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MAP-based Problem-Agnostic diffusion model for Inverse Problems

Pingping Tao, Haixia Liu, Jing Su, Xiaochen Yang, Hongchen Tan

TL;DR

This work addresses inverse problems with diffusion models by introducing a training-free MAP-based guided term estimation that splits the conditional score into an unconditional component from a pretrained model and a MAP-derived guided term. The guided term is estimated using a Gaussian-type prior on natural images, enabling a problem-agnostic, plug-and-play approach that still aligns with Bayes' rule. The method delivers competitive performance against DDRM, DPS, $\Pi$GDM, DMPS, and MCG across super-resolution, denoising, and inpainting, while preserving structural content in challenging regions. This approach broadens the applicability of diffusion-based inverse problem solvers by leveraging image-structure priors without problem-specific retraining, with practical impact on real-world restoration tasks.

Abstract

Diffusion models have indeed shown great promise in solving inverse problems in image processing. In this paper, we propose a novel, problem-agnostic diffusion model called the maximum a posteriori (MAP)-based guided term estimation method for inverse problems. To leverage unconditionally pretrained diffusion models to address conditional generation tasks, we divide the conditional score function into two terms according to Bayes' rule: an unconditional score function (approximated by a pretrained score network) and a guided term, which is estimated using a novel MAP-based method that incorporates a Gaussian-type prior of natural images. This innovation allows us to better capture the intrinsic properties of the data, leading to improved performance. Numerical results demonstrate that our method preserves contents more effectively compared to state-of-the-art methods--for example, maintaining the structure of glasses in super-resolution tasks and producing more coherent results in the neighborhood of masked regions during inpainting.

MAP-based Problem-Agnostic diffusion model for Inverse Problems

TL;DR

This work addresses inverse problems with diffusion models by introducing a training-free MAP-based guided term estimation that splits the conditional score into an unconditional component from a pretrained model and a MAP-derived guided term. The guided term is estimated using a Gaussian-type prior on natural images, enabling a problem-agnostic, plug-and-play approach that still aligns with Bayes' rule. The method delivers competitive performance against DDRM, DPS, GDM, DMPS, and MCG across super-resolution, denoising, and inpainting, while preserving structural content in challenging regions. This approach broadens the applicability of diffusion-based inverse problem solvers by leveraging image-structure priors without problem-specific retraining, with practical impact on real-world restoration tasks.

Abstract

Diffusion models have indeed shown great promise in solving inverse problems in image processing. In this paper, we propose a novel, problem-agnostic diffusion model called the maximum a posteriori (MAP)-based guided term estimation method for inverse problems. To leverage unconditionally pretrained diffusion models to address conditional generation tasks, we divide the conditional score function into two terms according to Bayes' rule: an unconditional score function (approximated by a pretrained score network) and a guided term, which is estimated using a novel MAP-based method that incorporates a Gaussian-type prior of natural images. This innovation allows us to better capture the intrinsic properties of the data, leading to improved performance. Numerical results demonstrate that our method preserves contents more effectively compared to state-of-the-art methods--for example, maintaining the structure of glasses in super-resolution tasks and producing more coherent results in the neighborhood of masked regions during inpainting.
Paper Structure (17 sections, 2 theorems, 24 equations, 10 figures, 4 tables, 1 algorithm)

This paper contains 17 sections, 2 theorems, 24 equations, 10 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Works
  3. Overview of Score-based Models
  4. MAP-based Guided Term Estimation
  5. Estimation of True Image
  6. Estimation of Guided Term
  7. Numerical Experiments
  8. Experimental Setup
  9. Super Resolution
  10. Denoising
  11. Inpainting
  12. Runtime
  13. Conclusion
  14. Derivation of the estimated value
  15. The computation of spatial derivative
  16. ...and 2 more sections

Key Result

Theorem 4.1

Let $\mathbf{x}_t$ be the $t$-th latent image in the backward process of diffusion model and the neural network $\mathbf{S}_{\theta}(\mathbf{x}_t, t)$ be an approximation of score function $\nabla_{\mathbf{x}_t}\log p(\mathbf{x}_t)$. Define $\bar{\alpha}_t=1-\zeta_t$, then the estimation of $\mathbf where $q_1$ and $q_2$ are parameters such that $\mathbf{S}_{\theta}(\hat{\mathbf{x}},0)=\mathbf{S}_

Figures (10)

  • Figure 1: Illustrations of the forward process (First row), unconditional backward process (Second row), and conditional backward process (Last row), respectively.
  • Figure 2: The results for super-resolution. The first column is the input image, the second one is the Ground Truth (denoted as GT) and the third to eighth columns are our proposed method, DDRM, DPS, $\Pi$GDM, DMPS and MCG, respectively.
  • Figure 3: The robustness analysis results for SR on FFHQ 256×256-1k (first row) and CelebA-HQ 256×256-1k (second row) validation sets with different parameters. Columns 1-3 are the plots of PSNR and LPIPS values versus the changes of parameters $q_1$, $q_2$, and $\eta$, with the other two fixed.
  • Figure 4: The results for denoising. All measurements are with Gaussian noise $\sigma=0.5$, where GT stands for Ground Truth.
  • Figure 5: The results for Inpainting (Box).
  • ...and 5 more figures

Theorems & Definitions (6)

  • Theorem 4.1
  • proof
  • Remark 4.2
  • Theorem 4.3
  • proof
  • proof : Proof of Theorem \ref{['theo:estimation_x']}