# Self-supervised conformal prediction for uncertainty quantification in Poisson imaging problems

Bernardin Tamo Amougou, Marcelo Pereyra, Barbara Pascal

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Bernardin Tamo Amougou, Marcelo Pereyra, Barbara Pascal. Self-supervised conformal prediction for uncertainty quantification in Poisson imaging problems. IEEE Statistical Signal Processing Workshop, Jun 2025, Edinburg, Scotland, United Kingdom. hal-04979165v2

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# Self-supervised conformal prediction for uncertainty quantification in Poisson imaging problems

Bernardin Tamo Amougou

School of Mathematical and Computer Sciences
& Maxwell Institute for Mathematical Sciences

Heriot-Watt University, Edinburgh, UK
& Université de Paris Cité, Paris, France

bt2027@hw.ac.uk

Marcelo Pereyra

School of Mathematical and Computer Sciences
& Maxwell Institute for Mathematical Sciences

Heriot-Watt University, Edinburgh, UK

m.pereyra@hw.ac.uk

Barbara Pascal

Nantes Université, École Centrale Nantes

CNRS, LS2N, UMR 6004
F-44000 Nantes, France

barbara.pascal@cnrs.fr

**Abstract**—Image restoration problems are often ill-posed, leading to significant uncertainty in reconstructed images. Accurately quantifying this uncertainty is essential for the reliable interpretation of reconstructed images. However, image restoration methods often lack uncertainty quantification capabilities. Conformal prediction offers a rigorous framework to augment image restoration methods with accurate uncertainty quantification estimates, but it typically requires abundant ground truth data for calibration. This paper presents a self-supervised conformal prediction method for Poisson imaging problems which leverages Poisson Unbiased Risk Estimator to eliminate the need for ground truth data. The resulting self-calibrating conformal prediction approach is applicable to any Poisson linear imaging problem that is ill-conditioned, and is particularly effective when combined with modern self-supervised image restoration techniques trained directly on measurement data. The proposed method is demonstrated through numerical experiments on image denoising and deblurring; its performance are comparable to supervised conformal prediction methods relying on ground truth data.

**Index Terms**—Conformal Prediction, Uncertainty Quantification, Image Restoration, Stein's Unbiased Risk Estimator, Poisson noise.

## I. INTRODUCTION

Poisson imaging problems are widely encountered across various scientific and engineering disciplines, with notable prevalence in fields such as astronomy and microscopy [1], [2]; common examples include Poisson image denoising, deblurring and computed tomography [3]–[5]. Such problems are often not well-posed, meaning that there exists a wide range of possible solutions that are in agreement with the observed data<fnref n="1" />. Accounting for this inherent uncertainty is essential when using the reconstructed images as evidence for science or critical decision-making [6]. Unfortunately, most methods for solving Poisson imaging problems currently available are unable to accurately quantify the uncertainty in the delivered solutions.

Bayesian statistical inference strategies, which rely on prior knowledge about the likely solutions to the problem as encoded by a probability distribution, constitute the predominant approach to uncertainty quantification in imaging problems (see, e.g., [3], [7]–[9]). Bayesian techniques can deliver a wide range of inferences, and the most accurate solutions to Poisson imaging problems are currently obtained by using Bayesian imaging methods combining ideas from optimisation, deep learning and stochastic sampling [9]. However, even the most advanced Bayesian imaging strategies currently available struggle to provide accurate uncertainty quantification on structures larger than a few pixels in size [10]. This has stimulated research on other frameworks for performing uncertainty quantification in imaging inverse problems. Namely, bootstrapping and conformal prediction have

This work was supported by French National Research Agency (ANR) (ANR-23-CE48-0009, “OptiMoCSI”); and by UKRI Engineering and Physical Sciences Research Council (EPSRC) (EP/V006134/1, EP/Z534481/1).

recently emerged as promising frameworks for delivering accurate uncertainty quantification for large image structures.

With regards to bootstrapping, we note the recent equivariant bootstrapping method [11], a statistical resampling technique that leverages known symmetries of the problem to produce excellent confidence regions, even for large image structures and for problems with no ground truth available [11], [12]. Equivariant bootstrapping is effective in problems that are severely ill-posed, but it often performs poorly for problems which are ill-conditioned but not ill-posed [11].

