# Stability and Generalizability in SDE Diffusion Models with Measure-
Preserving Dynamics

Weitong Zhang
Imperial College London
&Chengqi Zang
University ot Tokyo
&Liu Li
Imperial College London
&Sarah Cechnicka
Imperial College London
&Cheng Ouyang
University of Oxford
&Bernhard Kainz
Imperial College London
University of Erlangen-Nuremberg

###### Abstract

Inverse problems describe the process of estimating the causal factors from a
set of measurements or data. Mapping of often incomplete or degraded data to
parameters is ill-posed, thus data-driven iterative solutions are required,
for example when reconstructing clean images from poor signals. Diffusion
models have shown promise as potent generative tools for solving inverse
problems due to their superior reconstruction quality and their compatibility
with iterative solvers. However, most existing approaches are limited to
linear inverse problems represented as Stochastic Differential Equations
(SDEs). This simplification falls short of addressing the challenging nature
of real-world problems, leading to amplified cumulative errors and biases. We
provide an explanation for this gap through the lens of measure-preserving
dynamics of Random Dynamical Systems (RDS) with which we analyse Temporal
Distribution Discrepancy and thus introduce a theoretical framework based on
RDS for SDE diffusion models. We uncover several strategies that inherently
enhance the stability and generalizability of diffusion models for inverse
problems and introduce a novel score-based diffusion framework, the Dynamics-
aware SDE Diffusion Generative Model (D3GM). The Measure-preserving property
can return the degraded measurement to the original state despite complex
degradation with the RDS concept of stability. Our extensive experimental
results corroborate the effectiveness of D3GM across multiple benchmarks
including a prominent application for inverse problems, magnetic resonance
imaging. Code and data will be publicly available.

##  1 Introduction

Diffusion probabilistic models [54, 55, 53] have demonstrated impressive
performance across various image generation tasks, primarily by modeling a
diffusion process and then learning an associated reverse process. Among the
many commonly used approaches [64], diffusion models that incorporate the
concept of score functions [29, 56] can capture the intrinsic random
fluctuations of the forward diffusion process, positioning them as a good
choice for in-depth analysis. Score-based generative models (SGMs) entail
gradually diffusing images towards a noise distribution, and then generating
samples by chaining the score functions at decreasing noise levels with score-
based sampling approaches. One such example of an SGM with a score-based
sampling technique, known as score matching [55], has gained popularity for
density estimation [16]. It employs methods such as Langevin dynamics [21, 44,
31] and SDEs [30, 38, 32, 72] to simulate the underlying probability
distribution of training samples. However, vanilla unconditional SGMs can be
extended to inverse problems by leveraging an implicit prior distribution,
based on the available counterpart measurement, subjected to corruption and/or
noise. To this end transitionary SGMs enable an iterative recovery of the data
from this noisy counterpart, instead of relying on Gaussian white noise as a
starting point [67, 38, 40, 12, 20, 35].

Intuitively, leveraging priors and a generative capacity into transitionary
SGMs offers the possibility of exploring high quality reconstruction and
restoration and gaining better performance. However, current transitionary
SGMs that incorporate priors have largely overlooked the unreliable quality of
the prior and its measurement. Empirically, transitionary SGMs cannot always
be trusted in terms of stability and efficiency, especially in a regime of
non-uniformly distributed noise or corrupted signal quality [9, 67, 38].
Hence, the exploitation of transitional learning within SGMs does not come
without costs as their advantages vanish in limited data quality settings.
Theoretical understanding is notably lacking in this field with the following
fundamental open problem: Can we realize reliable transitionary diffusion
processes in practice for inverse problems with a theoretical guarantee?

While recent works have started to lay down a theoretical foundation for these
models, a detailed understanding is still lacking. Current best practice
advocates for smaller initialisation values (_e.g._ , noise schedule [67, 20],
instead of large values [15, 62] to ensure that the forward dynamics brings
the diffusion sufficiently close to a known prior and simple noise
distribution. However, a proper choice of the values conditioned on the prior
within a theoretical framework should be preferred for a better approximation
of the score-matching objective and higher computational efficiency. To fully
facilitate the power of reversion and generation of transitionary SGMs and to
mitigate the influence of low-quality measurements for solving inverse
problems, this paper provides a measure-preserving dynamics of random
dynamical system (RDS) perspective as a promising way to obtain reliable
reversion and generation. Notably, our ‘measure’ is not only the observations
(_e.g._ , degraded images), but also represents the invariant probability
measure (distribution) of the RDS. This allows to consider the concept of an
RDS _stability_ and to frame challenging degradation learning within a
measure-preserving dynamical system. Thus, we can start from a transitionary
SGM interpretation of diffusion models and connect RDS to the SDE in
transitionary SGMs. The pitfalls (_e.g.,_ Instability) are discussed in Sect.
3 and further implications can be found in the Appendix. Transitionary SGMs
have not been fully explored before, and we provide a theoretical
interpretation of a stationary process as a possible solution.

Our D3GM framework is abstracted from transitionary SGMs. The key to our
framework is a stationary process following measure-preserving dynamics to
ensure the stability and generalizability of the diffusion, as well as
reducing the influence of accumulated error, distribution bias and degradation
discrepancy. Our contributions can be summarised as follows:

1\. Temporal Distribution Discrepancy: We conduct a rigorous theoretical
examination of the instability issue of transitionary SGMs, measured as
Temporal Distribution Discrepancy (_i.e._ , lower bound of modeling error).
This analysis sheds light on critical aspects related to stability and
generalizability111Generalizability refers to the extent to which out-of-
distribution and domain shift impacts the fidelity of the restoration process.
Stability in SDE diffusion models is demonstrated by its resilience to
degradation beyond its domain and its consistent ability to restore high-
quality images., effectively addressing an unexplored fundamental gap in the
understanding of solving challenging inverse problems with SDEs.

2\. D3GM Framework: We propose a solution, D3GM, and an explanation from
measure-preserving dynamics of Random Dynamical Systems (RDS). ‘Measure’
includes both measurements (degraded image) and invariant measures
(distribution) of RDS, which allows complex degradation learning and enhances
restoration and reconstruction accuracy.

3\. Thorough Evaluation: Our contributions are substantiated by extensive
validation. We demonstrate the practical benefits of our D3GM framework across
various benchmarks, including challenging tasks such the reconstruction of
Magnetic Resonance Imaging (MRI) data.

We address the instability of diffusion models for inverse problems under
domain shift and concept drift (unknown and heterogeneous degradation). This
leads to what we believe is a completely novel view on the theoretical
foundation of how the degradation process is modelled. The result is an
approach that is more in line with the original intention of the theory of
diffusion. We chose inverse problems as a relevant application area to
demonstrate our ideas but also included a variety of challenging problem
settings to explore the generalizability of $D^{3}GM$. To the best of our
knowledge, no other method can handle a diverse range of challenging tasks
like real-world dehazing, compressed MRI reconstruction, blind MRI super-
resolution, etc. with a unified underlying theoretical framework. In Tab. 1 we
illustrate the key differences of $D^{3}GM$ compared with SGMs and
transitionary SGMs.

##  2 Preliminaries

SGMs. We will follow the typical construction of the diffusion process
$\boldsymbol{x}(t)$, $t\in[0,T]$ with $\boldsymbol{x}(t)\in\mathbb{R}^{d}$.
Concretely, we want $\boldsymbol{x}(0)\sim p_{0}(\boldsymbol{x})$, where
$p_{0}=p_{\text{data }}$, and $\boldsymbol{x}(T)\sim p_{T}$, where $p_{T}$ is
a tractable distribution that can be sampled. In this work, we consider the
score-based diffusion form of the SDE [56]. Consider the following Itô
diffusion process defined by an SDE:

$\vspace{-1mm}d\boldsymbol{x}={\boldsymbol{f}}(\boldsymbol{x},t)dt+g(t)d\boldsymbol{W},$
(1)

where ${\boldsymbol{f}}:\mathbb{R}^{d}\mapsto\mathbb{R}^{d}$ is the drift
coefficient of $\boldsymbol{x}(t)$, $g:\mathbb{R}\mapsto\mathbb{R}$ is the
diffusion coefficient coupled with the standard d-dimensional Wiener process
$\boldsymbol{w}\in\mathbb{R}^{d}$. By carefully choosing $\bar{f},g$, one can
achieve a spherical Gaussian distribution as $t\rightarrow T$.

For the forward SDE in Eq. 1, there exists a corresponding reverse-time SDE
[3, 56]:

$d\boldsymbol{x}=[{\boldsymbol{f}}(\boldsymbol{x},t)-g(t)^{2}\underbrace{\nabla_{\boldsymbol{x}}\log
p_{t}(\boldsymbol{x})}_{\text{score function }}]dt+g(t)d\boldsymbol{W},$ (2)

where $dt$ is the infinitesimal negative time step, and ${\boldsymbol{w}}$ is
the Brownian motion running backwards. The score function
$\nabla_{\boldsymbol{x}}\log p_{t}(\boldsymbol{x})$ is in general intractable
and thus SDE-based diffusion models approximate it by training a time-
dependent neural network under a score function [58, 29].

Transitionary SGMs. [17, 38, 67, 35, 20, 12] leverage a transitionary
iterative denoising paradigm for the inverse problems. In inverse problems,
such as super-resolution, we have an (nonlinear, partial, and noisy)
observation $\boldsymbol{y}$ of the underlying high-quality signal
$\boldsymbol{x}$. The mapping $\boldsymbol{x}\mapsto\boldsymbol{y}$ is many-
to-one, posing an ill-posed problem. In this case, a strong prior on
$\boldsymbol{x}$ is needed for finding a realistic solution. Formally, the
general form of the forward (measurement) model is:

$\boldsymbol{y}=\mathcal{A}\left(\boldsymbol{x}\right)+\boldsymbol{n},\quad\boldsymbol{y},\boldsymbol{n}\in\mathbb{R}^{n},\boldsymbol{x}\in\mathbb{R}^{d},\vspace{-1mm}$
(3)

where $\mathcal{A}(\cdot)$: $\mathbb{R}^{d}\mapsto\mathbb{R}^{n}$222MRI
signals are defined on $\mathbb{C}^{n}$ and $\mathbb{C}^{d}$. We demonstrate
in Sect. 5 that our approach is applicable to MRI., oftentimes $\ n\ll d$ is
the forward measurement operator and $\boldsymbol{n}$ is the measurement
noise, assuming
$\boldsymbol{n}\sim\mathcal{N}\left(0,\sigma^{2}\boldsymbol{I}\right)$.

While sharing a similar aim of bridging $\boldsymbol{y}$ and $\boldsymbol{x}$
in transitionary SGMs, different mathematical frameworks have been used: [17]
employs Inversion by Direct Iteration; [38, 35, 20] model it as a Mean-
reverting SDE.

Transitionary SGM has become an increasingly important line of SDE research
due to the applicability on images with theoretical guarantees. However, they
often perfrom poorly in real-world scenarios. To provide a theoretical
investigation of this gap, we interpret _Transitionary SGM_ as _Ornstein-
Uhlenbeck (OU)_ process. This perspective allows us to understand the random
fluctuations in image degradation as stochastic processes, providing a
foundation to integrate random dynamical systems (RDS) with the diffusion
process as a natural extension of the SDE framework involving the OU process.

