# ENSURE: A General Approach for Unsupervised Training of Deep Image Reconstruction Algorithms

arXiv: https://arxiv.org/abs/2010.10631 · HTML: https://mlsanity.com/paper/2010.10631/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Oct 2020.

## Built on by (most cited first, 14 of 14)

- [Unsupervised MRI Reconstruction via Zero-Shot Learned Adversarial Transformers](https://mlsanity.com/paper/2105.08059/) (2021)
- [Consistent Diffusion Meets Tweedie: Training Exact Ambient Diffusion Models with Noisy Data](https://mlsanity.com/paper/2404.10177/) (2024)
- [GSURE-Based Diffusion Model Training with Corrupted Data](https://mlsanity.com/paper/2305.13128/) (2023)
- [SMRD: SURE-based Robust MRI Reconstruction with Diffusion Models](https://mlsanity.com/paper/2310.01799/) (2023)
- [Diffusion models for inverse problems](https://mlsanity.com/paper/2508.01975/) (2025)
- [Unsupervised reconstruction of accelerated cardiac cine MRI using Neural Fields](https://mlsanity.com/paper/2307.14363/) (2023)
- [Robust multi-coil MRI reconstruction via self-supervised denoising](https://mlsanity.com/paper/2411.12919/) (2024)
- [Benchmarking Self-Supervised Learning Methods for Accelerated MRI Reconstruction](https://mlsanity.com/paper/2502.14009/) (2025)
- [Unsupervised Detection of Distribution Shift in Inverse Problems using Diffusion Models](https://mlsanity.com/paper/2505.11482/) (2025)
- [Unsupervised Accelerated MRI Reconstruction via Ground-Truth-Free Flow Matching](https://mlsanity.com/paper/2502.17174/) (2025)
- [SVD-DIP: Overcoming the Overfitting Problem in DIP-based CT Reconstruction](https://mlsanity.com/paper/2303.15748/) (2023)
- [UPMRI: Unsupervised Parallel MRI Reconstruction via Projected Conditional Flow Matching](https://mlsanity.com/paper/2512.17493/) (2025)
- [Self-Supervised Uncertainty Estimation For Super-Resolution of Satellite Images](https://mlsanity.com/paper/2603.14074/) (2026)
- [Hankel Subspace Self-Supervised Learning for Parallel MRI Reconstruction](https://mlsanity.com/paper/2610.11560/) (2026)

## Abstract (arXiv)

Image reconstruction using deep learning algorithms offers improved reconstruction quality and lower reconstruction time than classical compressed sensing and model-based algorithms. Unfortunately, clean and fully sampled ground-truth data to train the deep networks is often unavailable in several applications, restricting the applicability of the above methods. We introduce a novel metric termed the ENsemble Stein's Unbiased Risk Estimate (ENSURE) framework, which can be used to train deep image reconstruction algorithms without fully sampled and noise-free images. The proposed framework is the generalization of the classical SURE and GSURE formulation to the setting where the images are sampled by different measurement operators, chosen randomly from a set. We evaluate the expectation of the GSURE loss functions over the sampling patterns to obtain the ENSURE loss function. We show that this loss is an unbiased estimate for the true mean-square error, which offers a better alternative to GSURE, which only offers an unbiased estimate for the projected error. Our experiments show that the networks trained with this loss function can offer reconstructions comparable to the supervised setting. While we demonstrate this framework in the context of MR image recovery, the ENSURE framework is generally applicable to arbitrary inverse problems.