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

## Abstract

Abstract Deep learning models often hallucinate, producing realistic artifacts that are not truly present in the sample. This can have dire consequences for scientific and medical inverse problems, such as MRI and microscopy denoising, where accuracy is more important than perceptual quality. Uncertainty quantification techniques, such as conformal prediction, can pinpoint outliers and provide guarantees for image regression tasks, improving reliability. However, existing methods utilize a linear constant scaling factor to calibrate uncertainty bounds, resulting in larger, less informative bounds. We propose QUTCC , a quantile uncertainty training and calibration technique that enables nonlinear, non-uniform scaling of quantile predictions to enable tighter uncertainty estimates. Using a U-Net architecture with a quantile embedding, QUTCC enables the prediction of the full conditional distribution of quantiles for the imaging task. During calibration, QUTCC generates uncertainty bounds by iteratively querying the network for upper and lower quantiles, progressively refining the bounds to obtain a tighter interval that captures the desired coverage. We evaluate our method on several denoising tasks as well as compressive MRI reconstruction. Our method successfully pinpoints hallucinations in image estimates and consistently achieves tighter uncertainty intervals than prior methods while maintaining the same statistical coverage.

## 1 Introduction

In recent years, deep learning models have become the dominant approach across many inverse problems, favored for their ability to learn powerful and complex priors from an abundance of data (Ongie et al., 2020; Alshardan et al., 2024; Barbastathis et al., 2019; Xue et al., 2019). However, these models are generally limited in their ability to represent uncertainty in their predictions, which has been a significant barrier to their use in scientific and medical applications, where identifying observations as out-of-distribution is crucial and the consequences of model hallucination can be severe (Begoli et al., 2019).

While significant progress has been made in estimating uncertainty in deep learning, many methods involve substantial computational demands, incorporate strong prior data assumptions, or may not provide formal statistical guarantees Psaros et al. (2023); Gal & Ghahramani (2016); Gal et al. (2017); Lakshminarayanan et al. (2017); Sun et al. (2024). These limitations have motivated research into conformal prediction methods for uncertainty quantification, which aim to overcome these challenges by offering more efficient and reliable approaches (Angelopoulos & Bates, 2021; Angelopoulos et al., 2021; Dewolf et al., 2023; Barber et al., 2023; Romano et al., 2019).

Figure: Figure 1: QUTCC Overview. a) Quantile Regression with Quantile Embedding: During training, a neural network with a quantile embedding predicts an image as a function of the measurement and quantile, $q$. The quantile embedding is randomly sampled ($q\in(0,1)$) and the value of $q$ determines the asymmetry of the pinball loss, enabling the model to learn a range of conditional quantiles. b) Conformal Calibration: During calibration, the predictive bounds ($q_{\text{lower}},q_{\text{upper}}$) are iteratively adjusted on a held-out dataset to satisfy the desired miscoverage level $\alpha$. c) Uncertainty Prediction on Unseen Data: At test time, the model is queried with $q_{\text{lower}}$, $q_{0.5}$, and $q_{upper}$ to produce the mean prediction and a corresponding pixel-wise uncertainty map. Querying the full range of quantile values enables the prediction of the pixel-wise distribution.
Refer to caption: x1.png

Specifically, recent methods apply the statistical rigor of conformal prediction to inverse problems and image regression tasks through multi-dimensional conformalized quantile regression (Angelopoulos et al., 2022; Romano et al., 2019; Koenker & Bassett Jr, 1978). This is typically done by modifying a neural network to output two additional predictions: a lower-bound and an upper-bound image, which together define a confidence interval for each pixel. A calibration step is then used to scale these bounds so that the interval captures the true intensity values with the desired coverage, resulting in pixel-wise intervals that are statistically guaranteed to contain the true values with a user-specified confidence level (Angelopoulos et al., 2022). Multi-dimensional conformalized quantile regression is simple to implement and computationally inexpensive, unlike ensemble methods, which often require training multiple models, while still providing formal coverage guarantees that are independent of the model choice or data distribution. This makes it particularly well-suited for scientific and medical imaging, which often has small datasets, and adaptive imaging, where inference speed is crucial (Ye et al., 2025).
However, prior methods use a linear pixel-wise scaling during conformal calibration to approximate a complex, nonlinear distribution, which leads to overly loose prediction intervals (Angelopoulos et al., 2022). Furthermore, existing methods learn fixed quantiles, which carry the risk of quantile crossing (He, 1997).

To address these challenges, we present Quantile Uncertainty Training and Conformal Calibration (QUTCC), a novel method for simultaneous quantile prediction and conformal calibration that enables efficient and accurate uncertainty quantification for imaging inverse problems (Fig [1](https://arxiv.org/html/2507.14760v1#S1.F1)). Building off past work in multi-quantile estimation (Rodrigues & Pereira, 2020; Tagasovska & Lopez-Paz, 2019), QUTCC uses a single neural network to estimate a distribution of quantiles. During the conformal calibration step, QUTCC applies a non-uniform, nonlinear scaling to the uncertainty bounds, compared to constant scaling used by prior methods. This results in smaller and potentially more informative uncertainty intervals. Additionally, because all quantiles are learned during training, QUTCC can query the full quantile range at inference time to construct a pixel-wise estimate of the underlying probability distribution. We evaluate QUTCC on several imaging inverse problems, including Gaussian/Poisson synthetic denoising, real noise denoising, and MRI upsampling. We show that QUTCC can highlight regions of hallucination and maintain smaller uncertainty interval sizes than prior methods, while satisfying the same statistical guarantees. Additionally, we show that QUTCC effectively approximates pixel-wise probability density functions, providing a richer representation of uncertainty that enables more informed downstream decision-making in scientific and medical imaging applications.

## 2 Related Work

### 2.1 Conformal Prediction

Conformal prediction is a robust statistical technique that constructs predictive intervals with a formal guarantee of statistical coverage (Angelopoulos & Bates, 2021; Correia et al., 2024; Romano et al., 2019). In general, conformal methods take heuristic or model-derived uncertainty estimates and refine them into statistically valid predictive intervals by leveraging a held-out calibration dataset. These intervals are constructed to satisfy a finite-sample marginal coverage guarantee at a user-specified level ($1-\alpha$), meaning that, with high probability, the true response lies within the predicted interval at least $(1-\alpha)\%$ of the time. For example, setting $\alpha=0.1$ yields prediction intervals that are guaranteed to contain the ground truth in at least $90\%$ of test cases, with violations occurring no more than $10\%$ of the time. Conformal prediction has become increasingly popular due to its simplicity, speed, formal guarantees, and lack of assumptions on model or data distribution (Angelopoulos et al., 2023). It has been applied to a variety of areas, including classification Angelopoulos et al. (2020); Ding et al. (2023); Angelopoulos et al. (2021), language modeling (Quach et al., 2023; Campos et al., 2024), robotics (Lindemann et al., 2023; Lekeufack et al., 2024), protein design (Fannjiang et al., 2022), and time series estimation (Xu & Xie, 2023; Zaffran et al., 2022). Recently, conformal prediction has been proposed for image-to-image regression tasks to enable pixel-wise uncertainty prediction (Angelopoulos et al., 2022), and thereafter applied to adaptive-microscopy (Ye et al., 2025). These methods alter a neural network to predict fixed upper and lower quantile estimates, in addition to the mean image, then uniformly scale these predictions by a constant factor using conformal risk control to ensure valid statistical coverage at test time (Angelopoulos et al., 2022). Instead of predicting fixed quantiles and uniformly scaling them during conformal calibration, we use one network to predict all possible conditional quantiles. This enables non-uniform, asymmetric scaling during conformal calibration and leads to tighter uncertainty intervals for image regression tasks and imaging inverse problems.