The Measure-preserving property is introduced from the perspective of RDS: The
distribution can still return to the original state despite severe
degradation. Our approach constructs a bridge from measure-preserving
dynamical system to transitionary SGM through measure-preserving dynamics, and
highlights the Temporal Distribution Discrepancy in Sect. 3. Subsequently, we
address this issue of instability: by incorporating a measure-preserving
strategy into the solution of inverse problems, which is detailed in Sect. 4.
This covers counterpart modeling, bridging a transition from uncertain
diffusion modeling to deterministic solutions, yielding significant
improvements in both performance and efficiency as demonstrated in Sect. 5.
More details can be found in Sect. 6 and Appendix.

Table 1: Differences between state-of-the-art SDE diffusion-based approaches.

Model | $p(X_{0})$ | $p(X_{1})$ | Theorem Foundation | Properties | TDD (Prop.2) | Attractor | Operator Dir. | Inverse problem solver
---|---|---|---|---|---|---|---|---
SGM [54] | $p_{A}$ | $\mathcal{N}(0,I)$ | SDE diffusion | No prior | Unsolvable | No subset attracts | one-sided |
IR-SDE [38] | $p_{A}$ | $p_{B}(\cdot|X_{0},\mu)$ | Mean-reverting SDE | Instability | Limited | Gaussian | one-sided | Lin/Non-Lin
I2SB [35] | $\delta_{a}$ | $p_{B}(\cdot|X_{0})$ | Schrodinger Bridge | Strict Prior | Limited | No subset attracts | one-sided | Lin/ Non-Lin
D3GM (ours) | $p_{A}$ | $p_{B}(\cdot|X_{0},\mu,\tau)$ | Measure-Preserving RDS | Stability | Robust | $\mathcal{N}(\mu,\tau^{2}\sigma^{2}I)$ | two-sided | Lin/ Non-Lin/Blind

##  3 Instability Analysis: Transitionary SGMs with Corrupted SDE Diffusion

Ornstein-Uhlenbeck (OU) process. An OU process is a common case in
transitionary SGMs, where $x_{t}$ is defined using an SDE:
$dx_{t}=-\theta_{t}x_{t}dt+\sigma_{t}dW_{t}$. $W_{t}$ is standard Brownian
motion. A drift term $\mu$ can be introduced:

$dx_{t}=\theta_{t}(\mu-x_{t})dt+\sigma_{t}dW_{t},$ (4)

where $\mu$ denotes state mean, reflecting the expected state of the
measurement (_e.g._ , corrupted image [38], noisy speech [60]) over time.
$\theta_{t}$, $\sigma_{t}$ are time-dependent parameters. The drift term
corrects deviations from the constant $\mu$, effectively pulling the process
towards $\mu$ ($t\rightarrow\infty$) with Stability (in Appx. G) as opposed to
pure noise in Eq. 1.

Measure-preserving Dynamics in SDE Diffusion. The solution of the above SDE
can be represented by a continuous-time random dynamical system $\varphi$
defined on a complete separable metric space $(X,d)$. (See precise definition
of RDS in Appx. C). More generally, we can extend the RDS to a two-sided
solution operator with a flow map. The base flow driven by Brownian motion can
be written as
$W\left(t,\vartheta_{s}(\omega)\right)=W(t+s,\omega)-W(s,\omega)$.

###### Proposition 1

After extending the solution of the OU process to RDS, the measure-preserving
RDS $\varphi$ should meet the property
$\varphi(t,s;\omega)x=\varphi\left(t-s,0;\vartheta_{s}\omega\right)x$.
However, OU processes with time-varying coefficients usually do not satisfy
this property. In this situation, the system breaks the forward-reverse
processes, making it difficult to maintain stability.

Intuition 1. A two-sided measure-preserving random dynamical (MP-RDS) system
formulation enables us to use the Poincare recurrence theorem [46], (see
precise statement in Appx.D), intuitively, with a two-sided MP-RDS
$\varphi_{t}$, the Poincare recurrence theorem ensures that the system
$\varphi_{t}$ starts from terminal condition $x_{T}$, run backward in time,
will hit a region $(x_{0}-\epsilon,x_{0}+\epsilon)$ for small $\epsilon$ in
finite time, where $x_{0}$ is the high-quality image.

Example 1. Following Intuition 1. and Prop. 1, suppose that the OU’s
$\theta_{t}$ follows a cosine schedule, such that
$\theta_{t}=\operatorname{cos}(t)$ for $0\leq t\leq T$, then for some $0\leq
s\leq t\leq T$,
$\varphi(t,s;\omega)x\neq\varphi\left(t-s,0;\vartheta_{s}\omega\right)x$
because the change of $\theta_{t}$ w.r.t. time is not uniform. The OU-process
instability exists due to Temporal Distribution Discrepancy (Prop. 2).

At a high level, Proposition 1 can be extended to show that there exists a
compact attracting set at any $-\infty<t<\infty$, and this convention has
allowed us to characterize the attractor
$K(\omega)=\mathcal{N}(\mu,\sigma^{2}/2\theta)$. The closed-form distribution
for $\mathbf{y}$ can be complex and may not be tractable depending on the
particular scenarios of the actual image degradation process
$\boldsymbol{\mu}$. The modification of $\boldsymbol{\sigma}$ and
$\boldsymbol{\theta}$ is used to regularize the perturbation and attempt to
close the distribution. However, these injections might bypass the stationary
process. More details can be found in Appx. D.

Instability-Temporal Distribution Discrepancy. Given the process
${\operatorname{OU}}\left(x_{t},\mu;t,\theta\right)$ with Eq. 4, where
$x_{T}\neq x_{\infty}$ for finite $T$, indicates that the perturbed state
cannot move towards the degraded LQ image and fails in matching the
theoretical distribution. This inherent discrepancy further causes bias in the
estimation of ${\mu}_{t}$, which gradually accumulates into error in the
reverse process.

Temporal Distribution Discrepancy is illustrated by Proposition 2 (proof can
be found in Appx. E):

###### Proposition 2

Given Eq. 3 and Eq. 4, and assume that the score function is bounded by $C$ in
$L^{2}$ norm, then the discrepancy between the reference and the retrieved
data is, with probability $(1-\delta)$ at least:

$\displaystyle\|x_{0}-\operatorname{OU}(x_{0},\mu;T,\theta)\|_{2}^{2}\geq|\left((x_{0}-\mu)^{2}-\sigma_{T}^{2}/2\theta_{T}\right)\mathrm{e}^{-2\bar{\theta}_{T}}+\sigma_{T}^{2}/2\theta_{T}$
(5)
$\displaystyle-\sigma_{max}^{2}\left(C\sigma_{max}^{2}+d+2\sqrt{-d\cdot\log\delta}-2\log\delta\right)|,$

Intuition 2. Intuitively, Proposition 2 provides a theoretical measurement on
how the difference between finite iteration distribution and the asymptotic
distribution of the OU-process in $L^{2}$ could further enlarge the
discrepancy between the retrieved image and the actual HQ image. Discrepancies
are typically explained by complex degradation vs. monotonic modeling.

For a noisy inverse problem, the retrieved data with any finite $T$ depends on
$\sigma_{t},\mu_{t},\lambda,T,\bar{K}$ (Lipschitz constant). This proposition
also correlates to [40], where the lower bound of the distance between the
high quality image and retrieved image in $L^{2}$ norm in our model is further
enlarged by this discrepancy, which correlates to the term
$\left((x_{0}-\mu)^{2}-(\sigma_{T}^{2}/2\theta_{T})\right)e^{-2\bar{\theta}_{T}}+\sigma_{T}^{2}/2\theta_{T}$.

Another way to further minimize this bound is through the term
$e^{-2\bar{\theta}_{T}}$ with $[0,T]$ normalized to $[0,1]$. What we refer to
as $\theta$-schedule corresponds to the exact functional form of $\theta_{t}$,
several schedules can be set here, _e.g._ , constant, linear, cosine, and log.
At a high level, the discrepancy between the reference and the retrieved data
stems from the divergence between the forwarded final state and the low
quality image. Eq. LABEL:Eq6 can be factored into three constituent parts: the
data residual, the stationary disturbance, and the random noise. While
conditional diffusion generation entails a trade-off between variability and
faithfulness [67], the persistent discrepancy within the residual has a
significant impact on the generalizability of solving the transitionary tasks.
This also establishes a connection with SDEdit [40] and CCDF [12]. When
fitting inverse problems involving paired data into diffusion models, while
accounting for deviations and degradation, inserting them into Eq. 4 directly
may not be the most effective strategy.

During sampling and inference with the degraded input $y$, the discrepancy
identified in Prop. 2 intensifies. The complex degradations in $y$ exacerbate
the divergence from the expected $\mu$ distribution, significantly impacting
the accuracy of the restored data $\hat{x}_{0}$. More details are in Appx. F.

##  4 Towards Stability: Measure-preserving Dynamics in SDE Diffusion

Figure 1: Dynamics-aware SDE Diffusion Generative Model (D3GM). When extending
transitionary SDEs to random dynamical systems (RDS), their measure-preserving
property should be kept to maintain stability. This corresponds to driving the
SDE towards the drift term $\mu$ (LQ). There is a Temporal Distribution
Discrepancy which results from the gap between the forward estimation $x_{T}$
and the low quality image in the SDE. With the distribution aligned between
$x_{T}$ and $\mu$, the SDE can be made more robust to inverse problems.
Reconstruction results for low quality (LQ) images after application of our
D3GM method, on different tasks, compared to the ground truth (GT) on two
domains - The frequency domain: MRI Reconstruction (undersampling factor 8x,
16x, frequency masks are colored red); MRI Super-resolution (up-scaling factor
of X4, cross-domain evaluation). The image domain: Real Dense Haze Removal;
Rain Removal (light, heavy).

In Sect. 3, we extrapolate and theorize the Temporal Distribution Discrepancy
on $\mu$ and $x$ in the diffusion model for the inverse problem. Our key idea
is to combine the stationary process to alleviate the Temporal Distribution
Discrepancy problem following the measure-preserving dynamics from RDS. Recall
that our ‘measure’ is not only the measurements (_i.e._ , degraded image), but
also represents the invariant measure (distribution) of the RDS.

We begin by describing the forward and reverse processes of the D3GM, which
serves as a stable bridge between the quality data and the counterpart
measurement. We adapt score-based training methods to estimate this SDE.
Following this, we describe the essential constructions for preserving the
stationary process in the diffusion model and solving for Temporal
Distribution Discrepancy on an orthogonal basis compared to current
transitionary diffusion models.

Measure-preserving Dynamics with the Stationary Process. Following Prop. 1, in
a SDE Diffusion from 0 to $T$, the corresponding ‘attractors’ (states) can be
viewed as
$\mathcal{N}\left(\mu+(x_{0}-\mu)\mathrm{e}^{-\bar{\theta}_{t}},\sigma_{t}^{2}(1-\mathrm{e}^{-2\bar{\theta}_{t}})/2\theta_{t}\right)$.
We can guide SDE Diffusion towards a stable and robust solution based on the
properties of Measure-preserving in RDS. It can be extended to impose that for
every $t$, $\frac{\sigma_{t}^{2}}{2\theta_{t}}=\lambda^{2}$, where $\lambda$
is the variance of the designated stationary measure forward process. This
convention allows us to reduce the regularization on two variables
$\sigma_{t},\theta_{t}$ to just one variable to satisfy the property of the
measure-preserving dynamics in the asymptotic sense, _i.e._ ,
$\lim_{t\rightarrow\infty}\varphi(t,s;\omega)x=\lim_{t\rightarrow\infty}\varphi\left(t-s,0;\vartheta_{s}\omega\right)x$.
This convention allows us to characterize the attractor of the system as
$K(\omega)=\mathcal{N}(\mu,\lambda^{2})$.

