# Image-to-Image Regression with Distribution-Free Uncertainty Quantification and Applications in Imaging

arXiv: https://arxiv.org/abs/2202.05265 · HTML: https://mlsanity.com/paper/2202.05265/

## Key facts

- Cited by 107 papers in the map (data through 14 Oct 2026).
- Builds on 31 of its 82 references that are in the map.
- First posted on arXiv 10 Feb 2022.

## Builds on (most cited first, 30 of 31)

- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Image-to-Image Translation with Conditional Adversarial Networks](https://mlsanity.com/paper/1611.07004/) (2016)
- [Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning](https://mlsanity.com/paper/1506.02142/) (2015)
- [Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles](https://mlsanity.com/paper/1612.01474/) (2016)
- [Perceptual Losses for Real-Time Style Transfer and Super-Resolution](https://mlsanity.com/paper/1603.08155/) (2016)
- [Photo-Realistic Single Image Super-Resolution Using a Generative Adversarial Network](https://mlsanity.com/paper/1609.04802/) (2016)
- [Attention U-Net: Learning Where to Look for the Pancreas](https://mlsanity.com/paper/1804.03999/) (2018)
- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [Image Super-Resolution Using Deep Convolutional Networks](https://mlsanity.com/paper/1501.00092/) (2014)
- [Colorful Image Colorization](https://mlsanity.com/paper/1603.08511/) (2016)
- [A Gentle Introduction to Conformal Prediction and Distribution-Free Uncertainty Quantification](https://mlsanity.com/paper/2107.07511/) (2021)
- [Unpaired Image-to-Image Translation using Cycle-Consistent Adversarial Networks](https://mlsanity.com/paper/1703.10593/) (2017)
- [Conformalized Quantile Regression](https://mlsanity.com/paper/1905.03222/) (2019)
- [fastMRI: An Open Dataset and Benchmarks for Accelerated MRI](https://mlsanity.com/paper/1811.08839/) (2018)
- [Deep Ensembles: A Loss Landscape Perspective](https://mlsanity.com/paper/1912.02757/) (2019)
- [Classification with Valid and Adaptive Coverage](https://mlsanity.com/paper/2006.02544/) (2020)
- [Uncertainty Sets for Image Classifiers using Conformal Prediction](https://mlsanity.com/paper/2009.14193/) (2020)
- [Distribution-Free, Risk-Controlling Prediction Sets](https://mlsanity.com/paper/2101.02703/) (2021)
- [Least Ambiguous Set-Valued Classifiers with Bounded Error Levels](https://mlsanity.com/paper/1609.00451/) (2016)
- [Learn then Test: Calibrating Predictive Algorithms to Achieve Risk Control](https://mlsanity.com/paper/2110.01052/) (2021)
- [Neural Style Transfer: A Review](https://mlsanity.com/paper/1705.04058/) (2017)
- [Uncertainty Estimates and Multi-Hypotheses Networks for Optical Flow](https://mlsanity.com/paper/1802.07095/) (2018)
- [Deep Learning on Image Denoising: An overview](https://mlsanity.com/paper/1912.13171/) (2019)
- [A Conformal Prediction Approach to Explore Functional Data](https://mlsanity.com/paper/1302.6452/) (2013)
- [Fair Conformal Predictors for Applications in Medical Imaging](https://mlsanity.com/paper/2109.04392/) (2021)
- [Improving Conditional Coverage via Orthogonal Quantile Regression](https://mlsanity.com/paper/2106.00394/) (2021)
- [Cautious Deep Learning](https://mlsanity.com/paper/1805.09460/) (2018)
- [Quantifying Point-Prediction Uncertainty in Neural Networks via Residual Estimation with an I/O Kernel](https://mlsanity.com/paper/1906.00588/) (2019)
- [Private Prediction Sets](https://mlsanity.com/paper/2102.06202/) (2021)
- [Interval Neural Networks: Uncertainty Scores](https://mlsanity.com/paper/2003.11566/) (2020)

