Algorithm 2.

Jumpstarted Spectral DPS (JSDPS)

1: $T,T^{\prime }$: training steps, sampling steps, $T^{\prime }\ll T$

2: ${\eta }_t$: step size

3: $y$: spectral CT measurement

4: ${\widehat{x}}_0^f$: image-domain decomposition

5: $\mathit{ϵ}\sim 𝒩(0,I)$

6: $x_{T^{\prime }}={\widehat{x}}_0^{T^{\prime }}=\sqrt{{\overline{\alpha }}_{T^{\prime }}}{x}_0+\sqrt{1-{\overline{\alpha }}_{T^{\prime }}}\mathit{ϵ}$

7: for $t=T^{\prime }$ to 1 do:

8:    $z\sim 𝒩(0,\mathit{I})$

9:    ${\widehat{x}}_0=\frac{1}{\sqrt{{\overline{\alpha }}_t}}(x_t-\sqrt{1-{\overline{\alpha }}_t}{\mathit{ϵ}}_{\theta }(x_t,t))$

10:    $x_{t-1}^{\prime }=\frac{\sqrt{{\alpha }_t}(1-{\overline{\alpha }}_{t-1})}{1-{\overline{\alpha }}_t}{x}_t+\frac{\sqrt{{\overline{\alpha }}_{t-1}}{\beta }_t}{1-{\overline{\alpha }}_t}{\widehat{x}}_0+{\sigma }_{t}z$

11:    $x_{t-1}=x_{t-1}^{\prime }-{\eta }_t{\nabla }_{{\widehat{x}}_0}\Vert BS\exp (-QA{\widehat{x}}_0)-y{\Vert }_{K^{-1}}^2$

12: end for