The definition and constraint of the attractor are significant; without
imposing this constraint, the measure-preserving property cannot be
maintained, and the system would degrade into a Coefficient Decoupled SDE
(Coe. Dec. SDE), we also analyse this in Fig. 3 and Tab. H in Appx. H.

Example 2. When $\frac{\sigma}{\theta}\rightarrow\infty$, the attractor
becomes excessively large, reducing the significance of $\mu$. SDEs exhibiting
this behavior are defined as Coefficient Decoupled SDEs. In practice, $\mu$
demonstrates non-infinite properties as an input image, while the
corresponding sigma and theta are indeed unconstrained. In such decoupled
forms, the coefficients of the attractor size increases $\frac{\sigma}{\mu}$,
diminishing the significance of $\mu$.

Based on Prop. 2, since the temporal distribution discrepancy always exists as
long as the running time $T$ is finite, and we want the final state $x_{T}$ to
be as close as possible to the distribution of $x_{\infty}$. Therefore, we
introduce $\tau$ such that given $T$, the distribution of $x_{T}$ follows
$\mathcal{N}(\mu(1-e^{\bar{\theta}_{T}})+x_{0}e^{\bar{\theta}_{T}},\tau^{2}\lambda^{2}(1-e^{2\bar{\theta}_{T}}))$,
and $x_{\infty}$ follows $\mathcal{N}(\mu,\tau^{2}\lambda^{2})$, with
$\tau>1$, we increase the possibility of a sample $\tilde{x}_{T}$ from $x_{T}$
to become closer to the distribution of $x_{\infty}$, and thus serves as a
plausible initial state for the reverse process. We can control how much to
close the distributions, either by increasing the stiffness $\tau$ at the cost
of potentially destabilizing the reverse process, or by decreasing $\tau$ to
further smooth the density functions of both distributions at the cost of more
reverse iterations.

By connecting the inverse problem with the analysis above, we clarify the
discrepancy in the stationary modeling process from measure-preserving
dynamics and thereby improve the generalization of diffusion processes and the
accuracy of the reverse process. This is particularly important for
accommodating the diversity of degradation states and to ensure accurate
sampling.

Table 2: $\mu_{t}(x_{t},t)$, $v_{t}(x_{t},t)$ solutions with various $\theta_{t}$, $\sigma_{t}$. |  $\mu_{t}(x_{t},t)$ |  $v_{t}(x_{t},t)$
---|---|---
Lin |  $e^{\theta\frac{x^{2}}{2}}\mathbf{x}_{0}+\left(1-e^{\theta\frac{x^{2}}{2}}\right)\mathbf{y}$ |  $\tau^{2}\lambda^{2}\left(1-e^{\theta x^{2}}\right)\mathbf{y}$
Log |  $e^{\frac{\theta}{k}\log(1+e^{kt})}\mathbf{x}_{0}+\left(1-e^{\frac{\theta}{k}\log(1+e^{kt})}\right)\mathbf{y}$ |  $\tau^{2}\lambda^{2}\left(1-e^{\frac{2\theta}{k}\log(1+e^{kt})}\right)\mathbf{y}$
Cos |  $e^{-\theta\left(t-\frac{\sin(\theta t)}{\theta}\right)}\mathbf{x}_{0}+\left(1-e^{-\theta\left(t-\frac{\sin(\theta t)}{\theta}\right)}\right)\mathbf{y}$ |  $\tau^{2}\lambda^{2}\left(1-\mathrm{e}^{-2\theta\left(t-\frac{\sin(\theta t)}{\theta}\right)}\right)\mathbf{y}$
Quad |  $e^{\theta\frac{x^{3}}{3}}\mathbf{x}_{0}+(1-e^{\theta\frac{x^{3}}{3})}\mathbf{y}$ |  $\tau^{2}\lambda^{2}(1-e^{2\theta\frac{x^{3}}{3})}\mathbf{y}$

Forward Process. We describe the forward process as: $dx_{t}=\theta_{t}(\mu-
x_{t})dt+\tau\sigma_{t}dW_{t},$ parameterized by $\tau$ to calibrate the SDE
modeling, $\mu$ is the state mean. The parameters $\theta_{t}$ and
$\sigma_{t}$, both being time-dependent and strictly positive, correspond to
the rate of mean reversion and the stochastic volatility, respectively. The
selection of $\theta_{t}$ and $\sigma_{t}$ offers flexibility in Tab. 4,
_c.p._ , Sect. 3.

Considering the trade-off between complexity and effectiveness, Cos has been
chosen for both $\theta_{t}$ and $\sigma_{t}$. This aims at capturing complex
temporal dynamics in a computationally tractable manner, thereby optimizing
the balance between the performance and calculation convenience.

In the forward process, the mean $\mu_{t}$ approaches the low-quality image
with $\mathrm{E}\left(x_{t}\right)=\mu$, while the variance tends toward the
stationary variance
$\operatorname{var}\left(x_{t}\right)=\frac{\tau^{2}\sigma^{2}}{2\theta}$.
Essentially, the forward SDE transitions the high-quality image to a low-
quality counterpart infused with Gaussian noise. The discretized SDE for the
forward process is $x_{t_{i}}=x_{t_{i-1}}+\theta_{t_{i-1}}(\mu-
x_{t_{i-1}})\Delta t+\tau\sigma_{t_{i-1}}\Delta W_{i}$. We employ a transition
strategy utilizing a varied stationary variance. Additionally, we execute an
unconditional update, which operates without the need for matching in the
reverse process. These not only allow image corruption but also provides
effective adaptability for improvements.

Reverse Process. The reverse process aims at reconstructing the original image
by gradually denoising a low quality image. It utilizes the score of the
marginal distribution, denoted as $\nabla_{x}\log\hat{p}_{t}(x)$, and is
governed by:

$\mathrm{d}x_{t}=\left[\theta_{t}(\mu-
x_{t})-\tau^{2}\sigma_{t}^{2}\nabla_{x}\log\hat{p}_{t}(x)\right]\mathrm{d}t+\tau\sigma_{t}\mathrm{d}\widehat{W}_{t}.$
(6)

The reverse-time D3GM process of Eq. 6 can be found in Appx. D. This closely
mirrors the forward process and incorporates an additional drift term
proportional to the score of the marginal distribution. The ground truth score
for this process, necessary for training our generative model, is:

$\nabla_{x}\log\hat{p}_{t}(x\mid x_{0})=-\frac{x_{t}-\mu_{t}(x)}{v_{t}},$ (7)

where $\mu_{t}(x)$ represents the random attractor of the process at time $t$,
and $v_{t}$ is the variance. Our training objective is defined as the
minimization of the expected discrepancy between the predicted and true scores
over the data distribution:

$\displaystyle\theta^{*}=\underset{\theta}{\mathrm{arg\,min}}\;\mathbb{E}_{t,(\mathbf{x}_{0},\mathbf{y}),\mathbf{z},\mathbf{x}_{t}}\left[w\left\|S_{\theta}(\mathbf{x}_{t},\mathbf{y},t)-\nabla_{\mathbf{x}_{t}}\log
p_{0t}(\mathbf{x}_{t}\mid\mathbf{x}_{0},\mathbf{y})\right\|_{2}^{2}\right],$
(8)

where $w=-1/\tau^{2}$ is a time-dependent weighting function, and $S_{\theta}$
denotes the score network parameterized by $\theta$ which approximates the
score of the marginal distribution. The optimization is conducted over the
network parameters $\theta$, under the expectation with respect to the time
variable $t$, the initial image $\mathbf{x}_{0}$, noisy image
$\mathbf{x}_{t}$, and data $\mathbf{y}$.

##  5 Experiments

Experimental Settings:  We evaluate D3GM on various challenging restoration
and reconstruction problems. We initially analyze our method by examining its
performance with closely related diffusion formulation variants. Subsequently,
we benchmark D3GM against the state-of-the-art techniques in these domains.
For comprehensive evaluation across all experiments, we report the PSNR [26]
and SSIM [59] for pixel- and structural-level alignment, LPIPS [73] and FID
[24] for measuring perceptual variance. An in-depth description of our
implementation is provided in Appx. G.

###  5.1 Stability: Illustrations of the Measure-preserving Dynamics within
Diffusion Models

Simulated Deraining:  We evaluated D3GM together with the state-of-the-art
deraining strategies: (1) OU SDE method IR-SDE [38], Coefficient Decoupled
(CD) VPB [74] and other CNNs [65, 50, 69, 57]. We use two of the most renowned
synthetic raining datasets: Rain100H [66] and Rain100L [66]. Rain100H contains
1800 pairs of images with and without rain, along with 100 test pairs. As for
Rain100L, it consists of 200 pairs for training and 100 pairs for testing. We
present results based on the PSNR, SSIM, and LPIPS metrics.

Table 3: Quantitative results for Rain100H and Rain100L.(best in bold and
second best underlined)

| Rain100H | Rain100L
---|---|---
Method | PSNR$\uparrow$ | SSIM$\uparrow$ | LPIPS$\downarrow$ | PSNR$\uparrow$ | SSIM$\uparrow$ | LPIPS$\downarrow$
JORDER [65] | 26.25 | 0.835 | 0.197 | 36.61 | 0.974 | 0.028
IRCNN [38] | 29.12 | 0.882 | 0.153 | 33.17 | 0.958 | 0.068
PreNet [50] | 29.46 | 0.899 | 0.128 | 37.48 | 0.979 | 0.020
MPRNet [69] | 30.41 | 0.891 | 0.158 | 36.40 | 0.965 | 0.077
MAXIM [57] | 30.81 | 0.903 | 0.133 | 38.06 | 0.977 | 0.048
VPB (CD) [74] | 30.89 | 0.885 | 0.051 | 38.12 | 0.968 | 0.023
IR-SDE (OU) [38] | 31.65 | 0.904 | 0.047 | 38.30 | 0.981 | 0.014
D3GM | 32.41 | 0.912 | 0.040 | 38.40 | 0.982 | 0.013

Table 4: Quantitative results for O-HAZE and Dense-Haze.

| O-HAZE | Dense-Haze
---|---|---
Methods | PSNR$\uparrow$ | SSIM$\uparrow$ | PSNR$\uparrow$ | SSIM$\uparrow$
DCP [23] | 16.78 | 0.653 | 12.72 | 0.442
DehazeNet [6] | 17.57 | 0.770 | 13.84 | 0.430
GFN [51] | 18.16 | 0.671 | - | -
GDN [37] | 18.92 | 0.672 | 14.96 | 0.536
MSBDN [18] | 24.36 | 0.749 | 15.13 | 0.555
FFA-Net [47] | 22.12 | 0.770 | 15.70 | 0.549
AECR-Net [61] | - | - | 15.80 | 0.466
SGID-PFF [4] | 20.96 | 0.741 | 12.49 | 0.517
Restormer [68] | 23.58 | 0.768 | 15.78 | 0.548
Dehamer [22] | 25.11 | 0.777 | 16.62 | 0.560
MB-TF [48] | 25.31 | 0.782 | 16.44 | 0.566
D3GM | 26.23 | 0.786 | 15.85 | 0.551

(a) O-HAZE and Dense-Haze.

(b) Rain100H and Rain100L.

Figure 2: Qualitative results for (a) deraining and (b) dehazing.