### 2.2 Quantile Regression

Quantile regression is a general approach to estimate the conditional quantiles of a target distribution rather than the mean of a response variable (Koenker & Bassett Jr, 1978; Koenker & Hallock, 2001). This is often accomplished by leveraging an asymmetric loss function, called pinball loss (Fig. [1](https://arxiv.org/html/2507.14760v1#S1.F1), Eq. [3](https://arxiv.org/html/2507.14760v1#S3.E3)), tailored to the specified quantile level (Steinwart & Christmann, 2011). The estimated intervals obtained by quantile regression do not have formal guarantees on their own, but can be paired with conformal prediction to obtain coverage guarantees (Romano et al., 2019). Learning quantiles during neural network training can improve predictive performance by introducing a regularization effect, while enabling uncertainty estimation (Rodrigues & Pereira, 2020). One limitation of quantile-based methods is the tendency for quantile crossing to occur, a phenomenon wherein quantiles trained independently violate their natural ordering, resulting in lower quantiles exceeding higher ones (Das et al., 2019). Training a single network to simultaneously predict multiple quantiles, called ‘simultaneous quantile prediction’ can mitigate this issue, while also enabling the estimation of the entire conditional distribution (Sangnier et al., 2016; Liu & Wu, 2011; Tagasovska & Lopez-Paz, 2019; Rodrigues & Pereira, 2020). In our work, we leverage a single-network with shared parameters for simultaneous quantile prediction. We embed the quantile level as an explicit input parameter into a U-Net, which is well-suited for a variety of image regression and imaging inverse tasks. Furthermore, we pair our network with conformal prediction to achieve coverage guarantees. We show that our network mitigates the issue of quantile crossing while also maintaining overall prediction accuracy. We are the first to demonstrate that a single network trained for simultaneous quantile prediction can predict conformally calibrated uncertainty intervals for imaging inverse problems.

### 2.3 Predicting Tighter or More Interpretable Bounds

Achieving smaller interval lengths without sacrificing coverage guarantees reflects increased confidence in the model’s predictions, leading to more precise and reliable uncertainty quantification. Producing tighter bounds is a common objective across uncertainty estimation methods, not just conformal prediction (Xie et al., 2024). Several approaches have been proposed to enhance conformal prediction by targeting user-specified properties such as reduced interval length or improved conditional coverage (Xie et al., 2024; Chung et al., 2021); however, to date, none of these techniques have been applied to imaging tasks. On the other hand, several methods aim to improve the interpretability of uncertainty prediction for imaging tasks by moving away from per-pixel uncertainty estimates. These methods leverage principal components, posterior projected distribution, and spatial/topological relationships (Nehme et al., 2023; Yair et al., 2024; Belhasin et al., 2023; Gupta et al., 2023) to predict uncertainty in a more interpretable way. However, without incorporating conformal prediction, these methods lack statistical guarantees. Several methods pair inverse problems with downstream tasks, such as classification, to estimate the uncertainty in a more interpretable way (Cheung et al., 2024; Wen et al., 2024), and others represent uncertainty in a semantically-meaningful latent space (Sankaranarayanan et al., 2022). While these methods are promising, they are less general and often tied to a specific application. We present a more general method that can predict uncertainty for any imaging inverse problem while achieving smaller uncertainty interval lengths than previous image-to-image regression methods.

## 3 Methodology

### 3.1 Problem Overview

Given an unseen measurement image $\mathbf{y}$, our goal is to train a network $f_{\theta}$ that can be queried at any quantile level $q\in(0,1)$ to produce an estimate $\mathbf{\hat{x}}=f_{\theta}(\mathbf{y},q)$(^1^11Please note that we adopt the notation $\hat{x}=f(y)$, which is commonly used in the field of inverse problems instead of $\hat{y}=f(x)$, which is more common in the machine learning literature.). The image estimate is obtained by querying the network at $q=0.5$, which gives us the median image:

$$ $\hat{x}=f_{\theta}(\mathbf{y},q=0.5).$ (1) $$

The uncertainty interval is obtained by querying the network at a high quantile value, $q_{\text{upper}}$, and a low quantile value, $q_{\text{lower}}$. The upper quantile prediction serves as the upper bound of the uncertainty interval, and the lower quantile serves as the lower bound. These two quantities can be subtracted to get a notion of pixel-wise uncertainty: $\hat{\mathbf{u}}=f_{\theta}(\mathbf{y},q_{\text{upper}})-f_{\theta}(\mathbf{y}
,q_{\text{lower}}).$

During conformal calibration, we use a small calibration dataset, $\mathcal{D}_{c}=\{(\mathbf{x}_{i},\mathbf{y}_{i})\}_{i=1}^{N_{c}}$, to search over different values of the input parameters $q_{upper}$ and $q_{lower}$ until we reach the desired coverage. That is, the constructed interval, $\mathcal{C}\in[f_{\theta}(\mathbf{y},q_{\text{lower}}),f_{\theta}(\mathbf{y},q
_{\text{upper}})]$, contains at least $1-\alpha$ of the ground truth pixels with probability $1-\delta$ (Angelopoulos et al., 2022). Formally, this means that with probability of at least $1-\delta$:

$$ $E[\mathbf{x}_{\text{test}}\in C(\mathbf{y}_{\text{test}})]\geq 1-\alpha$ (2) $$

where $\mathbf{x}_{\text{test}}$, $\mathbf{y}_{\text{test}}$ are unseen test data from the same distribution as the calibration data. After calibration, the network can provide a pixel-wise uncertainty map for new, unseen data or can be queried to predict a conditional distribution for any pixel in the image.

Our method, Quantile Uncertainty Training and Conformal Calibration (QUTCC), is summarized in Fig. [1](https://arxiv.org/html/2507.14760v1#S1.F1). We elaborate on our network architecture, training procedure, and conformal calibration step below.

### 3.2 Network Architecture

Our proposed network, $f_{\theta}(\mathbf{y},q)$, takes the parameter $q$ as an input to predict the conditional quantile. To do this in practice, we propose to use an attention U-Net (Ronneberger et al., 2015; Oktay et al., 2018) with a quantile-embedding to condition the U-Net on a given $q$. This is inspired by the architecture of U-Nets used for diffusion models, which include a time-embedding (Ho et al., 2020; Zhang et al., 2023). Specifically, we encode randomly sampled quantiles ($q\in(0,1)$) as high-dimensional vectors during training, allowing the network to learn the distribution of the target variable for each specified quantile. This allows us to query an image prediction at any quantile during a forward pass. The neural network weights are shared across different quantile predictions, limiting quantile crossing. The proposed network is shown in Fig. [1](https://arxiv.org/html/2507.14760v1#S1.F1), and the full architecture and training details are described in the Supplement.

### 3.3 Simultaneous quantile regression

In order to train our neural network to predict an arbitrary quantile image, we use pinball loss ($L_{q}$), an asymmetric loss function commonly used in quantile regression (Eq. [3](https://arxiv.org/html/2507.14760v1#S3.E3)), where $\hat{x}$ denotes the predicted value, $x$ represents the ground truth, and $q\in(0,1)$ is the quantile of interest:

$$ $L_{q}(x,\hat{x})=\begin{cases}q\cdot\lvert x-\hat{x}\rvert&\text{if }x-\hat{x} \geq 0\\ (1-q)\cdot\lvert x-\hat{x}\rvert&\text{otherwise}.\end{cases}$ (3) $$

When $q\neq 0.5$, the pinball loss introduces asymmetry by penalizing overestimates and underestimates unequally. Specifically, when $q>0.5$, the loss assigns a greater penalty to underestimations (i.e., when $\hat{x}<x$), encouraging the model to predict higher values. Conversely, for $q<0.5$, overestimations (i.e., when $\hat{x}>x$) incur a larger penalty, biasing predictions downward. This asymmetry enables the loss function to model conditional quantiles of the target distribution, in contrast to losses like mean squared error (MSE), which are symmetric and designed to estimate the conditional mean. At each training step, the quantile parameter, $q$ is randomly sampled and used to both condition the network and as an input to the loss function. This allows the model to learn from different parts of the conditional distribution. The total loss is given by:

$$ $\mathcal{L}_{\text{total}}(\theta)=\sum_{i=1}^{N_{t}}\mathcal{L}_{q}(x_{i},f_{ \theta}(y_{i},q)),$ (4) $$

where $f_{\theta}$ is a neural network with parameters $\theta$, and $f_{\theta}(y_{i},q)$ is the output of the neural network given an input measurement, $y_{i}$ and quantile value $q$. This loss is minimized with the Adam optimizer using backpropagation. During training the neural network learns a full conditional quantile function, rather than a discrete, fixed quantile value as in prior image-to-image regression methods (Angelopoulos et al., 2022; Ye et al., 2025).

