Sorry for the delay. We would like to thank the reviewer for the fruitful discussion and the valuable comments that have helped improve our paper. Please find our detailed response below.
### Q10. Sparse-View CT in Figure 1
Thank you for pointing this out. We agree that the term "degraded" may not be the most accurate description. For the sparse-view CT, we adopted the same setting as [3] and used their official implementation available on GitHub (specifically, the "run\_CT\_recon.py" file) to generate the measurements, which are not in the image space. The sparsity level is set to 6. For an input tensor of size [1, 1, 256, 256], the sinogram has a size of [1, 1, 363, 30]. In Figure 1, we show the baseline reconstructions derived from these measurements (similar to Figure 4 in [3]).
### Q11. Section 3.2.3, Prior Work, Theoretical Analysis, Eq (19), Eq (21), Eq (22) [Part 1]
Here, we provide a more theoretical analysis of the approach discussed in Section 3.2.3 and comment on Eq (19), Eq (21), and Eq (22).
First, we want to note that the derivation in Section 3.2.3 is directly connected to our problem formulation as we aim to solve Eq (8):
$$
\hat{x} = \arg\min_{z: \|z\|^2 = N_x M_x} \| y - H(g(z)) \|^2
$$
One challenge in solving Eq (8) is that an iterative approach is computationally expensive, as a single evaluation of $g(z)$ may take minutes.
To address this, we introduce a family of generative processes $\text{DDIMReverse}(., \delta t) = g^{\delta t}$, parameterized by $\delta t \geq 1$. These processes are defined *not on all latent variables* $x_{1:T}$, but on a subset \{$x_0, x_{\delta t}, x_{2 \delta t}, \ldots, x_{T-2 \delta t}, x_{T-\delta t}, x_T$ \} of length $K$. The scalar $\delta t$ defines the jump in the generative process sampling.
We can show that, by carefully factorizing both the diffusion forward process and the corresponding generative process and choosing the appropriate marginals, for all $\delta t$, $g^{\delta t}$ constitutes a valid generative model from $p(x)$.
Here is a brief derivation:
We consider the following factorization of the diffusion forward process:
$$
q_{\delta t}(x_{1:T} | x_0) = q_{\delta t}(x_T| x_0) \prod_{i=1}^{K} q_{\delta t}(x_{(i -1) \delta t} | x_{i \delta t}, x_0) \prod_{t \notin \\{ 0, {\delta t}, {2 \delta t}, \ldots, {T-2 \delta t}, {T-\delta t}, T \\} } q_{\delta t}(x_t | x_0)
$$
where
$
q_{\delta t}(x_t | x_0) = \mathcal{N} \left( \sqrt{\bar{\alpha_t}} x_0, (1 - \bar{\alpha_t}) I \right)
$
and
$
q_{\delta t}(x_{(i -1) \delta t} | x_{i \delta t}, x_0) = \mathcal{N} \left( \sqrt{\bar{\alpha_{(i -1) \delta t}}} x_0 + \sqrt{1 - \bar{\alpha_{(i -1) \delta t}} - \sigma_{i \delta t}^2} \frac{x_{i \delta t} - \sqrt{\bar{\alpha_{i \delta t} }} x_0}{\sqrt{1 - \bar{\alpha_{i \delta t} }}}, \sigma_{i \delta t} ^2 I \right)
$
Specifically, $q_{\delta t}(x_{(i -1) \delta t} | x_{i \delta t}, x_0) $ is defined such that $q_{\delta t}(x_{i \delta t} | x_0) = \mathcal{N} \left( \sqrt{\bar{\alpha_{i \delta t}}} x_0, (1 - \bar{\alpha_{i \delta t}}) I \right)$ for all $i \in [1, K]$. (Here, all the marginals $q_{\delta t}(x_t | x_0)$ for $t \in [1, T]$ match those in the original diffusion formulation.)
We then define the corresponding generative process $p_{\theta, \delta t}$:
$$
p_{\theta, \delta t}(x_{1:T} | x_0) = p_{\theta, \delta t}(x_T) \prod_{i=1}^{K} p_{\theta, \delta t}(x_{(i -1) \delta t} | x_{i \delta t}) \prod_{t \notin \\{ 0, {\delta t}, {2 \delta t}, \ldots, {T-2 \delta t}, {T-\delta t}, T \\}} p_{\theta, \delta t}(x_0 | x_t)
$$
(Only the first part of the factorization is used to produce samples.)
We define
$
p_{\theta, \delta t}(x_0 | x_t) = \mathcal{N} \left( f_{\theta}^{(t)}(x_t), \sigma_t^2 I \right)
$
and
$
p_{\theta, \delta t}(x_{(i -1) \delta t} | x_{i \delta t}) = q_{\delta t}(x_{(i -1) \delta t} | x_{i \delta t}, f_{\theta}^{(i \delta t)}(x_{(i -1) \delta t}))
$
where
$
f_{\theta}^{(t)}(x_t) = \frac{x_t - \sqrt{1 - \bar{\alpha_t}} \cdot \epsilon_\theta^{(t)}(x_t)}{\sqrt{\bar{\alpha_t}}}
$
We optimize $\theta$ using the variational inference objective:
$$
J_{\delta t}(\epsilon_\theta) = \mathbb{E_{x_{0:T} \sim q_{\delta t} (x_{0:T})}} \left[ \log q_{\delta t} (x_{1:T} | x_0) - \log p_{\theta, \delta t} (x_{0:T}) \right]
$$
$$
= \mathbb{E_{x_{0:T} \sim q_{\delta t} (x_{0:T})}} \left[ \sum_{i=1}^{K} D_{KL} \left(q_{\delta t} (x_{(i -1) \delta t} | x_{i \delta t}, x_0) \| \| p_{\theta, \delta t} (x_{(i -1) \delta t} | x_{i \delta t}) \right) + \sum_{t \notin \\{ 0, {\delta t}, {2 \delta t}, \ldots, {T-2 \delta t}, {T-\delta t}, T \\}} D_{KL} \left(q_{\delta t} (x_t | x_0) \|\| p_{\theta, \delta t} (x_0 | x_t) \right) \right]
$$
where each KL divergence is between two Gaussians where only the mean depends on $\theta$.