Quantitative results from the two raining datasets are presented in Tab. 4.
Based on both distortion and perceptual metrics, D3GM is capable of generating
the most realistic and high fidelity results as shown in Fig. 2(b).

SGM, and Transitionary SGMs vs. D3GM:

Figure 3: Sampling trajectories of SGM, transitionary SGMs: Coef. Dec., OU
SDE, and D3GM.

We perform qualitative and quantitative analyses using variants of closely
related formulations for Prop. 1 and 2 and evaluate across (A) SGMs and (B)
transitionary SGMs. (A) uses a common score-based SDE, (B) uses a Coefficient
Decoupled SDE (_e.g._ , variance exploding SDE with the drift term $\mu$)
according to Prop. 1 and OU SDE, alongside our D3GM. Following Tab. 1, VPB
[74] can be regarded as a Coefficient Decoupled SDE, and IR-SDE [38] as an OU
SDE. Our results in Fig. 3 illustrate that D3GM converges stably towards the
expected distribution, unlike other methods which exhibit instability or
deviation. This highlights the reliance of other techniques, _e.g._ , score-
based SDEs, on retrospective measurement consistency corrections.

###  5.2 Generalizability: D3GM for real-world data

Case Study 1: Dehazing. We utilize the real-world datasets O-HAZE [2], Dense-
Haze [1], which contain 45 and 55 paired images, respectively. We use the last
5 images of each dataset as the testing set and the rest as the training set
following the common split of other methods. Results are shown in Tab. 4 and
Fig. 2(a). Our work improves results on O-HAZE both quantitatively and
qualitatively. Smaller improvements are observed on Dense-Haze. This can be
attributed to the severe signal corruption of the Dense-Haze data. A
combination with tailored task-specific, Transformer-based methods [48, 22]
might lead to further performance gains for such data. Such extensions are
beyond the focus of this paper. Qualitatively D3GM achieves excellent visual
results (Fig. 1 and Fig. 2(a)).

Case Study 2: MRI Reconstruction. MRI data is represented in the complex-
valued frequency domain, which is distinctly different from the natural image
domain. We utilized the fastMRI dataset [70], containing single-channel,
complex-valued MRI samples. Implementation details can be found in Appx. G.
For a robust comparison, we benchmarked against a diverse set of deep
learning-based state-of-the-art reconstruction methods. Although our method
does not have a task-specific design, we still get comparable performance
(more details and results are provided in Appx. H). Fig. 1 illustrates our
reconstruction results from masked k-space data for 8x and 16x acceleration,
_i.e._ , under-sampling for faster data acquisition.

Table 5: Quantitative results for fastMRI dataset with acceleration rates x8
and x16.

| Undersampling factor 8x | Undersampling factor 16x
---|---|---
Method | PSNR$\uparrow$ | SSIM$\uparrow$ | LPIPS$\downarrow$ | PSNR$\uparrow$ | SSIM$\uparrow$ | LPIPS$\downarrow$
ZeroFilling | 22.74 | 0.678 | 0.504 | 20.04 | 0.624 | 0.580
D5C5 [52] | 25.99 | 0.719 | 0.291 | 23.35 | 0.667 | 0.412
DAGAN [63] | 25.19 | 0.709 | 0.262 | 23.87 | 0.673 | 0.317
SwinMR [28] | 26.98 | 0.730 | 0.254 | 24.85 | 0.673 | 0.327
DiffuseRecon [45] | 27.40 | 0.738 | 0.286 | 25.75 | 0.688 | 0.362
CDiffMR [27] | 27.26 | 0.744 | 0.236 | 25.77 | 0.707 | 0.293
D3GM | 27.92 | 0.740 | 0.175 | 25.26 | 0.701 | 0.153

Table 6: Quantitative results for IXI MRI SR on unseen datasets.

X4 | HH | Guys | IOP
---|---|---|---
Methods | PSNR$\uparrow$ | SSIM$\uparrow$ | PSNR$\uparrow$ | SSIM$\uparrow$
IXI T2w | EDSR [34] | 23.03 | 0.700 | 25.10 | 0.727
SFM [19] | 23.28 | 0.711 | 25.18 | 0.731
PDM [71] | 22.89 | 0.709 | 27.93 | 0.851
ACT [49] | 22.80 | 0.707 | 26.38 | 0.826
CST [14] | 23.70 | 0.714 | 28.55 | 0.837
D3GM | 25.13 | 0.799 | 28.60 | 0.863

Case Study 3: MRI Super-resolution (SR). The IXI333http://brain-
development.org/ixi-dataset/ dataset is the largest benchmark considered in
our MRI SR evaluation. Clinical MRI T2-weighted (T2w) scans are collected from
three hospitals with different imaging protocols: HH, Guys, and IOP. For
investigating our cross-domain generalization and robustness, a challenging
task for both MRI SR and natural image restoration, we trained on HH data with
k-space truncation, and tested on Guys and IOP with kernel degradation with an
up-scaling factor of X4. More details can be found in Appx. G.

The methods are tested under unseen data conditions, including different
acquisition parameters, MRI scanners (different vendors and field-strengths)
and unseen degradations. With D3GM we are able to demonstrate varying degrees
of improvement, as well as generalizability to the discrepancy within the
training domain and across the test domain as shown in Tab. 6. Qualitative
results are shown in Fig. LABEL:fig:MRIrecon and further results across
domains in Appx. H.

##  6 Discussion and limitations

Other works, like VPB (CD) [74], I2SB [35] are based on diffusion bridges
assuming that clean and degraded images are already close. Thus, the
tractability of the reverse process heavily relies on the validity of the
assumed Dirac delta distribution. IR-SDE [38] employs the mean-reverting SDE
theorem based on running the reverse SDE with instability. Since unstable
errors accumulate in each step, this model will eventually become unable to
learn the transformation, _e.g._ , degradation. DPS [9] and CDDB [10] assume
that the degradation process is known, or linear operations are directly used
to simulate the degradation process, which limits the generalizability of the
method. In contrast, D3GM is built on the theorem of Measure-preserving RDS,
which bridges clean and degraded image distributions while taking both
degradation and measurements into account. Moreover, D3GM can be extended to a
two-sided solution operator (tractability) with a flow map according to Prop.
1.

Limitations. Even though our results are better than others when the
degradation process is very severly corrupted (e.g., real dehazing), the
overall quality of the restored image is still limited, which is consistent
with Prop. 2. This might be alleviated via guiding the sampling process with
priors and enhanced $\mu$, such as posterior sampling or degradation maps on
the data manifold, but such approaches are still limited as shown in Tab. 8
and Appx I.

Computational Complexity vs. Performance. Tab. 8 highlights that prior work is
often tailor-made for a specific subset of tasks and thus also generalisation-
limited for challenging environments in practice. D3GM’s focus on a generic
robust solutions from an RDS perspective can mitigate this, while maintaining
en-par performance with _task-specific_ approaches in Tab. 8.

Table 7: Comparison to tailor-made approaches.

Real Dehazing | Diffu. for Inverse Problems. | Transitionary SGMs.
---|---|---
Methods | DPS [9] | CDDB [10] | I${}^{2}SB$ [35] | IR-SDE [38] | D3GM
PSNR$\uparrow$ | 18.63 | 21.55 | 21.51 | 24.52 | 26.23
SSIM$\uparrow$ | 0.448 | 0.591 | 0.583 | 0.691 | 0.786

Table 8: D3GM vs. recent deraining works.

Deraining | rain200H | Model Complexity
---|---|---
Methods | PSNR$\uparrow$ | SSIM$\uparrow$ | Param. | FLOPs
DRSformer [7] | 32.17 | 0.933 | 33.7M | 242.9G
D3GM | 32.21 | 0.925 | 36.5M | 104.7G

##  7 Conclusion

The proposed D3GM framework enhances the stability and generalizability of
SDE-based diffusion methods for challenging inverse problems. Our approach,
grounded in measure-preserving dynamics of random dynamical systems, ensures
broad applicability and relevance. We demonstrate D3GM’s effectiveness across
various benchmarks, including challenging tasks like MRI reconstruction.

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## Appendix

##  Appendix A Broader Impacts

Our proposed method offers significant advancements in the restoration of
degraded images with minimal risk of hallucinations due to stability
guarantees and with applications including MRI reconstruction and super-
resolution. In the clinical domain, the adoption of our method for MRI
reconstruction must adhere to stringent regulatory and approval processes. The
results generated by our model should serve as an auxiliary tool to assist
healthcare professionals in their diagnostic and treatment decisions, rather
than as a standalone diagnostic tool.

##  Appendix B Intuition of Measure-Preserving Dynamics in SDE:

Consider the analogy of a stretched rubber band, which naturally seeks to
return to its original but does so with a lot of oscillations when released.
This elastic behavior parallels the dynamics of the OU process, where
deviations from a mean state are counteracted by a restorative force, guiding
the system back towards equilibrium (_i.e._ , final state), with random
perturbations.

Our process models the noise as a stochastic component that fluctuates around
a stationary process and improve the OU process with RDS. Measure-preserving
dynamics ensure that while the image undergoes transformations during the
denoising process, the overall statistical properties remain consistent
(_i.e._ , invariant image features), which cannot be satisfied by vanilla OU
processes or previous approaches (Tab. 1).

##  Appendix C Mathematical Preliminaries

Consider a probability space $(\Omega,\mathcal{F},P)$ accompanied by a
standard Brownian motion $W_{t}$. A stochastic process $x_{t}$ over the
interval $0\leq t\leq T$ can be formulated by the following stochastic
differential equation (SDE):

$dx_{t}=b(t,x_{t})dt+\sigma(t,x_{t})dW_{t}$ (9)

###### Definition 1

Filtration A collection of sigma-fields,

$\mathbb{F}:=\\{\mathcal{F}_{t},0\leq t\leq T\\},$

is termed a filtration if:

  1. 1.

$\mathcal{F}_{t}\subset\mathcal{F}$ is a sub-$\sigma$-field for every
$t\in[0,T]$;

  2. 2.

If $0\leq t_{1}<t_{2}\leq T$, then
$\mathcal{F}_{t_{1}}\subset\mathcal{F}_{t_{2}}$.

Here, $\mathcal{F}_{t}$ represents the information set at time $t$.

###### Definition 2

Strong Solution A process $x$, which is $\mathcal{F}$-progressively
measurable, is considered a strong solution to the SDE given by Eq. 1 if:

$\int_{0}^{T}(|b(t,x_{t})|^{2}+|\sigma(t,x_{t})|^{2})dt<\infty$

almost surely. This is captured by:

$x_{t}=x_{0}+\int_{0}^{t}b(s,x_{s})ds+\int_{0}^{t}\sigma(s,x_{s})dW_{s},\forall
t\in[0,T]$ (10)

###### Definition 3

Lipschitz Continuity For an $N$-dimensional stochastic process $x_{t}$ over
$t\in[0,\infty)$, adapted to the filtration $\mathcal{F}$, function $w(x_{t})$
exhibits Lipschitz continuity in $x$ if:

  1. 1.

$w(x_{t})$ is $\mathcal{F}$-measurable with the requisite dimensions.

  2. 2.

There exists a non-negative constant $K$ such that, for every $x_{t}$ and
$x_{s}$,

$|w(x_{t})-w(x_{s})|\leq K|x_{t}-x_{s}|.$

In numerous diffusion models, Lipschitz continuity is inherent, ensuring the
existence and uniqueness of the solution to the stochastic process. However,
for clarity in network design, the emphasis on Lipschitz continuity ensures
the foundation of neural networks remains consistent.