### 3.4 Conformal calibration

After training, the neural network can be queried to obtain the uncertainty interval predictions. However, these predictions may not be valid.
To ensure the statistical coverage in Eq. [2](https://arxiv.org/html/2507.14760v1#S3.E2), a conformal calibration step is necessary. Following the procedure in (Angelopoulos et al., 2022), we use a small, separate calibration dataset, $\mathcal{D}_{c}=\{(\mathbf{x}_{i},\mathbf{y}_{i})\}_{i=1}^{N_{c}}$ to adjust $q_{\text{lower}}$ and $q_{\text{upper}}$ until the desired coverage, 1-$\alpha$, is reached.
Our model’s risk over the calibration dataset is the number of miscovered pixels for each image normalized by the total number of pixels, $K$, and the size of the calibration dataset:

$$ $\hat{R}(q_{\text{lower}},q_{\text{upper}})=\frac{1}{N_{c}}\sum_{i=1}^{N}\frac{ 1}{K}\sum_{k=1}^{K}\mathbbm{1}\{\mathbf{x}_{i}(k)\notin\mathcal{C}(\mathbf{y}_ {i}(k),q_{\text{lower}},q_{\text{upper}})\},$ (5) $$

where $k$ is the index of the pixels in the image. This measures the miscoverage as a function of $q_{\text{lower}}$ and $q_{\text{upper}}$. We can decompose this total risk as a function of the miscoverage from the lower quantile (the number of ground truth pixels that are lower than the lower quantile) and the miscoverage from the upper bound (the number of ground truth pixels that are higher than the upper quantile)

$$ $\hat{R}(q_{\text{lower}},q_{\text{upper}})=\frac{1}{N_{c}}\sum_{i=1}^{N}\frac{ 1}{K}\sum_{k=1}^{K}\left[\mathbbm{1}\left\{\mathbf{x}_{i}(k)<f_{\theta}( \mathbf{y}_{i},q_{\text{lower}})\right\}+\mathbbm{1}\left\{\mathbf{x}_{i}(k)>f _{\theta}(\mathbf{y}_{i},q_{\text{upper}})\right\}\right].$ (6) $$

During conformal calibration, we calibrate the lower and upper quantile bounds independently. To satisfy a target total miscoverage rate of $\alpha$, the calibration process allocates half of this error budget to each bound. That is, the lower bound is adjusted to capture violations at a rate no greater than $\alpha/2$, and similarly for the upper bound. Thus, if $\mathbf{x}(k)$ denotes the ground truth at pixel $k$ and $[\hat{\mathbf{x}}_{lower}(k),\hat{\mathbf{x}}_{upper}(k)]$ the predicted interval, the goal is to ensure:

$$ $\mathbb{P}(\mathbf{x}(k)<\hat{\mathbf{x}}_{\text{lower}}(k))\leq\frac{\alpha}{ 2}\quad\text{and}\quad\mathbb{P}(\mathbf{x}(k)>\hat{\mathbf{x}}_{\text{upper}} (k))\leq\frac{\alpha}{2},$ $$

which implies:

$$ $\mathbb{P}(\hat{\mathbf{x}}_{\text{lower}}(k)\leq\mathbf{x}\leq\hat{\mathbf{x} }_{\text{upper}}(k))\leq\alpha.$ $$

By allowing the quantile bounds to vary independently and adaptively during calibration, our method supports non-uniform scaling as a function of image characteristics. Rather than scaling the quantile predictions by a constant, linear factor, as proposed in prior work (Angelopoulos et al., 2022), our quantile predictions can be scaled in a non-uniform, non-linear way as a function of the neural network, $f_{\theta}(\mathbf{y},q_{\text{lower}},q_{\text{upper}})$. In practice, this can result in smaller uncertainty intervals.

Pseudocode for this calibration process is provided in Algorithm 1. At each step, we compute the miscoverage from the quantile upper and lower bounds over the entire calibration dataset. If the violation rate for a bound exceeds the adjusted $\alpha$, we relax the corresponding bound; otherwise, we tighten it. This process proceeds via a binary search over the quantile space until the desired coverage is reached. Note that we adjust the error rate, $\alpha^{\prime}\leftarrow\alpha-\frac{1-\alpha}{N_{c}}$ to account for the finite calibration dataset size (Angelopoulos & Bates, 2021; Vovk, 2012). At the end of this procedure, we can obtain a constructed interval $\mathcal{C}\in[f_{\theta}(\mathbf{y},q_{\text{lower}}^{*}),f_{\theta}(\mathbf{
y},q_{\text{upper}}^{*})]$ that satisfies Eq. [2](https://arxiv.org/html/2507.14760v1#S3.E2).

Figure: Algorithm 1 Calibrating Quantile Bounds $q_{\text{lower}},q_{\text{upper}}$

#### 3.4.1 Estimating the conditional distribution

Since our network is trained to predict the full conditional quantile distribution rather than a single fixed quantile, we can recover an estimate of the entire conditional quantile function, $\hat{Q}_{k}(q)$, at each pixel $k$. This is accomplished by querying the network over a range of quantile levels $q\in(0,1)$. Recall that the quantile function $\hat{Q}_{k}(q)$ is the inverse of the cumulative distribution function for the conditional distribution at pixel $k$.

By approximating the derivative of the quantile function with respect to $q$, we can obtain the rate of change of the predicted values across quantile levels. Since the quantile function is the inverse of the cumulative distribution function, the reciprocal of this derivative provides an estimate of the conditional probability density function (PDF) at each pixel. This derivative is numerically approximated using finite differences on the discrete set of quantile predictions:

$$ $\frac{\partial\hat{\mathbf{x}}}{\partial q}\Big{|}_{q_{i}}\approx\nabla_{q} \hat{\mathbf{x}}\Big{|}_{q_{i}}=\frac{\hat{\mathbf{x}}^{(q_{i+1})}-\hat{ \mathbf{x}}^{(q_{i-1})}}{q_{i+1}-q_{i-1}}\quad\text{for }i=2,\dots,n-1,$ (7) $$

where $\hat{\mathbf{x}}^{(q_{i})}$ denotes the image prediction at quantile level $q_{i}$. The estimated PDF at each pixel is then given by the reciprocal of this derivative $\hat{p}=\left(\frac{\partial\hat{\mathbf{x}}}{\partial q}\right)^{-1}$. The full PDF is constructed by querying QUTCC at a range of desired risk levels. The corresponding calibrated bounds act as the quantiles that form the PDF.

The network is first queried at all desired quantile levels to obtain the conditional quantile function estimates. Next, the derivative of this function is approximated using finite differences, and finally, the pixel-wise PDF is computed as the inverse of the derivative. This approach not only enables accurate pixel-wise PDF estimation but, when combined with conformal calibration of multiple bounds, also provides statistically guaranteed coverage of the estimated density.

## 4 Results

To evaluate our proposed approach, we fit and calibrate a separate model, $f_{\theta}$, to four separate imaging inverse problems: accelerated MRI (Zbontar et al., 2018), denoising under real-noise, synthetic Poisson, and Gaussian noise  (Zhang et al., 2019). We compare against Im2Im-UQ, which is the leading conformal prediction approach for image-to-image regression. To ensure that our performance improvements come from our uncertainty quantification technique and not network improvements, we upgrade Im2Im-UQ to use the same architecture and depth as QUTCC, which we call Im2Im-Deep. In all cases, we set $\alpha=0.1$ during conformal calibration. Full training details are provided in the supplement. We evaluate predicted interval lengths, empirical risk, and model performance. We also visualize uncertainty and highlight the model’s ability to identify hallucinations, which are realistic features not present in the ground truth. Finally, we show that QUTCC infers pixel-wise PDFs without relying on prior distributional assumptions.

### 4.1 Uncertainty interval length and risk

We compare the uncertainty interval lengths and total risk on each imaging modality in Fig. [2](https://arxiv.org/html/2507.14760v1#S4.F2) and Table [1](https://arxiv.org/html/2507.14760v1#S4.T1).

