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RelativeFlow: Taming Medical Image Denoising Learning with Noisy Reference

Conference: CVPR 2026 arXiv: 2604.15459 Code: github.com/Deliver0/RelativeFlow Area: Medical Imaging Keywords: medical image denoising, flow matching, noisy reference, CT denoising, MR denoising

TL;DR

This paper proposes RelativeFlow, a flow matching-based framework that decomposes the absolute noise-to-clean mapping into relative noisier-to-noisy mappings. By incorporating a consistent transport constraint and a simulation-based velocity field, RelativeFlow learns a unified denoising flow from heterogeneous noisy references, overcoming the reference bias limitation.

Background & Motivation

Medical image denoising (MID) lacks access to absolutely clean images for supervision — only relatively high-quality references acquired under varying protocols and scanner configurations are available, with quality varying heterogeneously across categories. Three existing paradigms share notable limitations: (1) SimSDL naively treats noisy references as clean targets, leading to suboptimal convergence or reference-biased learning; (2) SSL relies on the independent noise assumption, which is difficult to satisfy in medical imaging; (3) SimSGL similarly treats noisy references as generative targets without correction. The core problem is how to learn a unified high-quality denoising mapping from heterogeneous noisy references.

Method

Overall Architecture

RelativeFlow decomposes the absolute denoising flow into a composition of multiple relative denoising flows. For each noisy reference \(x_t\), a degradation operator generates a noisier sample \(x_{t-\Delta t}\), forming a local relative denoising step. Two key components ensure the consistency and learnability of the relative flows.

Key Designs

  1. Consistent Transport (CoT) Displacement Mapping: A probability path is defined via linear interpolation in exponential time-space as \(p_t = \lambda p_{t_i} + (1-\lambda)p_{t_j}\), with weight \(\lambda = \frac{e^{-t}-e^{-t_j}}{e^{-t_i}-e^{-t_j}}\). This guarantees the transitivity of nested interpolation — the relative flow from any quality level \(t_i\) to \(t_j\) constitutes a component of the absolute flow \(\psi_{0 \to +\infty}\) and can be sequentially composed into it. Two properties are mathematically proven: (i) relative flows are components of the absolute flow, and (ii) consecutive relative flows can be progressively composed into the absolute flow.

  2. Simulation-based Velocity Field (SVF): Modality-specific degradation operators \(D_{\Delta t}\) (Poisson-Gaussian noise for CT, Rician noise for MR) are employed to simulate noisier samples from noisy references, yielding the velocity field training target \(u = \frac{x_t - D_{\Delta t}(x_t)}{e^{\Delta t} - 1}\). The training loss is the L2 distance between the neural network prediction and this target velocity.

  3. Progressive Step-size Curriculum: During training, the degradation step-size range \([\Delta t_{min}, \Delta t_{max}]\) is progressively expanded (multiplied/divided by a factor \(\alpha\) each epoch), enabling the model to gradually learn velocity fields from small to large degradations and cover the full quality spectrum. Multi-step Euler integration is used at inference.

Loss & Training

Training loss: \(\mathcal{L}_{RF} = \mathbb{E}_{x_t, \Delta t}\left[\left\|\mathcal{N}_\theta(D_{\Delta t}(x_t), \Delta t) - \frac{x_t - D_{\Delta t}(x_t)}{e^{\Delta t} - 1}\right\|_2^2\right]\). A progressive curriculum learning strategy is adopted throughout training.

Key Experimental Results

Main Results

RelativeFlow is compared against 10 methods (NID + MID) on CT and MR denoising tasks:

Task Metric Prev. SOTA Ours
CT Denoising PSNR/SSIM Runner-up Best
MR Denoising PSNR/SSIM Runner-up Best

RelativeFlow achieves state-of-the-art performance across all four metrics: PSNR, SSIM, RMSE, and LPIPS.

Ablation Study

  • The CoT consistency constraint is the key component for overcoming reference bias.
  • Modality-specific degradation modeling in SVF outperforms generic noise models.
  • Progressive step-size curriculum outperforms fixed step-size training.

Key Findings

  • RelativeFlow unifies images of varying quality levels into consistently high-quality outputs.
  • The mathematical properties of CoT (componentality and composability) provide theoretical guarantees for unified denoising.
  • Modality-specific degradation operators enable the framework to generalize across different medical imaging modalities.

Highlights & Insights

  • This work is the first to formally characterize the noisy reference problem and systematically address it via flow matching.
  • The mathematical analysis of CoT (proofs of componentality and composability) demonstrates exceptional theoretical depth.
  • Validation across both CT and MR modalities demonstrates the generality of the proposed framework.

Limitations & Future Work

  • Designing degradation operators requires domain knowledge (Poisson-Gaussian vs. Rician noise models).
  • Multi-step integration at inference is slower than discriminative approaches.
  • The convergence quality of the theoretical \(t \to +\infty\) limit in practice warrants further empirical validation.
  • The idea of composing relative flows offers inspiration for other learning scenarios lacking clean labels.
  • The CoT mathematical framework is generalizable to other progressive quality enhancement tasks.
  • The combination of modality-specific degradation modeling with general-purpose flow matching presents a paradigm worth broader adoption.

Rating

8/10 — Combining theoretical rigor with strong empirical performance, this work represents a significant methodological contribution to the medical image denoising literature.