###### Definition 4

Random Dynamical System A random dynamical system(RDS) consists of a base
flow, the "noise", and a cocycle dynamical system on the "physical" phase
space, we first discuss one fundamental element of our RDS, the base flow.
Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space, the noise space.
Define the base flow $\vartheta:\mathbb{R}\times\Omega\rightarrow\Omega$ as
follows: for each "time" $s\in\mathbb{R}$, let
$\vartheta_{s}:\Omega\rightarrow\Omega$ be a measure-preserving measurable
function: $\mathbb{P}(E)=\mathbb{P}\left(\vartheta_{s}^{-1}(E)\right)$ for all
$E\in\mathcal{F}$ and $s\in\mathbb{R}$
Suppose also that
1\. $\vartheta_{0}=\operatorname{id}_{\Omega}:\Omega\rightarrow\Omega$, the
identity function on $\Omega$;
2\. for all
$s,t\in\mathbb{R},\vartheta_{s}\circ\vartheta_{t}=\vartheta_{s+t}$.
That is, $\vartheta_{s},s\in\mathbb{R}$, forms a group of measure-preserving
transformation of the noise $(\Omega,\mathcal{F},\mathbb{P})$
Now we are ready to define the random dynamical system(RDS).
let $(X,d)$ be a complete separable metric space, the phase space. Let
$\varphi:\mathbb{R}\times\Omega\times X\rightarrow X$ be a
$(\mathcal{B}(\mathbb{R})\otimes\mathcal{F}\otimes\mathcal{B}(X),\mathcal{B}(X))$
measurable function such that 1\. for all
$\omega\in\Omega,\varphi(0,\omega)=\operatorname{id}_{X}:X\rightarrow X$, the
identity function on $X$; 2\. for (almost) all
$\omega\in\Omega,(t,x)\mapsto\varphi(t,\omega,x)$ is continuous; 3\. $\varphi$
satisfies the (crude) cocycle property: for almost all $\omega\in\Omega$,

$\varphi\left(t,\vartheta_{s}(\omega)\right)\circ\varphi(s,\omega)=\varphi(t+s,\omega)$

In the case of random dynamical systems driven by a Wiener process
$W:\mathbb{R}\times\Omega\rightarrow X$, the base flow
$\vartheta_{s}:\Omega\rightarrow\Omega$ would be given by

$W\left(t,\vartheta_{s}(\omega)\right)=W(t+s,\omega)-W(s,\omega).$

###### Theorem 1

Existence and Uniqueness If the initial condition $x_{0}\in\mathbb{L}^{2}$ is
a random variable that’s independent of $W$ and both $\mu(0,x_{0})$ and
$\sigma(0,x_{0})\in\mathbb{H}^{2}$, then, provided there exists a constant
$K>0$ that satisfies:

$\begin{split}&|b(t,x)-b(t,y)|+|\sigma(t,x)-\sigma(t,y)|\\\ &\leq
K|x-y|,\quad\forall t\in[0,T],x,y\in\mathbb{R}^{n}\end{split}$ (11)

(an attribute also recognized as Lipschitz continuity), a unique strong
solution to Eq. 1 exists in $\mathbb{H}^{2}$ for every $T>0$. Additionally:

$\mathbb{E}\left[\sup_{t\leq T}|x_{t}|^{2}\right]\leq
C(1+\mathbb{E}|x_{0}|^{2})e^{CT}$ (12)

holds true, where the constant $C$ depends on both $T$ and $K$.

##  Appendix D Preliminaries and Proof for Proposition 1

###  D.1 Proof for reverse-time D3GM process:

There is a one-to-one and onto correspondence between the stochastic
differential equation and the Kolmogorov equation for $p\left(x_{t},t\mid
x_{s},s\right),t\geqslant s$, which describes the evolution of the underlying
probability distribution. Consequently, there should be a one-to-one and onto
correspondence between a reverse-time equation for $\tilde{x}_{t}$ and a
Kolmogorov equation for $p(x_{t},t|x_{s},s),s\geqslant t$

$dx_{t}=\theta_{t}(\mu-x_{t})dt+\tau\sigma dW_{t}$

We have the corresponding Kolmogorov backward equation given by

$\displaystyle-\frac{\partial p\left(x_{s},s\mid x_{t},t\right)}{\partial t}=$
$\displaystyle\theta_{t}(\mu-x_{t})\cdot\frac{\partial p\left(x_{s},s\mid
x_{t},t\right)}{\partial x_{t}}$ (13)
$\displaystyle+\frac{1}{2}\tau^{2}\sigma_{t}^{2}\cdot\frac{\partial^{2}p\left(x_{s},s\mid
x_{t},t\right)}{\partial x_{t}^{2}},$

The unconditioned Kolmogorov forward equation is given by

$\displaystyle-\frac{\partial p\left(x_{s},s\mid x_{t},t\right)}{\partial t}=$
$\displaystyle\frac{\partial\left(\theta_{t}(\mu-x_{t})\cdot
p\left(x_{s},s\mid x_{t},t\right)\right)}{\partial x_{t}}$ (14)
$\displaystyle-\frac{1}{2}\cdot\frac{\partial^{2}\left(\tau^{2}\sigma_{t}^{2}\cdot
p\left(x_{s},s\mid x_{t},t\right)\right)}{\partial x_{t}^{2}},$

See [3] for more details on Kolmogorov equations.
Bayes rule gives

$p\left(x_{t},t,x_{s},s\right)=p\left(x_{s},s\mid
x_{t},t\right)p\left(x_{t},t\right)$

We plug this result into 13, which gives us the Kolmogorov equation

$\displaystyle-\frac{\partial}{\partial t}p\left(x_{t},t,x_{s},s\right)=$
$\displaystyle\frac{\partial}{\partial
x_{t}}\left[\bar{f}\left(x_{t},t\right)p\left(x_{t},t,x_{s},s\right)\right]$
(15)
$\displaystyle+\frac{1}{2}\frac{\partial^{2}\left[p\left(x_{t},t,x_{s},s\right)\cdot\tau^{2}\sigma_{t}^{2}\right]}{\partial
x_{t}^{2}}$

and the expression for $\bar{f}$ is given by

$\displaystyle\bar{f}\left(x_{t},t\right)$ $\displaystyle=\theta_{t}(\mu-
x_{t})-\frac{1}{p\left(x_{t},t\right)}\frac{\partial}{\partial
x_{t}}\left[p\left(x_{t},t\right)\tau^{2}\sigma_{t}^{2}\right]$ (16)
$\displaystyle=\theta_{t}(\mu-
x_{t})-\tau^{2}\sigma_{t}^{2}\log\frac{\partial}{\partial
x_{t}}\left[p(x_{t},t)\right]$

Therefore, we have that the reverse process corresponds to the Kolmogorov
equation 16 is given by

$dx_{t}=\theta_{t}(\mu-
x_{t})-\tau^{2}\sigma_{t}^{2}\log\nabla_{x}p_{t}(x_{t})+\tau^{2}\sigma_{t}^{2}d\bar{W}_{t}$

###### Definition 5

Different from deterministic dynamical systems, random dynamical systems
usually consider a pullback attractor rather than a forward attractor due to
the non-autonomousness introduced by the random noise. The pullback attractor
(or random global attractor) $\mathcal{A}(\omega)$ for the RDS $\varphi$ we
defined in 1 is a $\mathbb{P}$-almost surely unique random set such that:
1\. $\mathcal{A}(\omega)$ is a random compact set:
$\mathcal{A}(\omega)\subseteq X$ is almost surely compact and
$\omega\mapsto\operatorname{d}(x,\mathcal{A}(\omega))$ is a
$(\mathcal{F},\mathcal{B}(X))$-measurable function for every $x\in X$
2\. $\mathcal{A}(\omega)$ is invariant: for all
$\varphi(t,\omega)(\mathcal{A}(\omega))=\mathcal{A}\left(\vartheta_{t}\omega\right)$
almost surely;
3\. $\mathcal{A}(\omega)$ is attractive: for any deterministic bounded set
$B\subseteq X$,
$\lim_{t\rightarrow+\infty}\operatorname{d}\left(\varphi\left(t,\vartheta_{-t}\omega\right)(B),\mathcal{A}(\omega)\right)=0$
almost surely.
$\mathcal{B}(X)$ denotes the Borel $\sigma$-algebra generated by the space $X$
where the RDS is defined.

###### Definition 6

Poincare Recurrence Theorem
Let

$(X,\Sigma,\mu)$

be a finite measure space and let

$f:X\rightarrow X$

be a measure-preserving transformation. then we have that for any
$E\in\Sigma$, the set of those points $x$ of $E$ for which there exists
$N\in\mathbb{N}$ such that $f^{n}(x)\notin E$ for all $n>N$ has zero measure.
In other words, almost every point of $E$ returns to $E$. In fact, almost
every point returns infinitely often; i.e. $\mu\left(\left\\{x\in
E:\right.\right.$ there exists $N$ such that $f^{n}(x)\notin E$ for all
$\left.\left.n>N\right\\}\right)=0$.

###  D.2 Analysis of Proposition 1

Proposition 1 After extending the solution of the OU process to RDS, the
measure-preserving flow map of the solution should meet the property
$\varphi(t,s;\omega)x=\varphi\left(t-s,0;\vartheta_{s}\omega\right)x$.
However, OU processes with time-varying coefficients are usually not satisfied
for this property(can be referred to as time-homogeneity) and thus the
stability of the system breaks.
The forward process in SDE notation

$dx_{t}=\theta_{t}(\mu-x_{t})dt+\sigma_{t}dW_{t}$

and the solution to the above SDE is given by

$x_{t}=\mu+(x_{s}-\mu)\mathrm{e}^{-\bar{\theta}_{s:t}}+\int_{s}^{t}\sigma_{z}\mathrm{e}^{-\bar{\theta}_{z:t}}dW_{z}$

where $\bar{\theta}_{s:t}=\int_{s}^{t}\theta_{z}dz$.