Figure: Figure 2: QUTCC exhibits smaller uncertainty interval sizes: We compare the predictive interval sizes of Im2Im-Deep and QUTCC across all four inverse tasks. QUTCC consistently produces narrower uncertainty intervals. The black bolded line indicates the mean interval length.
Refer to caption: x2.png

QUTCC consistently produces smaller prediction intervals than Im2Im-Deep across all four modalities, while keeping the total risk under 0.1.
These results suggest that the observed differences in uncertainty estimates are not only attributed to model depth, but rather to the specific design elements of QUTCC. By achieving smaller interval lengths while exhibiting comparable risk, QUTCC demonstrates that its uncertainty quantification is both more precise and well-calibrated, effectively capturing predictive confidence without sacrificing coverage.

**Table 1: Interval Length and Total Risk**
| Metric | Method | Gaussian | Poisson | Real Noise | MRI |
| --- | --- | --- | --- | --- | --- |
| Interval Length | Im2Im-Deep | 0.063 $\pm$ 0.050 | 0.047 $\pm$ 0.038 | 0.038 $\pm$ 0.045 | 0.110 $\pm$ 0.056 |
| QUTCC | 0.059 $\pm$ 0.048 | 0.040 $\pm$ 0.029 | 0.037 $\pm$ 0.035 | 0.108 $\pm$ 0.057 |  |
| Total-Risk | Im2Im-Deep | 0.094 $\pm$ 0.065 | 0.049 $\pm$ 0.042 | 0.096 $\pm$ 0.040 | 0.099 $\pm$ 0.047 |
| QUTCC | 0.091 $\pm$ 0.046 | 0.093 $\pm$ 0.091 | 0.098 $\pm$ 0.029 | 0.100 $\pm$ 0.033 |  |

Furthermore, we observe that QUTCC consistently produces narrower uncertainty intervals compared to Im2Im-Deep, particularly in regions of high signal intensity Fig. [3](https://arxiv.org/html/2507.14760v1#S4.F3). As the noise level ($\sigma$) increases from 0.1 to 0.5, the difference in uncertainty interval lengths between the two models diminishes gradually. Nevertheless, even at $\sigma=0.5$, QUTCC retains a notable advantage by yielding shorter uncertainty intervals for high-intensity pixels (intensity > 0.8).

Figure: Figure 3: QUTCC exhibits smaller uncertainty intervals in regions of high pixel intensity: In a Gaussian denoising setting, we analyze how uncertainty interval lengths vary with pixel intensity across increasing noise levels ($\sigma=0.1,\ 0.3,\ 0.5$). Under low-noise conditions, QUTCC exhibits substantially narrower uncertainty intervals compared to the baseline. As noise increases to $\sigma=0.5$, this advantage becomes less pronounced overall. However, QUTCC continues to produce significantly shorter uncertainty intervals for high-intensity pixels (intensity > 0.8), even at higher noise levels. All confidence intervals were estimated over a set of 10 samples.
Refer to caption: x3.png

### 4.2 Uncertainty Visualizations

Next, we visualize the predicted pixel-wise uncertainty of Im2Im-Deep and QUTCC for an undersampled MRI image and an image with real noise (Fig. [4](https://arxiv.org/html/2507.14760v1#S4.F4)). We compare these against the true error which we obtain by taking the difference between the model prediction and the ground truth. Both Im2Im-Deep and QUTCC display regions of high uncertainty that align with areas of high reconstruction error. However, QUTCC produces a more informative uncertainty map by selectively highlighting only the most uncertain regions, whereas Im2Im-Deep tends to assign broad, uniform areas of elevated uncertainty

Figure: Figure 4: Hallucination Visualization: We show pixel-wise QUTCC and Im2Im-Deep uncertainty quantification for MRI and Real-Noise tasks. In both imaging scenarios, both models highlight regions of high uncertainty that correspond to regions of high error. In the MRI task, the arrows point to a hallucination that appears in the Im2Im-Deep and QUTCC model predictions that is not present in the ground truth. QUTCC produces tighter uncertainty intervals that can better pinpoint uncertainty and hallucinations compared to Im2Im-Deep, which highlights a larger region.
Refer to caption: x4.png

In the undersampled MRI example shown in Fig. [4](https://arxiv.org/html/2507.14760v1#S4.F4), arrows in the model prediction, uncertainty, and error maps indicate a hallucinated structure that appears in both the Im2Im-Deep and QUTCC predictions. QUTCC is able to accurately localize this hallucination with its uncertainty prediction, but Im2Im-Deep exhibits broader, less specific uncertainty across the surrounding structure.

### 4.3 Capturing Probability Density Distribution

Previous uncertainty quantification methods for image-to-image regression tasks aim to approximate the underlying probability density function (PDF) of pixel-wise predictions, with varying degrees of success. Im2Im-Deep is limited in this regard, as it only predicts discrete upper and lower bounds. Bayesian approaches, while capable of producing PDFs, rely on strong distribution assumptions, which may not hold in practice. Ensemble-based methods can approximate the PDF more flexibly by aggregating outputs from multiple independently trained models; however, their accuracy depends heavily on the number and diversity of models in the ensemble. Even with large ensembles, there is no formal guarantee that the resulting distribution faithfully represents the true underlying PDF.
We demonstrate that querying multiple quantiles from QUTCC enables construction of a conformalized, pixel-wise PDF  [5](https://arxiv.org/html/2507.14760v1#S4.F5). To provide statistical coverage guarantees, we calibrate the model across a range of miscoverage levels $\alpha$. By systematically varying $\alpha$ (e.g., from 0.1 to 0.9) and recording the corresponding quantile bounds, we obtain a collection of confidence intervals that, when aggregated, approximate the full cumulative distribution function (CDF). Differentiating this CDF yields a conformalized pixel-wise PDF with formal coverage guarantees at each risk level. In Figure [5](https://arxiv.org/html/2507.14760v1#S4.F5), we illustrate QUTCC ’s ability to model diverse pixel-wise predictive distributions. The blue pixel displays a noticeably right-skewed distribution. The green pixel approximates a symmetric, Gaussian-like distribution and in contrast, the red exhibits a left-skewed distribution.

Figure: Figure 5: QUTCC predicts diverse pixel-wise distributions: In a Gaussian denoising task with $\sigma=0.1$, QUTCC models varying pixel-wise distributions by querying across quantiles. The resulting PDFs are conformalized, ensuring formal statistical coverage guarantees. Unlike prior methods that rely on strong parametric assumptions or cannot construct full predictive distributions, QUTCC offers a flexible, assumption-free approach. At the blue, green, and red pixels, the model captures a right-skewed distribution, a Gaussian-like distribution, and left-skewed, respectively.
Refer to caption: x5.png

## 5 Limitations and Conclusion

We propose QUTCC, a new uncertainty quantification method for imaging inverse problems that can achieve tighter uncertainty estimates than previous methods while maintaining the same statistical coverage. QUTCC accomplishes this by training a U-Net with a quantile embedding simultaneously on $q\in(0,1)$ quantiles and then dynamically adjusting its quantile bound predictions during calibration until the desired risk is satisfied. We validated QUTCC on four imaging inverse problems - undersampled MRI as well as denoising under Gaussian, Poisson, and real noise, comparing its performance against prior conformal methods for image-to-image regression tasks. Our method exhibited tighter uncertainty intervals, on average, while still pinpointing model hallucinations and regions of high error. This can be attributed to our model applying a nonlinear and asymmetrical scaling to its pixel-wise uncertainty predictions. While quantifying model uncertainty remains a significant open challenge in the field of deep learning, we believe that QUTCC offers a simple, yet robust method of uncertainty quantification for imaging inverse problems and image-to-image regression tasks.

One limitation of our work is the need for paired data for both the training and conformal calibration steps to predict uncertainty intervals. Paired data may not be available or limited in many scientific and medical applications. Some of this can be mitigated by using synthetic data for training and a small dataset of real paired data for conformal calibration. In addition, our method does not consider motion or 3D effects, which can be present in real samples. An interesting future direction would be to consider the effects of sample movement and distribution shifts, as well as uncertainty across multiple measurements instead of considering uncertainty for a single-frame independently.

## 6 Ethics and Reproducibility

Ethical Considerations: Using machine learning methods for scientific and medical applications has inherent risk - producing realistic artifacts that are not truly present in the image can be catastrophic for discovery and medical diagnostics. As research in uncertainty quantification matures, we hope that some of these risks can be mitigated to enable more trustworthy imaging.

