⚡ electrical engineering

QUTCC: Quantile Uncertainty Training and Conformal Calibration for Imaging Inverse Problems

The paper introduces QUTCC, a novel framework combining quantile uncertainty training with spatially-adaptive conformal calibration to generate tighter, statistically valid uncertainty intervals for high-dimensional imaging inverse problems while effectively identifying model hallucinations.

Original authors: Cassandra Tong Ye, Shamus Li, Tyler King, Kristina Monakhova

Published 2026-05-26

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CC BY 4.0

Original authors: Cassandra Tong Ye, Shamus Li, Tyler King, Kristina Monakhova

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a doctor looking at a medical scan, or a scientist examining a microscopic image. You have a powerful AI assistant that can clean up blurry photos or reconstruct missing parts of an image. The AI is amazing; it produces sharp, clear pictures. But here's the catch: AI can sometimes "hallucinate." It might invent a tumor that isn't there or smooth over a crack that actually exists. It's confident in its answer, but it's wrong.

The problem is, how do you know when to trust the AI and when to be suspicious?

This paper introduces a new tool called QUTCC (pronounced "cute-see") that acts like a smart, adjustable safety net for these AI images. Instead of just giving you a picture, it tells you exactly where the AI is guessing and how wide its "margin of error" is.

Here is how it works, using simple analogies:

1. The Old Way: The "One-Size-Fits-All" Raincoat

Previous methods tried to measure uncertainty by putting a giant, uniform raincoat over the entire image.

The Problem: If it's drizzling in the kitchen but pouring in the living room, a single raincoat size doesn't work well. You might be too wet in the living room or too covered up in the kitchen.
In AI terms: Old methods applied the same "error bar" to every pixel in an image. They didn't realize that some parts of an image are easy to predict (like a blank wall) while others are hard (like a complex, blurry texture). This resulted in uncertainty intervals that were either too wide (wasting space) or too narrow (missing errors).

2. The New Way: QUTCC's "Smart, Stretchy Suit"

QUTCC is different. Instead of a static raincoat, it learns to wear a custom-fit, stretchy suit that changes shape depending on the terrain.

How it learns: Imagine training a tailor who doesn't just learn to make a "medium" shirt. Instead, the tailor learns to make a shirt for every possible size, from extra-small to extra-large, all at once. In the paper, this is called Simultaneous Quantile Regression. The AI is trained to predict not just the "average" image, but the entire spectrum of possibilities (the "quantiles").
The "Conformal" Calibration: Once the tailor makes the suit, they test it on a group of people they haven't seen before (the calibration dataset). They adjust the suit's tightness until it fits perfectly 90% of the time.
The Magic: Because the suit is "stretchy" and learned from the data, QUTCC can make the uncertainty interval tiny where the image is clear and wide where the image is blurry or noisy. It adapts to the specific features of the image, pixel by pixel.

3. What Does It Actually Show You?

When you use QUTCC, you get three things:

The Image: The AI's best guess at what the picture looks like.
The "Wiggle Room" Map: A visual map showing where the AI is unsure. If the AI hallucinates a fake feature (like a ghostly object), QUTCC lights up that specific spot with a "high uncertainty" warning. It's like a radar that beeps loudly when the AI is making things up.
The "Possibility Cloud": Unlike other methods that just say "the value is between X and Y," QUTCC can show you the shape of the probability. It can tell you, "For this specific pixel, the AI thinks it's likely to be dark, but there's a small chance it could be bright." It does this without assuming the data follows a simple bell curve (Gaussian distribution), allowing it to handle weird, real-world noise.

4. Why Is This Better?

The authors tested QUTCC on five different imaging challenges, including:

Denoising: Cleaning up grainy photos.
MRI: Reconstructing fast, blurry magnetic resonance scans.
Microscopy: Seeing tiny cells clearly.

