Noisier2Inverse: Self-Supervised Learning for Image Reconstruction with Correlated Noise

TL;DR

Noisier2Inverse addresses inverse problems corrupted by spatially correlated noise by proposing a one-step, self-supervised reconstruction method that operates in measurement space. It trains a network using noisier data $y+\eta$ and a fixed initial map $B^{\sharp}$ to minimize a loss that targets $2Y-Z$, thereby avoiding ill-posed extrapolation. Theoretical results connect the surrogate loss to the unknown clean image and practical realizations yield a data-domain objective with flexible Sobolev variants. Empirical CT experiments show significant improvements over existing self-supervised baselines, including robustness to noise-model variations and effectiveness under sparse data, highlighting its potential for real-world projection imaging tasks.

Abstract

We propose Noisier2Inverse, a correction-free self-supervised deep learning approach for general inverse problems. The proposed method learns a reconstruction function without the need for ground truth samples and is applicable in cases where measurement noise is statistically correlated. This includes computed tomography, where detector imperfections or photon scattering create correlated noise patterns, as well as microscopy and seismic imaging, where physical interactions during measurement introduce dependencies in the noise structure. Similar to Noisier2Noise, a key step in our approach is the generation of noisier data from which the reconstruction network learns. However, unlike Noisier2Noise, the proposed loss function operates in measurement space and is trained to recover an extrapolated image instead of the original noisy one. This eliminates the need for an extrapolation step during inference, which would otherwise suffer from ill-posedness. We numerically demonstrate that our method clearly outperforms previous self-supervised approaches that account for correlated noise.

Figures (7)

• Figure 1: Noisier2Inverse for CT reconstruction: We first create noisy sinograms $y$ by adding correlated noise $\xi$ (white noise convolved with a Gaussian), where $\sigma$ is the correlation parameter of the noise and $\delta$ its standard deviation. At each training iteration, we then add additional noise to generate $z = y + \eta$ The initial reconstruction $B^\sharp(z)$ serves as the network input, and subsequently, the loss is computed and minimized in the data domain by applying the forward operator $A$ to the network output.
• Figure 2: Reconstruction results performing inference on the noisier data $y$ for the Walnut testing set (top) and the heart CT testing set (bottom).
• Figure 3: PSNR values for the different methods and for two different stopping criteria evaluated for correlation parameter $\sigma$ of the noise.
• Figure 4: PSNR and SSIM for different correlation parameters $\sigma$ for the Walnut dataset (top) and the Heart CT dataset (bottom). We display the mean metrics for 5 different correlation parameter $\sigma$, while the network was trained on only one of them (2.0, 3.0, or 5.0).
• Figure 5: Sparse data results. Top: Results obtained on sparse projection data consisting of 64 angular directions using the $y-$prediction method. We can clearly observe the benefit of the one-step approaches compared to the two-step method. Bottom: PSNR and SSIM for the different methods obtained on sparse data examples (number of projection consisting of measurements from 64 different angular directions.

Theorems & Definitions (5)

• Theorem 2.1: Expected Prediction Error
• proof
• Remark 2.2: Regularization via early stopping
• Remark 2.3: Sobolev loss
• Remark 2.4: One-step Noisier2Noise