In contrast, conformal prediction is a fully data-driven strategy for constructing regions with almost exact marginal coverage [13]. Conformal prediction is agnostic to both the image observation model and our prior knowledge about likely solutions. However, in its original form, it requires abundant ground truth data for calibration, which is a main bottleneck to implementation for quantitative and scientific imaging problems where ground truth data is often scarce or unavailable. The need for ground truth can be partially mitigated by combining conformal prediction with Bayesian inference and by focusing on single pixels rather than regions [14]. Alternatively, the recent method [15] fully bypasses the need for ground truth data by focusing on Gaussian imaging problems and leveraging Stein's Unbiased Risk Estimator [16] to perform conformal prediction directly from the observed measurement data, in a self-supervised manner, with accuracy comparable to supervised conformal prediction.

There are many important imaging problems for which the Gaussian noise assumption underpinning [15] is not sufficiently accurate to deliver meaningful inferences. In particular, imaging sensors often exhibit so-called shot noise, a form of signal-dependent measurement error that exhibits Poisson statistics. With quantitative and scientific Poisson imaging problems in mind, this paper proposes a generalisation of the self-supervised conformal prediction framework [15] for high-dimensional linear inverse problems involving Poisson noise.

The remainder of the paper is organised as follows. The principles of conformal prediction for inverse problems involving Poisson noise are introduced in Section II. Then, the proposed self-supervised conformal prediction framework is detailed in Section III. This framework is demonstrated on Poisson image denoising and deblurring experiments in Section IV. Concluding remarks and future research directions are provided in Section V.

## II. CONFORMAL PREDICTION UNDER POISSON MODELS

In the context of image estimation, conformal prediction aims to identify a set of statistically likely values for an unknown image $x^* \in \mathbb{R}^n$ from some noisy measurements $y \in \mathbb{R}^m$. Assuming $x^*$ and $y$ are realizations of random variables $X$ and $Y$, for a confidence level $\alpha \in (0, 1)$, we seek a region $C(Y) \subset \mathbb{R}^n$ verifying

$$P_{(X,Y)}(X \in C(Y)) \geq 1 - \alpha \quad (1)$$

<footnote n="1">¹A problem is well-posed if its solution is unique and stable w.r.t. data, ill-conditioned if the solution is not stable, ill-posed if solutions are not unique.</footnote>
where the probability is taken w.r.t. the joint distribution of $(X, Y)$. Conformal prediction is a general framework leveraging a non-conformity measure $s: \mathbb{R}^n \times \mathbb{R}^m \to \mathbb{R}$, which reflects how plausible is an image $x$ given measurements $y$, to provide such confidence sets [13]. Once chosen $s$, the set $C(y)$ is constructed by computing the top $(1 - \alpha)$-quantile $q_\alpha$ of the statistic $s(X, Y)$, i.e.,

$$C(y) := \{x \in \mathbb{R}^n \mid s(x, y) \le q_\alpha\} \quad \text{for all } y \in \mathbb{R}^m. \quad (2)$$

which, by construction, satisfies:

$$\mathbb{P}_{(X,Y)}(X \in C(Y)) = \mathbb{P}_{(X,Y)}(s(X, Y) \le q_{\alpha}) \ge 1 - \alpha, \quad (3)$$

for any confidence level $\alpha \in (0, 1)$. Provided that $s$ has been suitably designed and that enough samples $\{x_i, y_i\}_{i=1}^M$ under the joint distribution of $(X, Y)$ are available to calibrate $q_\alpha$, then (2) yields regions $C(y)$ that contain $x^*$ with the prescribed probability.

In this paper, we consider a linear Poisson, i.e., shot noise, observation model, in which $X$ and $Y$ are related by

$$(Y | X = x^*) \sim \mathcal{P}(\gamma A x^*),$$

where $\mathcal{P}$ denotes the Poisson distribution, $A$ is a linear measurement operator, and $\gamma > 0$ is related to the strength of the shot noise (the larger $\gamma$, the easier the image restoration problem). We henceforth assume $A$ and $\gamma$ are known, and that $A$ is full rank but possibly severely ill-conditioned. To design $C(y)$, we assume the availability of some estimator of $x^*$, henceforth denoted $\hat{x}(y)$.