The Results:

Tighter Intervals: QUTCC produced uncertainty intervals that were smaller and more precise than previous methods. This means it gives you a clearer picture of where the AI is confident and where it isn't, without sacrificing safety.
Spotting Hallucinations: In tests, when the AI invented a fake feature, QUTCC correctly flagged that specific area as "high uncertainty," helping humans catch the mistake.
Speed: It achieves this high level of precision much faster than other advanced methods that require running the AI model dozens of times.

Summary

Think of QUTCC as a super-smart, self-adjusting ruler for AI images. It doesn't just tell you the answer; it tells you exactly how much you can trust that answer, pixel by pixel. It learns to shrink the "error bars" where the image is clear and expand them where the image is tricky, ensuring that scientists and doctors can spot when the AI is hallucinating and avoid serious mistakes.

Technical Summary: QUTCC – Quantile Uncertainty Training and Conformal Calibration for Imaging Inverse Problems

Problem Statement

Deep learning has revolutionized scientific and medical imaging by enabling the extraction of meaningful signals from sparse, noisy, and underdetermined measurements (e.g., fluorescence microscopy, accelerated MRI, and black hole recovery). However, these models frequently produce "hallucinations"—predictions that appear compelling and high-resolution but do not coincide with reality. Identifying these failures is difficult, posing significant risks for downstream scientific and clinical applications.

While uncertainty estimation techniques like conformal prediction offer a solution by generating statistically valid error bars with finite-sample coverage guarantees, existing methods are ill-suited for high-dimensional image-valued problems. Current approaches (e.g., Im2Im, K-RCPS) typically rely on linear, constant, or group-wise scaling factors applied uniformly across an image. These methods fail to account for spatial correlations and local image content, resulting in uncertainty intervals that are larger than necessary to satisfy coverage requirements.

Methodology

The authors propose QUTCC (Quantile Uncertainty Training and Conformal Calibration), a framework designed to produce tighter, spatially-adaptive uncertainty intervals for imaging inverse problems. The method consists of three core components:

1. Simultaneous Quantile Regression with Quantile Embeddings

Instead of training separate models for different quantiles or predicting only a mean, QUTCC employs a single neural network (an Attention U-Net) conditioned on a quantile parameter $q \in (0, 1)$ .

Architecture: The network $f_\theta(y, q)$ takes a measurement $y$ and a quantile embedding $q$ as input. The quantile is embedded as a continuous scalar (similar to time embeddings in diffusion models) and propagated through self-attention layers.
Training: The model is trained using pinball loss (asymmetric loss), where the quantile parameter $q$ is randomly sampled from a uniform distribution $U(0, 1)$ at each step. This allows the network to learn the full conditional quantile distribution $f_\theta(y, q)$ rather than a fixed point estimate.
Output: The median image is obtained by querying at $q=0.5$ .

2. Spatially-Adaptive Conformal Calibration

Unlike prior methods that calibrate a global scalar $\lambda$ or a set of $K$ group-wise scalars, QUTCC calibrates the input quantile parameters ( $q_{lo}, q_{hi}$ ) of the neural network.

Process: Using a held-out calibration dataset, the method performs a binary search to find the optimal lower ( $\hat{q}_{lo}$ ) and upper ( $\hat{q}_{hi}$ ) quantile bounds that satisfy the desired miscoverage level $\alpha$ (split as $\alpha/2$ for each tail).
Mechanism: Because the network is non-linear and conditioned on $q$ , adjusting the input quantiles results in a continuous, non-linear, and spatially-adaptive scaling of the prediction intervals. The effective scaling factor varies per pixel based on local image features, unlike the uniform scaling of Im2Im or the fixed partition scaling of K-RCPS.
Guarantees: This process ensures pixel-marginal coverage guarantees (the true value falls within the interval with probability $1-\alpha$ ) without assuming a specific underlying distribution.