Reproducibility:
We describe the different imaging inverse tasks in more detail in  [A.1](https://arxiv.org/html/2507.14760v1#A1.SS1). In this section, we also include the model epochs that we used to evaluate in this paper. Additionally explanation of QUTCC’s model architecture can be found in  [A.2](https://arxiv.org/html/2507.14760v1#A1.SS2), where we describe the quantile embedding in more detail. Code reproducing the experiments can be found in the GitHub repository.

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## Appendix A Appendix

### A.1 Experiment Details

Im2Im-Deep and QUTCC were each trained for 50 epochs on their respective datasets. Following training, we conducted a model selection sweep to identify the epoch that yielded the narrowest uncertainty intervals while satisfying a target risk level ($\alpha=0.1$).

#### A.1.1 Real Noise Task

For the real noise task, we used the Fluorescence Microscopy Denoising (FMD) dataset (Zhang et al., 2019), which contains experimentally acquired fluorescence microscopy images spanning 12 wide-field, confocal, and two-photon modalities. The model was trained on 10,000 images, with 500 confocal mouse images used for calibration and an additional 500 for validation. The epochs used for Im2Im-Deep and QUTCC are 5 and 10, respectively.

#### A.1.2 Gaussian and Poisson Noise Task

For both the Gaussian and Poisson tasks, we synthetically introduced varying levels of noise to the FMD ground truth images. The dataset was split into 180 ground truth images for training, 40 for calibration, and 20 for validation. The pseudocode for generating gaussian and poisson
noise are shown below [2](https://arxiv.org/html/2507.14760v1#alg2), [3](https://arxiv.org/html/2507.14760v1#alg3).

Figure: Algorithm 2 Add Gaussian Noise

For Gaussian noise, we set $\sigma_{\max}=0.5$. For Poisson noise, the noise level range $(\lambda_{\min},\lambda_{\max})$ was set to $(50,100)$.
In each iteration, with a batch size of 16, we generated 25 random noise levels uniformly sampled between the specified minimum and maximum values. The number of training epochs for Im2Im-Deep and QUTCC with Gaussian noise were 15 and 20, respectively. For Poisson noise, Im2Im-Deep was trained for 10 epochs, while QUTCC was trained for 35 epochs.

#### A.1.3 Magnetic Resonance Imaging (MRI) Task

Data used for the MRI task was obtained from the NYU fastMRI Initiative database (fastmri.med.nyu.edu) (Knoll et al., 2020; Zbontar et al., 2018). The primary goal of fastMRI is to test whether machine learning can aid in the reconstruction of medical MRI images. To train our models, we split the FastMRi dataset into 700 volumes for training, 200 volumes for calibration, and 200 for validation. It is important to note that a single volume contains multiple MRI images. The epochs used for Im2Im-Deep and QUTCC are 25 and 30, respectively.

To simulate the forward model in MRI, we start with a fully-sampled 3D volume composed of multiple 2D image slices. Each 2D image slice is transformed into its frequency domain representation using the 2D Fourier Transform, producing its k-space data. To simulate undersampled acquisition, we apply a 4x undersampling mask to the k-space. The resulting undersampled k-space is then transformed back into the image domain using the inverse Fourier Transform, yielding an aliased or artifact-corrupted image that serves as the input for the models.

### A.2 Model Architecture

Our method, QUTCC, is based on a U-Net backbone augmented with self-attention mechanisms, where quantile embeddings are propagated through the self-attention layers to guide the network’s quantile predictions. In our design, the target quantile level is embedded as a continuous scalar, analogous to the time-step embeddings in diffusion models. A core part of this architecture is the integration of self-attention layers within the U-Net, implemented as AttentionBlock modules. These blocks allow the model to capture global dependencies across spatial dimensions, enabling each pixel or feature location to attend to all other locations within its feature map. Specifically, attention layers are incorporated at various downsampling resolutions within both the encoder and decoder paths, as well as at the bottleneck of the U-Net.

The ability to condition the model’s output on a given quantile is achieved through the quantile embedding mechanism. A quantile value between (0, 1) is chosen for each image sample at each iteration and then transformed into a high-dimensional vector representation using a positional encoding scheme, employing sinusoidal functions to create generalizable embeddings. This initial embedding is then processed by a small multi-layer perceptron, which is then used throughout to condition the network. By introducing this conditioning, the model learns to generate outputs that are responsive to the entire range of quantile levels. This mechanism works in tandem with the self-attention layers, which are specified using the attention_resolutions parameter.

In our experiments, QUTCC was trained using attention_resolutions configured as $[16,8,4,2,1]$. This means for 512 x 512 images, using the specified configuration results in attention layers within the encoder path (at resolutions 512x512, 256x256, 128x128, 64x64, and 32x32), one central attention layer in the middle block (at 8x8 resolution), and additional attention layers distributed across the decoder path (at resolutions 32x32, 64x64, 128x128, 256x256, and 512x512). The attention_resolutions parameter dictates the spatial scales at which these attention mechanisms are introduced. Further model configurations can be found in models/model_config.yaml.

### A.3 Model Analysis

In Fig. [4](https://arxiv.org/html/2507.14760v1#S4), we restricted our comparisons to the Im2Im-Deep and QUTCC models. The original Im2Im-UQ model was excluded due to its comparatively shallow architecture, resulting in decreased performance. However, for the subsequent analysis, we reintroduce Im2Im-UQ for completeness. In this section, we assess the model’s mean predictive performance and the quantile crossing occurrences.

#### A.3.1 Mean Prediction Performance

Does QUTCC produce tighter intervals because it is simply a better image prediction network? To investigate this, we compare the predictive performance of Im2Im-UQ, Im2Im-Deep and QUTCC. In Table [2](https://arxiv.org/html/2507.14760v1#A1.T2), we compare the performance of Im2Im-Deep and QUTCC using standard image reconstruction metrics: MSE, SSIM, PSNR, and LPIPS. For QUTCC, the mean prediction was obtained by setting the quantile level to $q=0.5$. The results indicate that all models achieve nearly identical performance in terms of MSE, with only minor differences observed in SSIM, PSNR, and LPIPS. These variations are not substantial enough to suggest that QUTCC provides a significantly better mean prediction. These findings suggest that QUTCC ’s improved uncertainty quantification predictions are not attributed to better mean prediction performance. Rather, its ability to more effectively characterize uncertainty appears to come from the explicit learning of quantiles during training.

**Table 2: Image reconstruction performance of Im2Im-UQ, Im2Im-Deep, and QUTCC: For each metric, the arrow indicates the direction of better performance.**
| Metric | Model | Gaussian | Poisson | Real Noise | MRI |
| --- | --- | --- | --- | --- | --- |
| MSE ($\downarrow$) | Im2Im-UQ | 0.0007 $\pm$ 0.0006 | 0.0003 $\pm$ 0.0003 | 0.0030 $\pm$ 0.0004 | 0.003 $\pm$ 0.002 |
| Im2Im-Deep | 0.0006 $\pm$ 0.0006 | 0.0003 $\pm$ 0.0002 | 0.0004 $\pm$ 0.0002 | 0.001 $\pm$ 0.002 |  |
| QUTCC | 0.0006 $\pm$ 0.0005 | 0.0003 $\pm$ 0.0003 | 0.0002 $\pm$ 0.0001 | 0.001 $\pm$ 0.002 |  |
| SSIM ($\uparrow$) | Im2Im-UQ | 0.852 $\pm$ 0.102 | 0.931 $\pm$ 0.038 | 0.803 $\pm$ 0.016 | 0.668 $\pm$ 0.127 |
| Im2Im-Deep | 0.856 $\pm$ 0.107 | 0.937 $\pm$ 0.035 | 0.959 $\pm$ 0.006 | 0.707 $\pm$ 0.139 |  |
| QUTCC | 0.865 $\pm$ 0.102 | 0.941 $\pm$ 0.036 | 0.957 $\pm$ 0.008 | 0.708 $\pm$ 0.139 |  |
| PSNR ($\uparrow$) | Im2Im-UQ | 32.739 $\pm$ 3.535 | 36.163 $\pm$ 3.059 | 25.833 $\pm$ 0.734 | 26.867 $\pm$ 2.923 |
| Im2Im-Deep | 33.557 $\pm$ 4.018 | 37.062 $\pm$ 2.968 | 34.038 $\pm$ 1.837 | 29.711 $\pm$ 3.138 |  |
| QUTCC | 33.660 $\pm$ 4.143 | 37.498 $\pm$ 3.797 | 37.350 $\pm$ 1.936 | 29.833 $\pm$ 3.156 |  |
| LPIPS ($\downarrow$) | Im2Im-UQ | 0.420 $\pm$ 0.092 | 0.294 $\pm$ 0.071 | 0.360 $\pm$ 0.033 | 0.343 $\pm$ 0.033 |
| Im2Im-Deep | 0.414 $\pm$ 0.103 | 0.299 $\pm$ 0.071 | 0.297 $\pm$ 0.029 | 0.324 $\pm$ 0.043 |  |
| QUTCC | 0.408 $\pm$ 0.102 | 0.284 $\pm$ 0.072 | 0.312 $\pm$ 0.026 | 0.323 $\pm$ 0.040 |  |