## Built on by (most cited first, 30 of 107)

- [Confident Adaptive Language Modeling](https://mlsanity.com/paper/2207.07061/) (2022)
- [Conformal Risk Control](https://mlsanity.com/paper/2208.02814/) (2022)
- [Class-Conditional Conformal Prediction with Many Classes](https://mlsanity.com/paper/2306.09335/) (2023)
- [Improved Online Conformal Prediction via Strongly Adaptive Online Learning](https://mlsanity.com/paper/2302.07869/) (2023)
- [Training Uncertainty-Aware Classifiers with Conformalized Deep Learning](https://mlsanity.com/paper/2205.05878/) (2022)
- [Estimating Epistemic and Aleatoric Uncertainty with a Single Model](https://mlsanity.com/paper/2402.03478/) (2024)
- [How to Trust Your Diffusion Model: A Convex Optimization Approach to Conformal Risk Control](https://mlsanity.com/paper/2302.03791/) (2023)
- [Federated Conformal Predictors for Distributed Uncertainty Quantification](https://mlsanity.com/paper/2305.17564/) (2023)
- [Achieving Risk Control in Online Learning Settings](https://mlsanity.com/paper/2205.09095/) (2022)
- [Improving Trustworthiness of AI Disease Severity Rating in Medical Imaging with Ordinal Conformal Prediction Sets](https://mlsanity.com/paper/2207.02238/) (2022)
- [Conformal Prediction Sets Improve Human Decision Making](https://mlsanity.com/paper/2401.13744/) (2024)
- [Predictive Inference with Feature Conformal Prediction](https://mlsanity.com/paper/2210.00173/) (2022)
- [Conffusion: Confidence Intervals for Diffusion Models](https://mlsanity.com/paper/2211.09795/) (2022)
- [On the Posterior Distribution in Denoising: Application to Uncertainty Quantification](https://mlsanity.com/paper/2309.13598/) (2023)
- [Fast yet Safe: Early-Exiting with Risk Control](https://mlsanity.com/paper/2405.20915/) (2024)
- [Principal Uncertainty Quantification with Spatial Correlation for Image Restoration Problems](https://mlsanity.com/paper/2305.10124/) (2023)
- [Calibrated Uncertainty Quantification for Operator Learning via Conformal Prediction](https://mlsanity.com/paper/2402.01960/) (2024)
- [Conformal Prediction with Temporal Quantile Adjustments](https://mlsanity.com/paper/2205.09940/) (2022)
- [Conformal Contextual Robust Optimization](https://mlsanity.com/paper/2310.10003/) (2023)
- [What's in a Prior? Learned Proximal Networks for Inverse Problems](https://mlsanity.com/paper/2310.14344/) (2023)
- [Conformal Prediction: A Data Perspective](https://mlsanity.com/paper/2410.06494/) (2024)
- [Adaptive Bounding Box Uncertainties via Two-Step Conformal Prediction](https://mlsanity.com/paper/2403.07263/) (2024)
- [Generative Uncertainty in Diffusion Models](https://mlsanity.com/paper/2502.20946/) (2025)
- [Semantic uncertainty intervals for disentangled latent spaces](https://mlsanity.com/paper/2207.10074/) (2022)
- [Conformal Semantic Image Segmentation: Post-hoc Quantification of Predictive Uncertainty](https://mlsanity.com/paper/2405.05145/) (2024)
- [Integrative conformal p-values for powerful out-of-distribution testing with labeled outliers](https://mlsanity.com/paper/2208.11111/) (2022)
- [Equivariant Bootstrapping for Uncertainty Quantification in Imaging Inverse Problems](https://mlsanity.com/paper/2310.11838/) (2023)
- [Safe Perception-Based Control under Stochastic Sensor Uncertainty using Conformal Prediction](https://mlsanity.com/paper/2304.00194/) (2023)
- [Uncertainty Quantification via Neural Posterior Principal Components](https://mlsanity.com/paper/2309.15533/) (2023)
- [Fortuna: A Library for Uncertainty Quantification in Deep Learning](https://mlsanity.com/paper/2302.04019/) (2023)

## Abstract (arXiv)

Image-to-image regression is an important learning task, used frequently in biological imaging. Current algorithms, however, do not generally offer statistical guarantees that protect against a model's mistakes and hallucinations. To address this, we develop uncertainty quantification techniques with rigorous statistical guarantees for image-to-image regression problems. In particular, we show how to derive uncertainty intervals around each pixel that are guaranteed to contain the true value with a user-specified confidence probability. Our methods work in conjunction with any base machine learning model, such as a neural network, and endow it with formal mathematical guarantees -- regardless of the true unknown data distribution or choice of model. Furthermore, they are simple to implement and computationally inexpensive. We evaluate our procedure on three image-to-image regression tasks: quantitative phase microscopy, accelerated magnetic resonance imaging, and super-resolution transmission electron microscopy of a Drosophila melanogaster brain.