3. Conditional Probability Density Estimation

A unique capability of QUTCC is the ability to estimate the full conditional probability density function (PDF), $p(x|y)$ , for each pixel without distributional assumptions (e.g., Gaussian).

Reconstruction: By querying the network at multiple quantile levels $\{q_j\}$ , the authors reconstruct the quantile function.
PDF Estimation: The conditional PDF is estimated via the inverse of the derivative of the quantile function: $\hat{p}(x|y) \approx (\frac{\partial \hat{f}}{\partial q})^{-1}$ . This is computed using finite differences between adjacent quantile predictions.
Calibration: The PDF can be "conformalized" by calibrating multiple quantile pairs to ensure the reconstructed distribution maintains statistical coverage guarantees.

Key Contributions

Simultaneous Quantile Regression for Imaging: The introduction of a practical method using a U-Net with quantile conditioning to learn a full conditional quantile distribution for high-dimensional imaging inverse problems.
Spatially-Adaptive Calibration: A novel calibration strategy that leverages the learned non-linear function to produce pixel-wise adaptive scaling, resulting in significantly tighter uncertainty intervals compared to global or group-wise scaling methods.
Distribution-Free Conditional PDFs: A method to estimate and calibrate pixel-wise conditional probability density functions without Gaussian assumptions, allowing for the visualization of complex, unknown uncertainty structures.
Comprehensive Evaluation: Validation across five imaging inverse problems (synthetic Gaussian/Poisson denoising, real-noise denoising, accelerated MRI, and quantitative phase imaging), demonstrating state-of-the-art performance in interval tightness.

Results

The authors evaluated QUTCC against posterior-sampling baselines (MC-Dropout, Deep Ensembles) and conformal baselines (Im2Im, Im2Im-Asymm, K-RCPS) across five tasks.

Interval Tightness: QUTCC consistently produced the smallest mean uncertainty interval lengths among all conformal methods. In high-intensity regions (signal-rich areas), QUTCC intervals were significantly tighter than baselines, while maintaining comparable performance in low-intensity background regions.
Computational Efficiency: While posterior-sampling methods (MC-Dropout, Deep Ensembles) offered competitive interval lengths, they incurred substantial computational costs (e.g., 28–47 $\times$ slower inference for MC-Dropout, 10 $\times$ more training time for Deep Ensembles). QUTCC achieves similar or better tightness using a single model and only two forward passes (for bounds) or a few passes (for PDFs).
Hallucination Detection: Visualizations on MRI and microscopy data showed that high-uncertainty regions predicted by QUTCC correlated strongly with reconstruction errors and model hallucinations (features present in the prediction but absent in the ground truth). QUTCC provided more localized uncertainty maps than Im2Im.
PDF Accuracy: The estimated conditional PDFs correctly reflected noise characteristics (e.g., skewness in Poisson noise, symmetry in Gaussian noise) and broadened appropriately as noise levels increased.

Significance and Claims

The paper claims that QUTCC addresses a critical gap in trustworthy AI for scientific imaging by providing tighter, spatially-adaptive uncertainty intervals without sacrificing statistical validity.

Practicality: The method is fast and practical, requiring only a single trained model and minimal forward passes, making it suitable for real-world deployment where computational resources are limited.
Trustworthiness: By identifying regions of high uncertainty that coincide with hallucinations, the method aids scientists and doctors in recognizing when a model's prediction may be incorrect.
Flexibility: The ability to generate distribution-free conditional PDFs offers a new tool for probing the underlying uncertainty of imaging inverse problems, going beyond simple error bars to visualize the full spectrum of plausible solutions.

The authors acknowledge limitations, noting that QUTCC requires paired training/calibration data and currently offers only pixel-marginal coverage (not joint spatial guarantees). They also note that the method may struggle with rare events or out-of-distribution data where the model predicts low uncertainty despite high error. However, they posit that advances in uncertainty quantification like QUTCC are essential steps toward mitigating the risks of deploying deep learning in critical scientific and medical domains.