#### A.3.2 Quantile Crossing Performance

To ensure the validity of the predicted quantiles, specifically to avoid the issue of quantile crossing, we quantified the number of quantile crossing occurrences between $q=[0.1,0.2,0.3,...,0.9,0.99]$ in QUTCC (Table [3](https://arxiv.org/html/2507.14760v1#A1.T3)).

**Table 3: Quantile crossing occurrences for QUTCC**
| Task | Crossed Pixels | Total Pixels | % Crossing |
| --- | --- | --- | --- |
| Gaussian | $242\,197$ | $1\,887\,436\,800$ | $0.000\,128\,320\,58$ |
| Poisson | $4620$ | $1\,887\,436\,800$ | $0.000\,002\,447\,764\,08$ |
| Real Noise | $39\,250$ | $1\,179\,648\,000$ | $0.000\,033\,272\,637\,3$ |
| MRI | 0 0 | $654\,336\,000$ | 0 0 |

Quantile crossing can undermine the interpretability of our uncertainty estimates, as it contradicts the notion that quantile functions should be non-decreasing/non-overlapping. The results indicate that across all four imaging tasks, the percentage of quantile crossing occurrences is minimal.

### A.4 Additional Uncertainty Interval Evaluations

#### A.4.1 Size-Stratified Risk

We observed the size-stratified risk of all four inverse tasks between Im2Im-Deep and QUTCC [6](https://arxiv.org/html/2507.14760v1#A1.F6). To calculate size-stratified risk, the prediction intervals are first binned into different sizes, ranging from smallest to largest. Then the risk is calculated across all the bins to ensure that the model’s uncertainty estimates are well-calibrated across different levels of confidence. While both Im2Im-Deep and QUTCC have bins that exceed the $\alpha$, generally, most bins fall under the chosen risk.

Figure: Figure 6: Size-Stratified Risk of Im2Im-Deep vs. QUTCC: We evaluate the size-stratified risk of Im2Im-Deep and QUTCC across four tasks- Poisson, Gaussian, Real Noise, and MRI. Overall, neither model exhibits a strong relationship between interval width and empirical risk, suggesting limited sensitivity to interval size. However, in the Gaussian and Poisson settings, both models display a mild trend toward improved calibration, or lower risk, for narrower prediction intervals.
Refer to caption: x6.png

#### A.4.2 Additional Visualizations

We also provide visualizations of the remaining imaging tasks not included in the main results (Fig. [7](https://arxiv.org/html/2507.14760v1#A1.F7)). For the two sample cases, one with a Real Noise measurement and the other with a Poisson measurement at $\lambda=60$, both QUTCC and Im2Im-Deep effectively highlight regions with high reconstruction error. However, QUTCC demonstrates a more focused identification of areas with high uncertainty that align closely with the true error, indicating its greater precision in uncertainty estimation. While Im2Im-Deep is capable of identifying regions of error, it tends to assign elevated uncertainty across larger portions of the sample, making it challenging for downstream tasks to prioritize regions based on uncertainty interval sizes. This trend is consistent across all four imaging tasks.

Figure: Figure 7: Additional Uncertainty Visualizations: We visualize both the full and zoomed-in regions of images reconstructed from Poisson and Real Noise measurements. Consistent with observations presented in the results section, QUTCC produces more precise uncertainty estimates that closely align with localized regions of high reconstruction error. In contrast, Im2Im-Deep tends to highlight broader regions of uncertainty and lacks specificity, making it hard to distinguish areas of importance. This highlights QUTCC ’s ability predict more informative uncertainty maps.
Refer to caption: x7.png

#### A.4.3 Bound Asymmetry

Figure: Figure 8: QUTCC produces narrower intervals through asymmetric bounds: We analyze the pixel-wise uncertainty bounds predicted by Im2Im-UQ and QUTCC and observe that QUTCC exhibits asymmetric behavior in its interval estimates. In Im2Im-UQ, both the upper and lower bounds are uniformly scaled by a global factor $\lambda$ to satisfy coverage constraints, which limits flexibility in adapting to signal-specific uncertainty. In contrast, QUTCC learns to predict quantiles directly, enabling it to independently modulate upper and lower bounds based on signal characteristics. This results in more adaptive and efficient uncertainty intervals. For instance, in the red boxed region, QUTCC matches Im2Im-UQ’s lower bound but produces a significantly tighter upper bound. Conversely, in the green boxed region, both models share an upper bound, yet QUTCC yields a tighter lower bound. Samples shown are Gaussian images with $\sigma=0.1$.
Refer to caption: x8.png

How does QUTCC achieve tighter confidence intervals while maintaining the same level of risk?
As shown in Fig. [8](https://arxiv.org/html/2507.14760v1#A1.F8), QUTCC produces asymmetric predictive intervals—its upper and lower bounds are adjusted independently based on localized uncertainty. In contrast, Im2Im-UQ applies a single global scaling factor $\lambda$ uniformly to both bounds, which can be suboptimal in cases where only one side of the interval requires adjustment. This limitation often leads to unnecessarily widened intervals. In the red boxed region of Fig. [8](https://arxiv.org/html/2507.14760v1#A1.F8), both QUTCC and Im2Im-UQ share a similar lower bound, yet QUTCC predicts a significantly tighter upper bound. Similarly, in the green boxed region, both methods align on the upper bound, but QUTCC yields a tighter lower bound. These examples highlight QUTCC’s ability to adaptively adjust its interval predictions, leading to more efficient and precise interval estimates.

### A.5 Additional PDF results

We demonstrate various non-Gaussian pixel-wise probability density function (PDF) distributions in Section [4](https://arxiv.org/html/2507.14760v1#S4). This section presents additional PDF analysis results. In Fig. [9](https://arxiv.org/html/2507.14760v1#A1.F9)a, we sample high and low intensity pixels from a Gaussian image $\sigma=0.4$. Detailed views of the corresponding pixel-wise PDFs are presented in Fig.  [9](https://arxiv.org/html/2507.14760v1#A1.F9)b for the low-intensity pixel and Fig.  [9](https://arxiv.org/html/2507.14760v1#A1.F9)c for the high-intensity pixel.
Each pixel analysis displays two distinct PDFs: the blue PDF represents the quantile predictions from the uncalibrated model, while the green PDF represents the conformally calibrated quantiles that provide finite-sample statistical coverage guarantees. The conformal calibration procedure adjusts the quantile levels to ensure valid coverage properties. For instance, while the 25th and 75th percentiles theoretically provide $50\%$ coverage, conformal calibration determines that the 20.7th and 69.2nd percentiles are required to achieve statistically guaranteed $50\%$ coverage for this specific dataset and model.

