{+},w_{+})-(u,v,w)\mid \mid }{\mid \mid K\left(\right(u_{+},v_{+},w_{+})-(u,v,w\left)\right)\mid \mid }\right)$
19:	τ+ ← σ+/η2
20:	(u, v, w) ← (u+, v+, w+)
21:	until Stopping criterion fulfiled
22:	return (u+, v+)
23:	end function