This solution above can be represented by a continuous random dynamical system
(RDS) $\varphi$ defined on a complete separable metric space $(X,d)$, where
the noise is chosen from a probability space
$(\Omega,\mathcal{F},\mathbb{P})$. More details can be found in [13].
More generally, we can extend the RDS to two-sided, infinite time, define a
flow map or (solution operator)
$\varphi:\mathbb{R}\times\Omega\times\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$
by $\varphi\left(t,s,|\omega,x_{0}\right):=x\left(t,s|\omega,x_{0}\right)$
with $\omega\in\Omega$, $-\infty<s\leqslant t<\infty$. The base flow driven by
Brownian motion can be explicitly written as
$W\left(t,\vartheta_{s}(\omega)\right)=W(t+s,\omega)-W(s,\omega)$.
Now suppose that:
1\. The flow map $\vartheta_{t},t\in\mathbb{R}$ is a measure-preserving
transformations of $(\Omega,\mathcal{F},P)$, with the property that for all
$s<t$ and $x\in X$,

$\varphi(t,s;\omega)x=\varphi\left(t-s,0;\vartheta_{s}\omega\right)x,\quad
P\text{-a.s. }$ (17)

2\. (i) $\varphi(t,r;\omega)\varphi(r,s;\omega)x=\varphi(t,s;\omega)x$ for all
$s\leqslant r\leqslant t$ and $x\in X$;
(ii) $\varphi(t,s;\omega)$ is continuous in $X$, for all $s\leqslant t$.
(iii) for all $s<t$ and $x\in X$, the mapping

$\omega\mapsto\varphi(t,s;\omega)x$

is measurable from $(\Omega,\mathcal{F})$ to $(X,\mathcal{B}(X))$; and
(iv) for all $t,x\in X$, and $P$-a.e. $\omega$, the mapping
$s\mapsto\varphi(t,s;\omega)x$ is right continuous at any point.
Where $\mathcal{B}(X)$ denotes the $\sigma$-algebra generated by X.
Under assumptions (i), (ii), (iii), (iv) and suppose that for $P$-a.e.
$\omega$ there exists a compact attracting set $K(\omega)$ at time 0, _i.e._ ,
such that for all bounded sets $B\subset X$,

$d(\varphi(0,s;\omega)B,K(\omega))\rightarrow 0\quad\text{ as }\quad
s\rightarrow-\infty$

We can see that the attractor of this system is defined in the pullback sense,
such that time is rewind backward before iterating forward.
Moreover, the reverse process with any starting time $t$ to $s$ is defined as
the RDS going backward in time

$\varphi(s,t|\vartheta_{t}\omega,x_{t})$

start from the time $t$ realization and run backwards to $s$.

the above proposition can be extended to show that there exists a compact
attracting set at any $-\infty<t<\infty$. and this convention has allowed us
to characterize the attractor $K(\omega)=\mathcal{N}(\mu,\lambda^{2})$, when
the time becomes finite, for example from 0 to $T$, the random attractors can
be abstractly viewed as
$\mathcal{N}\left(\mu+(x_{s}-\mu)\mathrm{e}^{-\bar{\theta}_{0:t}},\lambda(1-\mathrm{e}^{-\bar{\theta}_{0:t}})\right)$.

Moreover, an important assumption in the 17 is usually not satisfied for OU
processes with time-varying coefficients, therefore, we impose that for every
$t$, $\frac{\sigma_{t}^{2}}{2\theta_{t}}=\lambda^{2}$, where $\lambda$ is a
constant, and will be the asymptotic variance of the forward process. This
convention has allowed us to reduce the regularization on two variables
$\sigma_{t},\theta_{t}$ to just one variable to satisfy 17; and this
convention has allowed as to characterize the attractor
$K(\omega)=\mathcal{N}(\mu,\lambda^{2})$, when the time becomes finite, for
example from 0 to $T$, the random attractors can be abstractly viewed as the
Gaussian measure
$\mathcal{N}\left(\mu+(x_{s}-\mu)\mathrm{e}^{-\bar{\theta}_{0:t}},\lambda(1-\mathrm{e}^{-\bar{\theta}_{0:t}})\right)$.

##  Appendix E Proof for Proposition 2

Proposition 2 Given Eq. 3 and Eq. 4, and assume that the score function is
bounded by $C$ in $L^{2}$ norm, then the discrepancy between the reference and
the retrieved data is, with probability at least $(1-\delta)$:

$\displaystyle\|x_{0}-\operatorname{OU}(x_{0},\mu;T,\theta)\|_{2}^{2}$ (18)
$\displaystyle\geq|\left((x_{0}-\mu)^{2}-\sigma_{T}^{2}/2\theta_{T}\right)\mathrm{e}^{-2\bar{\theta}_{T}}+\sigma_{T}^{2}/2\theta_{T}$
$\displaystyle-\sigma_{max}^{2}\left(C\sigma_{max}^{2}+d+2\sqrt{-d\cdot\log\delta}-2\log\delta\right)|,$

where $x_{0}$ $\hat{x}_{0}$ are the quality reference and sampling data. For a
noisy inverse problem scenario, the retrieved data with any finite $T$ always
indicates difference depends on $\sigma_{t},\mu_{t},\lambda,T,\bar{K}$, where
$\bar{K}$ is the Lipschitz constant for the reverse process.

we have that the absolute value between the theoretical expectation and actual
expectation after $T$ period is given by

$\|\mu-\mathbb{E}(\hat{x}_{T})\|=\|(x_{0}-\mu)e^{-\bar{\theta}_{T}}\|>0$

Similarly, the difference between theoretical variance and T-period variance
also has a strictly positive difference. where
$\bar{\theta}_{t}=\int_{0}^{t}\theta_{s}ds$.
Therefore, with finite $T$, the final state of the forward process can only
reach a $\hat{x}_{T}$ rather than the theoretical stationary distribution,
which we denote by $x_{\infty}$, we denote the retrieved image after T-periods
from theoretical stationary distribution by $\hat{x}_{0}$, $x_{0}$ by the
ground truth HQ image, $\hat{x}_{T}$ the true distribution after T iteration,

$\displaystyle\|x_{Q}-f_{\operatorname{OU}}n(x_{Q},\mu;t_{0},)\|$
$\displaystyle=\|\hat{x}_{T}-x_{\infty}-(\hat{x}_{0})-x_{\infty}\|_{2}^{2}$
(19)
$\displaystyle\geq\|\|\hat{x}_{T}-x_{\infty}\|_{2}^{2}-\|\hat{x}_{0}-x_{\infty}\|_{2}^{2}\|_{2}^{2}$

Inside the norm, the first term is bounded below, since both $\hat{x}_{T}$ and
$x_{\infty}$ both follow a normal distribution and are independent of each
other, the difference between those two random variables, we denote by
$z_{T}$, that follows a normal distribution

$\displaystyle\mathcal{N}\left(\mu+(x_{0}-\mu)\mathrm{e}^{-\bar{\theta}_{t}}-\mu,\lambda^{2}(1-e^{-2\bar{\theta}_{t}})+\lambda^{2}\right)$
(20) $\displaystyle=$
$\displaystyle\mathcal{N}\left((x_{0}-\mu)\mathrm{e}^{-\bar{\theta}_{t}},\lambda^{2}(2-e^{-2\bar{\theta}_{t}})\right)$

Therefore, we could rewrite the equality as:

$\displaystyle\|x_{Q}-f_{\text{OU}}(x_{Q},\mu;t_{0})\|$ (21)
$\displaystyle\geq$
$\displaystyle\left\|\|z_{T}\|_{2}^{2}-\left\|\int_{T}^{0}-\frac{d\sigma_{t}^{2}}{dt}\nabla_{x}\log
p_{t}(x)\,dt+d\overline{\mathbf{w}}_{t}\right\|_{2}^{2}\right\|_{2}^{2}$
$\displaystyle=$
$\displaystyle\left\|(x_{0}-\mu)^{2}e^{-2\bar{\theta}_{t}}+\lambda^{2}(2-e^{-2\bar{\theta}_{t}})-C\sigma_{\text{max}}^{4}-\left\|\int_{T}^{0}\sqrt{\frac{d\sigma_{t}^{2}}{dt}}\,d\overline{w}_{t}\right\|_{2}^{2}\right\|_{2}^{2}$

Since we require $\lambda=\frac{\sigma_{t}^{2}}{2\theta_{t}}$, we can find a
$\sigma_{max}$ such that $\sigma_{t}<\sigma_{max}$ for all $t$. The last term
only concerns the random noise, according to [33], we have that the last term
is equivalent to the squared $L_{2}$ norm of a random variable from a Wiener
process at time $t=0$, with marginal distribution being
$\epsilon\sim\mathcal{N}\left(\mathbf{0},\sigma^{2}_{T}\mathbf{I}\right)$. The
squared $L_{2}$ norm of $\epsilon$ divided by $\sigma^{2}_{T}$ is a
$\chi^{2}$-distribution with $d$-degrees of freedom, we have the following
one-sided tail bound, according to [33]

$\operatorname{Pr}\left(\|\epsilon\|_{2}^{2}/\sigma^{2}\left(t_{0}\right)\geq
d+2\sqrt{d\cdot-\log\delta}-2\log\delta\right)\leq\exp(\log\delta)=\delta$

Therefore, with probability $1-\delta$, the HQ image and retrieved image has a
lower bound of,(d is the number of dimension for $x_{0}$),

$\displaystyle|\left((x_{0}-\mu)^{2}-\lambda^{2}\right)\mathrm{e}^{-2\bar{\theta}_{T}}+2\lambda^{2}$
$\displaystyle-$
$\displaystyle\sigma_{max}^{2}\left(C\sigma_{max}^{2}+d+2\sqrt{-d\cdot\log\delta}-2\log\delta\right)|$

A SDE modeling that has suffered complex degradation is considered
’corrupted’:  Example 1. Based on the Prop. The conditional SDE diffusion is
composed of three general stages, forward, backward, and sampling. Both time-
reversal processes have been declared unstable under the degradation process
$\boldsymbol{\mu}$ according to the Prop. 2. The following advantages would be
offered by our measure-preserving dynamical system when formulated as SDEs:
Intuition 1. A two-sided measure-preserving random dynamical(MP-RDS) system
formulation enables us to use the Poincare recurrence theorem, intuitively,
with a two-sided MP-RDS $\varphi_{t}$, the Poincare recurrence theorem ensures
that the system $\varphi_{t}$ starts from terminal condition $x_{T}$, run
backwards in time, will hit a region $(x_{0}-\epsilon,x_{0}+\epsilon)$ for
small $\epsilon$ in finite time, where $x_{0}$ is the high-quality image.

##  Appendix F Temporal Distribution Discrepancy during Sampling

###### Theorem 2

Suppose that both the drift $b_{t}(x)$ and diffusion $\sigma_{t}(x)$ term of a
stochastic process $x_{t}$ is Lipschitz continuous with some constant $K$,
moreover, $x\in\mathbb{L}^{2}\left(\mathbb{F},\mathbb{R}\right)$ is a solution
to the SDE

$dx_{t}=b(t,x)dt+\sigma(t,x)dW_{t}$

and initial condition $b_{0},\sigma_{0}$ then we have that

$\mathbb{E}\left[\left|x_{T}-x_{0}\right|^{2}\right]\leq CI_{0}^{2},\\\ $ (22)

such that

$I_{0}^{2}:=\mathbb{E}\left[\left(\int_{0}^{T}\left|b_{0}\right|dt\right)^{2}+\int_{0}^{T}\left|\sigma_{0}\right|^{2}dt\right]$

and C depends only on $T,K$, which is the running time and Lipschitz constant

Firstly, we have the following relationship:

$\displaystyle x_{T}$
$\displaystyle\leq\left|x_{0}\right|+\int_{0}^{T}\left|b(t,x)\right|\,dt+\sup_{0\leq
t\leq T}\left|\int_{0}^{t}\sigma(s,x)\,dW_{s}\right|,$ (23)
$\displaystyle\left|x_{T}-x_{0}\right|$
$\displaystyle\leq\int_{0}^{T}\left|b(t,x)\right|\,dt+\sup_{0\leq t\leq
T}\left|\int_{0}^{t}\sigma(s,x)\,dW_{s}\right|.$

Squaring both sides, taking expectations, and applying the Burkholder-Davis-
Gundy inequality, we get:

$\displaystyle\mathbb{E}\left[\left|x_{T}-x_{0}\right|^{2}\right]$ (24)
$\displaystyle\leq
C\mathbb{E}\left[\left(\int_{0}^{T}\left|b(t,x_{t})\right|\,dt\right)^{2}+\sup_{0\leq
t\leq T}\left(\int_{0}^{t}\sigma(s,x_{s})\,dB_{s}\right)^{2}\right]$
$\displaystyle\leq
C\mathbb{E}\left[\left(\int_{0}^{T}\left[\left|b_{0}\right|+\left|x_{t}\right|\right]\,dt\right)^{2}+\int_{0}^{T}\left|\sigma(t,x_{t})\right|^{2}\,dt\right]$
$\displaystyle\leq
C\mathbb{E}\left[\left(\int_{0}^{T}\left|b_{0}\right|\,dt\right)^{2}+\int_{0}^{T}\left[\left|\sigma_{0}\right|^{2}+\left|x_{t}\right|^{2}\right]\,dt\right].$

Remark: It should be noted that the constant $C$, which depends on $T$ and
$K$, varies from line to line.