Figure: Figure 9: QUTCC predicts different pixel-wise PDFs based on different siganl intensity a) Comparison of pixel-wise PDFs for representative low-intensity and high-intensity pixels in a Gaussian measurement ($\sigma=0.4$). b) Detailed view of the low-intensity pixel PDF, exhibiting a narrow, high-density distribution concentrated around few intensity values, indicating low predictive uncertainty. c) Detailed view of the high-intensity pixel PDF, showing a broader, lower-density distribution with increased spread, reflecting higher predictive uncertainty in bright image regions.
Refer to caption: x9.png

Figure: Figure 10: PDF broadens as noise increases: We observe the PDF of a single pixel under varying noise levels. At $\sigma=0.1$, the noise is low, and the PDF is compact. As the noise increases to $\sigma=0.3$ and $\sigma=0.5$, the PDF gradually broadens, while the mean prediction value remains relatively unchanged. This broadening occurs due to the increased uncertainty introduced by higher noise levels. QUTCC successfully predicts this increased uncertainty as the noise increases.
Refer to caption: x10.png

Additionally, Fig. [10](https://arxiv.org/html/2507.14760v1#A1.F10) illustrates the evolution of the pixel-wise PDF for a fixed pixel coordinate under varying Gaussian noise levels $\sigma\in\{0.1,0.3,0.5\}$. As the noise standard deviation increases, the PDF widens. This widening corresponds to increased epistemic uncertainty in the model’s predictions, as higher noise levels reduce the information content available for accurate pixel intensity estimation. This then directly widens prediction intervals to maintain the desired coverage guarantees.

## Peer Reviews

confidence: 5
contribution: 3
initial_rating: 8
presentation: 3
questions: Questions and suggestions for the authors:
1. Could the authors comment on the potential complementarity between multi-modal prediction frameworks (e.g., multi-head or mixture density networks) and quantile-based uncertainty estimation? It would be interesting to understand whether quantile conditioning could be integrated into architectures designed for multi-hypothesis prediction.
2. In Equation (8), shouldn’t the inequality be $\mathbb{P}(\\hat{\\mathbf{x}}\_{lower}(k)\\le\\mathbf{x}\\le\\hat{\\mathbf{x}}\_{upper}(k)) \\ge 1 - \\alpha$? Can you clarify this?
3. The calibration procedure in Algorithm 1 appears similar in spirit to the approach used in Im2Im. Could the proposed conformal calibration, in principle, have been implemented within that framework as well?
4. What practical insight or downstream utility is gained from the pixel-wise marginal distributions shown in Figure 5? As presented, these marginals appear difficult to interpret or use for actionable uncertainty quantification.
5. In Table 1, is the comparison between QUTCC around the same predictor? QUTCC reports median-based estimates, is Im2Im also doing that? or is it using mean prediction? Aligning the central statistic across methods is crucial and could affect the reported results.
6. How sensitive is the proposed model to the quantile sampling strategy used during training? For example, do uniform versus importance-weighted quantile samples impact calibration quality, coverage, or stability during optimization?
soundness: 4
strengths: The paper presents a principled and rigorous approach to uncertainty estimation in imaging inverse problems. Its key contribution—the quantile-conditioned U-Net trained with a pinball loss and paired with nonlinear conformal calibration—represents a natural yet nontrivial extension of prior conformalized quantile regression methods such as Im2Im. The methodology is conceptually elegant and statistically grounded, providing a unified framework for pixel-wise uncertainty quantification, coverage guarantees, and conditional density estimation. The paper is clearly written, with informative figures that effectively illustrate both the conceptual flow and empirical results. Experimental validation across multiple imaging modalities and noise models (MRI, QPI, Gaussian, Poisson, and real noise) convincingly demonstrates robustness and generality, highlighting the method’s potential significance for reliable deep imaging.
summary: The paper introduces QUTCC, a framework for uncertainty quantification in imaging inverse problems. It proposes a quantile-conditioned U-Net that predicts conditional quantiles as a function of a sampled quantile level, trained using a pinball loss akin to quantile regression. Similar to how diffusion models condition on timesteps, QUTCC conditions on quantiles, enabling the network to predict the full conditional distribution of image intensities. A nonlinear, non-uniform conformal calibration step then adjusts quantile bounds to guarantee statistical coverage while producing tighter uncertainty intervals. The model can be queried at test time to yield median predictions, pixel-wise confidence intervals, and even per-pixel probability density functions (PDFs). Across multiple imaging tasks—including denoising, MRI, and quantitative phase imaging—QUTCC achieves smaller, sharper uncertainty bounds and better hallucination detection compared to prior methods like Im2Im-UQ.
weaknesses: While the paper is technically sound and well presented, several aspects could be improved to strengthen its contribution. First, the related work section omits an important body of literature on multi-modal prediction and multi-hypothesis uncertainty estimation using multi-head or mixture-based networks (e.g., Learning in an Uncertain World: Representing Ambiguity Through Multiple Hypotheses by Rupprecht et al., Hierarchical Uncertainty Exploration via Feedforward Posterior Trees by Nehme et al.). These works share conceptual overlap with simultaneous quantile prediction and discussing them in the related work section can better contextualize the paper’s novelty. Second, the interpretation of pixel-wise marginal distributions raises conceptual concerns: modeling uncertainty independently for each pixel neglects spatial correlations that are fundamental to imaging tasks. It remains unclear how these marginal PDFs can be practically leveraged beyond visualization, and Figure 5 does not convincingly demonstrate their utility. Finally, the quantitative results in Table 1, while consistent, show only marginal improvements over the baseline, and stronger ablations or uncertainty-coverage tradeoff analyses would help substantiate the claimed performance gains.

---

confidence: 3
contribution: 3
initial_rating: 2
presentation: 4
questions: Are there any other benefits of estimating the full quantile function over a few fixed quantile values apart from obtaining tighter prediction intervals? 

Are there any differences in the model’s internal representations between using QUTCC vs. fixed quantiles?
soundness: 3
strengths: Novelty and relevance: While the idea of adding conditioning mechanisms to enable a quantile regression network to learn the full quantile function is not novel (see [1, 2]), its application for uncertainty prediction in inverse imaging problems is novel, relevant, and timely. 

Extensive experiments: The authors conduct a thorough evaluation of their proposed method on five different datasets, selecting an appropriate baseline for comparison.  

References: 
[1] Ostrovski, Georg, Will Dabney, and Rémi Munos. "Autoregressive quantile networks for generative modeling." International Conference on Machine Learning. PMLR, 2018.
[2] Dabney, Will, et al. "Implicit quantile networks for distributional reinforcement learning." International conference on machine learning. PMLR, 2018.
summary: The authors propose quantile uncertainty training and conformal calibration (QUTCC), a method for training quantile regression neural networks to estimate the full quantile function (as opposed to a fixed set of quantiles, e.g., 0.05 and 0.95), and how to obtain conformalized prediction bounds using this approach. While the proposed method is general, the authors focus on its application for image-to-image regression. Extensive experiments are conducted across five imaging inverse problems and compared against an appropriate baseline (Im2Img-UQ). The authors claim that the proposed method consistently estimates narrower uncertainty intervals compared to the baseline, while achieving the desired level of risk.
weaknesses: Major weaknesses: 
Insufficient improvements over baseline: My main criticism of this work is its minor (possibly zero?) improvements over the baseline Im2Im-UQ. The results shown in Table 1 and Figure 2 suggest nearly identical performance. Even in Figure 4, the selected outputs of the baseline and proposed methods are extremely similar. Consequently, it is difficult to justify that predicting the full quantile function versus 2 fixed quantile values (0.05 and 0.95) offers any real benefit. 

Minor weaknesses: 
Alternate architectures: The authors chose to fix the UNet architecture (ResNet-18 backbone) used for the experiments to ensure a fair comparison between the proposed method (QUTCC) and the baseline method (Im2Im-UQ). While I do think this is completely appropriate, it would be nice to see this comparison for a variety of architectures. Nevertheless, this is a very minor point, and I don’t believe it would substantially alter the main results of this work.

