Image Restoration Driven by Non-Markovian Noise
Recent advances in generative diffusion models have explored replacing the standard Brownian driving noise with non-Markovian fractional Brownian motion (fBM) or heavy-tailed Lévy processes. In image restoration, the Generalized Ornstein–Uhlenbeck Bridge (GOUB) model maps low-quality to high-quality images by extending the stochastic dynamics of generative bridge models to a time-varying Ornstein-Uhlenbeck (OU) process, while retaining standard Brownian motion as the driving noise. We combine these two directions by investigating how GOUB can be extended to a driving fractional noise, leveraging a Markov approximation of fBM (MA-fBM) for tractable simulation. We show that extending GOUB to fractional noise is inherently difficult: the resulting noise process will depend on the starting value at terminal time, making it infeasible to initialize the backward process during sampling. To nevertheless enable non-Markovian noise trajectories for low-to-high image translation, we employ instead of GOUB the recently proposed Fractional Diffusion Bridge Model (FDBM) for paired data translation and extend its application to image restoration. Empirically, FDBM improves reconstruction quality across a broad range of metrics compared to a Brownian baseline with an identical diffusion constant, but still falls short of purely Brownian-driven GOUB dynamics with a time-dependent diffusion coefficient. Interestingly, FDBM performs best when Brownian motion is modified only minimally toward a non-Markovian process.