Next, we show that for any $\varepsilon>0$, there exists a constant
$C_{\varepsilon}>0$ such that:

$\sup_{0\leq t\leq
T}\mathbb{E}\left[\left|x_{t}\right|^{2}\right]\leq\varepsilon\mathbb{E}\left[\left|x_{T}^{*}\right|^{2}\right]+C_{\varepsilon}I_{0}^{2}.$
(25)

Applying Itô’s formula, we get:

$d\left|x_{t}\right|^{2}=\left[2x_{t}b(t,x_{t})+\left|\sigma(t,x_{t})\right|^{2}\right]\,dt+2x_{t}\sigma(t,x_{t})\,dB_{t}.$
(26)

Considering the martingale property of the third term, integrating, and taking
expectation on both sides, we have:

$\displaystyle\mathbb{E}\left[\left|x_{t}\right|^{2}\right]$
$\displaystyle=\mathbb{E}\left[|x_{0}|^{2}+\int_{0}^{t}\left[2x_{s}b(s,x_{s})+\left|\sigma(s,x_{s})\right|^{2}\right]\,ds\right]$
(27)
$\displaystyle\leq\mathbb{E}\left[|x_{0}|^{2}+\int_{0}^{t}\left[C\left|x_{s}\right|^{2}+2\left|x_{s}\right|\left|b_{0}\right|+C\left|\sigma_{0}\right|^{2}\right]\,ds\right]$
$\displaystyle\leq
C\int_{0}^{t}\mathbb{E}\left[\left|x_{s}\right|^{2}\right]\,ds+2\mathbb{E}\left[x_{T}\int_{0}^{T}\left|b_{0}\right|\,ds\right]+CI_{0}^{2}.$

Using Gronwall’s inequality, we can prove (25) with the result above.
Substituting (25) into (LABEL:x_t) completes the proof, showing that the
distance between $x_{T}$ and $x_{0}$ in the $L^{2}$-sense is bounded by
$C(K,T)$.

With theorem 1 in hand, since the OU process is Lipschitz continuous, then the
reverse process for the OU process is also Lipschitz continuous.
Now, suppose that the Lipschitz constant for the reverse process is given by
$\bar{K}$. then we have that in $L^{2}$-norm, the distance between any final
state $x_{T}\in\mathcal{N}(\mu,\lambda)$ and the initial state(HQ image) is
bounded by a constant that only depends on time $T$ and Lipschitz constant
$\bar{K}$, and the initial condition for the drift and diffusion term, where
we denote as $C(\bar{K},T,\mu,\lambda)$, which will be written as
$C(\bar{K},T)$ for short.
Now, since the time $T$ is finite, where theoretically only when
$T\rightarrow\infty$ would $x_{T}$ converge to the theoretical stationary
distribution, thus, if we denote the sample from $T$ time steps by
$\hat{x}_{T}$, and $x_{T}$ by the sample from the theoretical stationary
distribution, then we have
$\mathbb{E}[x_{T}-\hat{x}_{T})]>\epsilon(T,K,\mu,\lambda)$, note that the $K$
here is the Lipschitz constant for the forward process, and this distance
strictly decrease in $T$.
Suppose that in inference, when the ground truth $x_{0}$ is unknown, the
distance between ground truth $x_{0}$ and sample from the theoretical
distribution $x_{T}$ is bounded from below by

$||x_{0}-x_{\infty}||>C(\bar{K},T)I^{2}_{0}+\epsilon$ (28)

Then, we can see that the gap between $x_{T}$ and $\hat{x}_{T}$ increases such
bound, which is

$\displaystyle||x_{0}-\hat{x}_{T}||\geq||x_{0}-x_{\infty}-(\hat{x}_{T}-x_{\infty})||$
$\displaystyle>C(\bar{K},T)I^{2}_{0}+\epsilon(T,K,\mu,\lambda)$

where $T$ is the time step for the forward process, and $\epsilon$ is some
strictly positive constant. Now suppose that $\hat{x_{t}}$ is the solution to
the reverse process, which runs for $T$ periods in total, then we have that
for $t\in[0,T]$, denote $x_{0},\hat{x}_{0}$ as the original HQ image and the
final state of the reverse process, respectively.

$\displaystyle\left\|x^{Quality}-\operatorname{OU}\left(x^{Quality},\mu;t_{0},\theta\right)\right\|_{2}^{2}=$
(29)
$\displaystyle\|x_{0}-\hat{x}_{0}\|=\|x_{0}-\hat{x}_{T}-(\hat{x}_{0}-\hat{x}_{T})\|\geq$
$\displaystyle\|\|x_{0}-\hat{x}_{T}\|-\|(\hat{x}_{0}-x_{T})\|\|>$
$\displaystyle\epsilon+C(\bar{K},T)I^{2}_{0}-C(\bar{K},T)I^{2}_{0}=\epsilon$

Therefore, the bias created in inference depends on the LQ image $\mu$,
stationary variance, Lipschitz constant, and the time steps T.

##  Appendix G Stable in Probability

###### Definition 7

Stable in Probability
Given a probability space $(\Omega,\mathcal{F},P)$ and a standard Brownian
motion $W_{t}$, a general form of SDE for a stochastic process $x_{t},0\leq
t\leq T$ is given by

$dx_{t}=b(t,x)dt+\sigma(t,x)dW_{t}$ (30)

Such that the Lipschitz condition is satisfied, for both $b(t,x),\sigma(t,x)$,
A solution $x(t,\omega)\equiv 0$ is said to be stable in probability for
$t\geq 0$ if for any $s\geq 0$ and $\varepsilon>0$

$\lim_{x_{0}\rightarrow
0}\mathbf{P}\left\\{\sup_{t>s}\left|x^{s,x}(t)\right|>\varepsilon\right\\}=0.$
(31)

It says that the sample path of the process issuing from a point $x$ at time
$s$ will always remain within any prescribed neighbourhood of the origin with
probability tending to one as $x\rightarrow 0$. In practice, this property
ensures that the perturbation from the initial state caused by a stable
process is bounded for all $t$ with probability one.
For example, the OU process 4 admits a unique unconditional stationary
solution provided by theorem 19, however, in this example, without specifying
the value that determines the stationary variance, _i.e._ , $\sigma,\theta$.
If for large $\sigma$ and a sample $\hat{x}$ from the stationary distribution
of the OU process, we have that

$\mathbb{P}(\hat{x}>\mu\pm\frac{\sigma}{2\theta})>0$ (32)

This means that a sample from the stationary distribution of the forward
process could deviate largely from $\mu$, thus making the result no different
from the traditional VE(variance exploding) diffusion models that are defined
as

$dx_{t}=\sigma_{t}dW_{t}$

because the variance could be set arbitrarily large if no restriction is
specified. Therefore, how should such a problem be approached most easily? The
Lyapunov theorem for stability has provided an easy way, without explicitly
solving the SDEs, to ensure stability just from the coefficients.
Then we provide the main theorem that ensures the stability of SDE, first, we
give the definition of positive definite in the Lyapunov sense

###### Definition 8

Let $K$ denote the family of all continuous nondecreasing functions
$\mu:R_{+}\rightarrow R_{+}$such that $\mu(0)=0$ and $\mu(r)>0$ if $r>0$. For
$h>0$, let $S_{h}=\left\\{x\in R^{n}:|x|<h\right\\}$. A continuous function
$V(x,t)$ defined on $S_{h}\times\left[t_{0},\infty\right)$ is said to be
positive-definite (in the sense of Lyapunov) if $V(0,t)\equiv 0$ and, for some
$\mu\in K$,

$V(x,t)\geq\mu(|x|)$

for all $(x,t)\in S_{h}\times\left[t_{0},\infty\right)$

Then we will use the convention of the Lyapunov quadratic function, such that

###### Definition 9

Lyapunov quadratic function $V$ is given

$V\left(x_{t}\right)=x_{t}^{T}Qx_{t},$

where $Q$ is a symmetric positive-definite matrix.

###### Theorem 3

The function $LV$

$\displaystyle LV\left(x_{t}\right)=$ $\displaystyle
x_{t}^{T}Qb\left(t,x_{t}\right)+b\left(t,x_{t}\right)^{T}Qx_{t}+$ (33)
$\displaystyle\sigma\left(t,x_{t}\right)^{T}Q\sigma\left(t,x_{t}\right),$

is negative-definite in some neighbourhood of $x_{t}=0$ for $t\geq t_{0}$,
with respect to system 30. Then the trivial solution of equation 30 is
stochastically asymptotically stable.

Since this theorem is important and the proof will be intuitive in explaining
why such a condition could ensure stability, the proof will be put here.
Proof:
First, we compute $dV(x)$, which is the instantaneous growth of the Lyapunov
quadratic function, gives

$\displaystyle dV\left(x_{t}\right)=$ $\displaystyle
V\left(x_{t}+dx_{t}\right)-V\left(x_{t}\right)$ $\displaystyle=$
$\displaystyle\left(x_{t}^{T}+dx_{t}^{T}\right)Q\left(x_{t}+dx_{t}\right)-x_{t}^{T}Qx_{t}$
$\displaystyle=$ $\displaystyle
x_{t}^{T}Qb\left(t,x_{t}\right)dt+x_{t}^{T}Q\sigma\left(t,x_{t}\right)dB_{t}+$
$\displaystyle
b\left(t,x_{t}\right)^{T}dtQx_{t}+\sigma\left(t,x_{t}\right)^{T}dB_{t}Qx_{t}+$
$\displaystyle\sigma\left(t,x_{t}\right)^{T}Q\sigma\left(t,x_{t}\right)dt$

Then, take expectation, we can get

$\displaystyle E\left\\{dV\left(x_{t}\right)\right\\}$
$\displaystyle=x_{t}^{T}Qb\left(t,x_{t}\right)dt+b\left(t,x_{t}\right)^{T}Qx_{t}dt+$
$\displaystyle\sigma\left(t,x_{t}\right)^{T}Q\sigma\left(t,x_{t}\right)dt$
$\displaystyle=LV\left(x_{t}\right)dt$

Then, if we assume that $LV(x)$

$-LV\left(x_{t}\right)\geq kV\left(x_{t}\right)$

such that $k$ is a constant, then

$\displaystyle\frac{d}{dt}E\left\\{V\left(x_{t}\right)\right\\}\leq-
kE\left\\{V\left(x_{t}\right)\right\\},$ (34) $\displaystyle
E\left\\{V\left(x_{t}\right)\right\\}\leq\exp(-kt).$

As can be seen from the proof, the operator $LV$ as the function of the SDE
$x_{t}$ is the expectation of the $dV(x_{t})$, and the negative semi-
definiteness can be regarded as requiring $dV(x_{t})$ to be a contraction.
This can be understood from 34 such that

$\frac{d}{dt}E(V(x_{t}))/E(V(x_{t}))<-k$

for $k>0$

##  Appendix H Implementation details

Model Implementation: Our exploration into mitigating Temporal Distribution
Discrepancy in diffusion models employs two neural network architectures, each
catering to different dataset complexities.