---

confidence: 5
contribution: 2
initial_rating: 2
presentation: 4
questions: 1. Why place a quantile embedding after each layer instead of picking a particular layer? An ablation study would be interesting to see. 
2. In the supplementary, QUTCC requires more training epochs. Does the QUTCC training regime (sampling q randomly) require more careful hyperparameter tuning compared to baselines?
3. The conditional PDF is estimated using finite differences. How sensitive is the shape of the resulting PDF to the number and spacing of the quantile queries used to build the CDF?
soundness: 2
strengths: 1. The manuscript is approachable and well written. The figures are clear.
2. The quantile embedding for image-to-image regression tasks is novel.
3. Estimating the conditional distribution and being able to mitigate quantile crossing offer practical utility.
summary: The authors propose a quantile embedding to enable nonlinear and non-uniform scaling of quantile predictions, thereby allowing for tighter pixel-based conformal interval lengths. The authors demonstrate their method on inverse problems using a U-Net, are able to estimate the full conditional distribution of quantiles for each image, and estimate the conditional CDF.
weaknesses: 1. The performance improvements compared to Im2Im-Deep (Table 1 and Table 2) are not entirely convincing and are marginal at best.
2. The crux of the method is the quantile embedding. However, the paper does not compare Im2Im-Deep on an equal playing field due to the different formulations (Im2Im-Deep uses symmetric, while QUTCC uses asymmetric). Specifically, it is unclear whether the performance gains are due to the asymmetric formulation, the quantile embedding, or both. It would make sense that the asymmetric formulation would perform better for higher pixel intensities, because NN models tend to accurately capture the more moderate/prevalent intensities well at the expense of less prevalent intensities. An apples-to-apples comparison between Im2Im-Deep with an asymmetric formulation would be beneficial to clarify this issue. I would be more than happy to increase my score with greater clarity and an additional baseline.
3. The manuscript would benefit from a more rigorous statement of the conformal guarantee.
4. The generation of the conformalized PDF is interesting, but it can be viewed as repeatedly applying the main calibration procedure at different $\alpha$ levels (brute force) rather than a fundamentally new technique.

---

confidence: 4
contribution: 3
initial_rating: 2
presentation: 2
questions: **Conformal Risk Control vs. Risk-Controlling Prediction Sets**

In its current formulation, there appears to be a disconnect between the guarantee mentioned in the text, the one written in Eq. (2), and the one provided by the algorithm. If I understand correctly:

* $x,y$ are high-dimensional objects, let us say $\mathbb{R}^d$ (this should be specified clearly in the text)
* Lines 183-185 state that the prediction set should "contain at least $1 - \alpha$ of the ground-truth pixels with probability $1- \delta$". This is the notion of risk-controlling prediction set of [1] (Definition 1)

$$\mathbb{P}[ \mathbb{E}[\frac{1}{d} \sum_{j \in [d]} \mathbb{I}[x_j \notin C_{\lambda}(y)_j]] \leq \alpha]\geq 1 - \delta.$$

* However, Eq. (2) requires the **entire** input to be contained at once in the prediction set, which is a stronger guarantee. In practice, Eqs. (5) and (6) refer to the risk-controlling notion, not Eq. (2).

* Finally, the algorithm implements a conformal risk control procedure that does not provide a high-probability claim, but in expectation only (Algorithm 1 does not depend on $\delta$). That is, the guarantee provided by conformal risk control is: 

$$\mathbb{E}[\frac{1}{d} \sum_{j \in [d]} \mathbb{I}[x_j \notin C_{\lambda}(y)_j]] \leq \alpha.$$

In fact, the submission uses the same procedure described in [2], Eq. (4). Note that to achieve the stronger notion of high-probability risk control, binary search would not be a valid algorithm, but other strategies, such as backtracking or fixed hypothesis testing might be used [3].

Could the authors clarify this apparent inconsistency?

**Multiple quantile estimation and quantile crossing**

The submission acknowledges that for the proposed method to be statistically valid, the risk must be monotonically non-increasing with the quantile level. This implies that no quantile crossing can happen, but the training procedure proposed in the submission does not prevent quantile crossing by design. In fact, Table 3 in the appendix shows that quantile crossing does happen (even if minimally). 

Experimentally, this does not seem to affect the validity of the calibration procedure, but it renders the guarantees of the proposed method vacuous in the general sense.

Could the authors propose technical ways to address this robustly? For example, see [4], who also propose to learn multiple quantiles simultaneously in order to estimate conformalized histograms of the posterior distribution (related to the goal of Sec. 3.4.1, although different in methodology and application).

Could the authors include a formal statement of the guarantees provided by the proposed method? This will improve clarity of presentation, even if the result might follow from standard properties of conformal prediction.

**Definition of the loss function**

In its current formulation, the loss in Eq. (4) does not faithfully represent the sampling of the quantile levels from a uniform distribution.

Could the authors state the loss function as an expectation over a population, clearly stating the distribution quantile levels are sampled from at training time?

**Estimating the conditional distribution**

It might be helpful to distinguish between the true quantile function, the empirical quantile function, and the estimated quantile from the trained network. Note that in the general sense, it is not true that the quantile function is the inverse of the CDF, as the latter might not be strictly increasing. 

I am not sure I understand what coverage guarantee the estimated PDF provides. Could the authors expand on this point and clearly state the guarantee(s)? Is this similar to the concept of conformal predictive distributions [5]?

As an aside, it might be worth mentioning existing Bayesian approaches that provide estimates of these distributions by design, with, for example, Gaussian processes.

**Experimental results**

Table 1 and Figure 2 provide weak evidence that the proposed method significantly provides shorter intervals compared to im2im, as all differences are well within one standard deviation of the results. The result in Figure 3 is more convincing. Could the authors provide the equivalent of Figure 2 stratified by pixel intensity? That might be a more compelling example of the advantages provided by the proposed method.

If the main advantage of the proposed method is to provide shorter interval lengths, it is important to compare with existing works that address this in imaging inverse problems [6,7].

I am not sure I follow the choice of training two different models, one for im2im and one for QUTCC. If I understand correctly, one could use im2im to calibrate the predictions of a network trained with multiple quantile regression as well. Could the authors expand on the choice of using different networks instead of keeping the network fixed and changing the calibration procedure only? That might provide a more straightforward comparison between the two methods.

---

**Minor comments**

* "Where accuracy is more important than perceptual quality" may be too broad in its current formulation. Image quality evaluation is a nuanced and complementary field of research, especially in medical applications. Uncertainty quantification is a well-motivated endeavor even without this comment.
* The emoji in the title seems unrelated to the contribution of the paper? I would consider removing unless it contributes to the submission.
* Typo in Eq. (8), $\geq 1-\alpha$?
* In its current formulation, does Algorithm 1 always terminate? Since the step size does not change, isn't there a chance the thresholds will start oscillating?
* The claim that no existing methods have addressed minimal interval length or conditional coverage in imaging inverse problems is too strong as there exist prior works that have proposed ways to tackle parts of these issues [6,8].
* The claim that prior works use linear scaling of the interval bounds might also be too broad, as several works have used additive parametrizations that do not scale intervals bounds multiplicatively, or different strategies such as masking.

---

**References**

[1] Angelopoulos et al. "Image-to-Image Regression with Distribution-Free Uncertainty
Quantification and Applications in Imaging" (2022).

[2] Angelopoulos et al. "Conformal Risk Control", (ICLR 2024).

[3] Angelopoulos et al. "Learn then Test: Calibrating Predictive Algorithms to Achieve Risk Control", (2022).

[4] Sesia and Romano "Conformal Prediction using Conditional Histograms", (2021).

[5] Vovk et al. "Conformal Predictive Distribution with Kernels" (2018).

[6] Teneggi et al. "How to Trust Your Diffusion Model: A Convex Optimization Approach to Conformal Risk Control" (2023).

[7] Belhasin et al. "Principal Uncertainty Quantification with Spatial Correlation for Image Restoration Problems" (2023).

[8] Fischer et al. "Subgroup-Specific Risk-Controlled Dose Estimation in Radiotherapy" (2024).
soundness: 2
strengths: * Uncertainty volume minimization in high-dimensional inverse problems is an important problem.
* Training a quantile-conditional network is an interesting idea.
* Estimating the full posterior density using quantile regression is novel in the context of imaging inverse problems.
summary: The submission focuses on conformal risk control for imaging inverse problems. The submission proposes to train a quantile-conditional neural network to jointly estimate all quantiles $q \in [0,1]$ of the pixel-wise conditional distribution of the ground-truth image given a noisy observation. With this network, the submission introduces an algorithm to estimate the conformalized quantile $\hat{q}$ that guarantees coverage (in the conformal risk control sense) of the ground-truth image.

Experiments compare the proposed method with one existing baseline method.
weaknesses: * Presentation of theory is hand-wavy.
* Experimental results provide limited evidence in support of the proposed method.

I will expand on a few questions below and I am looking forward to discussing with the authors!


Decision: Reject