We utilize the adopted UNet, a staple in DDPM [25] and DDIM[54] frameworks,
chosen for its widespread use and strong benchmarking capabilities. By solving
the discrepancy through this established structure, we achieve state-of-the-
art results on synthetic data, showcasing the potential of improving
transanary SDE diffusion models in terms of Temporal Distribution Discrepancy
and stationary process. Additionally, the use of this prevalent architecture
allows for comprehensive analysis and discussion.

Real-world data with its inherent complexity, such as combined degradations,
large resolutions, and extensive interdependencies, requires an architecture
beyond the conventional UNet. Our improved model incorporates Squeeze-and-
Excitation [39] and NAF [8], explicitly designed to capture intricate feature
interrelations. While these models do not seek to innovate the architectural
paradigm, it provides a solid baseline that provides better feature extraction
ability than UNet performance in demanding scenarios.

Table 9: Datasets parameters and split setting. IOP details of the scan
parameters are not available.

Datasets | Data setting | Collection | Sequence parameters
---|---|---|---
Source domain | Subjects | Slices | Hospitals | Scanner | Repetition time | Echo train length | Matrix size | Receiver coil
HH (IXI Brain) | 184 | 60 | Hammersmith | Philips 3T | 5725 | 16 | 192 x 187 | Single
Target domain |  |  |  |  |  |  |  |
Guys (IXI Brain) | 30 | 60 | Guy’s Hospital | Philips 1.5T | 8178 | 16 | Unnormalized | Single
IOP (IXI Brain) | 30 | 60 | Institute of Psychiatry | GE 1.5T | Unknown | Single

Method |  PSNR$\uparrow$ |  SSIM$\uparrow$ |  LPIPS$\downarrow$
---|---|---|---
WD UNet |  27.31 |  0.8322 |  0.227
SN UNet |  28.63 |  0.8687 |  0.144
UNet |  31.03 |  0.9001 |  0.058

Table 10: Evaluation of the Lipschitz continuity.

Additional details:  The U-Net we adopted is similar to DDPM as described in
[38, 11, 9], where the improved model incorporates Squeeze-and-Excitation [39]
to replace the Attention module within NAF [8]. EDSR [34] is employed as the
base model for TTA-based comparison methods in MRI super-resolution. In
different downstream tasks, we follow the common setting of the latest
compared methods: Deraining [38], Real Dehazing [48], MRI Reconstruction [27],
and MRI super-resolution [14] and [34]. Below are more specific details about
MRI reconstruction and MRI superresolution. For most of the experiments, the
training patch-size is set to 128x128 with a batch size of 16. We utilize the
Adam optimizer with $\beta_{1}=0.9,\beta_{2}=0.999$, a learning rate of
$10^{-4}$ with a decay strategy. Our models are trained on three RTX 6000 GPUs
for about four days, each with 40GB of memory. Random seed is 42. All
mathematical variants follow the cosine schedule as per [43]. The variance
$\lambda$ is set to 10 for the OU and stationary processes. In the coef.
decoupled SDE, $\sigma$ is kept decoupled and does not vary with $\theta$. We
observed that the adaptability of $\tau$ to tasks is contingent upon the
task’s corruption for model stability and generalizability.

$\tau$ is a hyperparameter as one of the ways to introduce Measure-preserving
Dynamics to shape more stable SDE diffusion. It is informed by the deviation
between the degraded image and the expected high-quality image. For tasks with
moderate intra-domain deviations, $\tau$ is set to 2, while for tasks with
substantial cross-domain and degradation discrepancies, a larger $\tau$ is
used. This metric is based on RDS settings of different degradations, rather
than learned. A MLP can be used to learn $\tau$ as adaptive metric, although
it is not comparable with obvious improvements.

SGM, and Transitionary SGMs vs. D3GM: We perform qualitative 3 and
quantitative H analyses using variants of closely related formulations for
Prop. 1 and 2 and evaluate across (A) SGMs and (B) transitionary SGMs. (A)
uses a common score-based SDE, (B) uses a Coefficient Decoupled SDE (_e.g._ ,
variance exploding SDE with the drift term $\mu$) according to Prop. 1 and OU
SDE, alongside our D3GM.

Table 11: Exemplary deraining results of (A) SGM, (B) transitionary SGMs:
Coefficent Decoupled SDE, OU SDE, and D3GM.

|  Method |  PSNR$\uparrow$ |  SSIM$\uparrow$ |  LPIPS$\downarrow$
---|---|---|---|---
A |  SGM |  27.27 |  0.840 |  0.144
|  Coef. Dec. SDE |  26.18 |  0.826 |  0.205
B |  OU SDE |  30.58 |  0.900 |  0.051
|  D3GM (ours) |  32.41 |  0.912 |  0.040

MRI Reconstruction:  We applied 584 proton-density weighted knee MRI scans
without fat suppression. These were subsequently partitioned into a training
set (420 scans), a validation set (64 scans), and a testing set (100 scans).
For each scan, we extracted 20 coronal 2D single-channel complex-valued
slices, predominantly from the central region with the uniform size of 320 x
320. Our test set differs from the 200 reported in [27], possibly as a result
of the change in the official versions of FastMRI. We subjected all
experiments to Cartesian under-sampling masks with undersampling factor 8x and
16x. In the undersampled MRI scans, the acquisition process entails sampling a
fractional subset of the Fourier-space (k-space), typically governed by a mask
along the dimension of undersampling rate. Undersampling inherently induces
aliasing artifacts in the resultant images. Considering the domain deviation,
we did not supplement the extra test set into the final 100 samples.
Otherwise, we stayed consistent with the details of the paper. For a robust
comparison, we benchmarked against a diverse set of deep learning-based state-
of-the-art reconstruction methods, including CNN-based approaches such as D5C5
[52] and DAGAN [63], the Transformer-based SwinMR [28], diffusion model-
inspired DiffuseRecon [45], and the CDiffMR [27]. Quantitative results in Tab.
6 exhibit a differing trend from the image domain. The task-specific diffusion
models achieve better results and are more capable of capturing the complex
degradation that occurs in the frequency domain.

Generally, MRI uses a mask in the phase-encoding direction (the shortest
anatomical direction [42]) to model the complex degradation caused by
undersampling. In the knee data, based upon the uncertainty of clinical
diagnosis (longitudinal artifacts can significantly confuse the diagnosis of
meniscal injury), we also masked the frequency direction, which will cause
resolution reduction, deterioration of image features, and longitudinal
artifacts, more qualitative results can be found in the next section.

MRI Super-resolution:  IXI444http://brain-development.org/ixi-dataset/
contains clinical T1- and T2-weighted scans from three hospitals with
different imaging protocols: HH, Guys, and IOP. We selected 184 HH T2 subjects
as the source-domain data [train/val/test ratio: 7:1:2], and 30 subjects each
from Guys and IOP as two target-domain datasets without degradation,
acquisition parameters, datasets, and patient-wise crossovers. The central 60
slices were selected.

We consider two benchmark ideas: (1) Blind SR degradation methods in frequency
SFM [19] and spatial PDM [71] domain. (2) Source-free SR adaptation ACT [14]
using external priors and test-time adaptation proposed initially for MRI CST
[14] with data consistency in the source-domain. We are concerned that
existing work on robustness and generalization focuses on medical image
segmentation and natural images, and rarely on super-resolution and
reconstruction of medical images. Employing the available evaluation criteria
in this inevitable problem in practice makes it difficult to reflect the
performance of D3GM. Therefore, domain-aware datasets isolation standard for
public datasets is designed for the SRR adaptation to cross-domain data rather
than same-domain in a source-free manner. The publicly available data were
split into several subsets based on hospitals, scanner, acquisition
parameters, modality, and anatomy as illustrated in Table 9. Explicit
reference standards implicitly correspond to various degradation patterns,
thus enabling the isolation of natural degradation patterns in source training
domain and target-testing domain. In addition to this, different artificial
degradation patterns for the subsets in the training and testing domains are
employed: K-space truncation downsampling was applied to obtain LR data in the
source domain and a kernel degradation [5] was applied in target domain.

We reproduce two types of benchmark ideas on the top of EDSR [34] backbone to
achieve the multi-purpose goal: (1) Repurposed blind SR (BSR) for cross-domain
data: We utilized BSR degradation methods in frequency SFM [19] and spatial
PDM [71] domain. (2) Test-time adaptation (TTA): We compare to a source-free
TTA method ACT [14] using external priors and a second TTA method proposed
initially for MR reconstruction CST [14] with cycle consistency at source-
domain. The settings and adaptation strategies of the comparison methods were
used directly.

Lipschitz Continuity for Stationary Process:

We hypothesize that ensuring Lipschitz continuity of the neural network is
pivotal for the convergence and stability of the diffusion process,
particularly in the context of achieving a stationary process. Theoretically,
Lipschitz continuity offers two key benefits: (1) it mitigates the impact of
small perturbations in input data or model parameters, thus safeguarding
against excessive output variability which could cause numerical instabilities
or result in an ill-posed problem, and (2) it guarantees the existence of a
unique solution to the diffusion process, underpinning the reliability and
convergence of the numerical methods employed to solve these equations.

To instantiate these theoretical benefits within our architecture, we
integrated spectral normalization (SN) [41] and weight decay (WD) [36] into a
U-Net structured score network. In the case of SN, it is achieved by rescaling
each layer’s weight matrix $W^{(l)}$ by its spectral norm,
$\sigma_{max}(W^{(l)})$, to obtain a normalized weight
$\tilde{W}^{(l)}=\frac{W^{(l)}}{\sigma_{max}(W^{(l)})}$. By doing so, we
intend to control the overall Lipschitz constant of the network for robust
score matching within our diffusion model framework. A quantitive result can
be seen in Tab. 10.

##  Appendix I Additional Insights

Basin of Attraction in Reverse Sampling:  Within our diffusion framework, we
establish a forward process with variety to accommodate a wide range of
potential corruptions, ensuring the desired final distribution close to the
expected distribution. We consider that our diffusion models inherently
encompass the forward operator $\mathcal{A}$ within their structure. The
transition from high to low-quality images is implicitly encoded in the
diffusion pathway, hence the additional forward operator guidance might not
contribute supplementary information, which we initially hypothesized would
enhance the reverse process.

Figure 4: Reverse Initialization with Basin of attraction.

Thus, the keypoint transfers from the $\mathcal{A}$ to the initial
$y^{\prime}$ (Detailed analysis based on Prop. 2 is provided in Appx. F). Also
we found this problem fall into the Basin of Attraction (BA) in dynamical
system as shown in Fig. 4. BA can be interpreted as the quality of the
attractor of the degraded image in the reverse process here. A BA-guided
initialization might be a more effective approach for posterior sampling in
transitional SDE diffusion models. By initializing the reverse process closer
to the expected solution, we may bypass the initial hurdle of distribution
mismatch.

##  Appendix J More Qualitative Results

Figure 5: Deraining results with light rain images of our method. Figure 6:
Deraining results with heavy rain images of our method. Figure 7: Dehazing
results with real hazy images of our method. Figure 8: MRI reconstruction
results with undersampling rate x8 and x16, on Frequency-encoding and Phase-
encoding directions. Figure 9: MRI super-resolution results with in-domain
images of our method. Figure 10: MRI super-resolution results with cross-
domain (different imaging devices and degradation methods) images of our
method.