To gain an intuition for (1), it is useful to consider a specific example. Suppose that $x^*$ is a high-resolution Positron Emission Tomography (PET) scan of an adult brain which is only accessible in practice through a noisy degraded measurement $y$. Measurements are processed by the estimator $\hat{x}(y)$ to yield an estimate of $x^*$. In this setting, $x^*$ can be viewed as a realization of the random variable $X$ describing the distribution of generic noise-free and resolution-perfect brain PET scans of adult individuals within a studied population. Though, rather than a point estimate, for diagnosis purpose one might be preferably interested in all plausible estimates of $x^*$, by definition constituting the set $C(y)$ of Equation (2). Then, if a large number of brain PET scans are considered, at least a proportion $1 - \alpha$ of the true images $x^*$ should fall in the plausible sets $C(y)$.

Split conformalisation is the most widely used implementation of conformal prediction. It consists in estimating the top $\frac{[(M+1)(1-\alpha)]}{M}$ quantile $\hat{Q}_\alpha$ of $\{s(x_i, y_i)\}_{i=1}^M$ from $M$ training samples $\{x_i, y_i\}_{i=1}^M$ of independent (or exchangeable) realizations of $(X, Y)$. Then, given a new measurement $Y_{\text{new}}$ stemming from some unknown image $X_{\text{new}}$, the set yielded by split conformal prediction is given by

$$\hat{C}(Y_{\text{new}}) := \{X_{\text{new}} \in \mathbb{R}^n \mid s(X_{\text{new}}, Y_{\text{new}}) \le \hat{Q}_\alpha\},$$

which satisfies the condition<fnref n="2" />:

$$\mathbb{P}_{(X,Y)^{M+1}}(X_{\text{new}} \in \hat{C}(Y_{\text{new}})) \ge 1 - \alpha, \quad (4)$$

where $(X, Y)^{M+1}$ denotes the joint distribution of the $M$ training samples and the new observation. We refer the reader to [13] for a comprehensive introduction to the conformal prediction framework.

One major advantage of the conformal prediction framework lies in its flexibility, notably in choosing the non-conformity measure. Different choices for $s$ yield infinitely many different regions $\hat{C}(y)$

satisfying (4). Though, in practice, not all these plausible sets are useful neither relevant. Indeed, as $\hat{C}(y)$ consists in a sub-level set of the function $x \mapsto s(x, y)$ in high-dimension, many of the plausible sets might be overly large, up to encompassing most of the support of $X$, and hence poorly informative to practitioners. Hence a careful design of $s$ is key to obtain compact, thus useful, sets. In particular, $s$ should be constructed so as to minimize the variability of $s(X, Y)$. To that aim, normalized non-conformity measures [17] of the form

$$s(x, y) = \|x - \hat{x}(y)\|_{\Lambda(y)}^2 = (x - \hat{x}(y))^{\top} \Lambda(y) (x - \hat{x}(y)), \quad (5)$$

where $\Lambda(y)$ is a positive definite matrix of size $n \times n$ are particularly useful, notably when $\Lambda(y)$ enables significant reduction of variability in $s(X, Y)$. An efficient strategy consists in choosing $\Lambda(y)$ close to the inverse covariance matrix of the estimation error $X - \hat{x}(Y)$ [17].

A major practical bottleneck in the implementation of conformal prediction strategy is the difficulty, if not the impossibility, to get an abundance of reliable data $\{s(x_i, y_i)\}_{i=1}^M$; indeed, obtaining the ground truth $x_i$ from measurements $y_i$ is precisely the overarching goal of an imaging problem. Also, relying on a training dataset might result in inaccurate prediction sets when a distribution shift occurs. In the example of brain PET scans, this corresponds to a situation in which the calibration dataset has been done on a population that represents poorly the population encountered during deployment.

To bypass the need for ground-truth data, Everink et al. [15] recently proposed a self-supervised conformal prediction method for linear imaging problems with Gaussian noise. This paper generalises this approach to perform conformalised uncertainty quantification in linear Poisson imaging problems without ground truth data available.

### III. SELF-SUPERVISED UNCERTAINTY QUANTIFICATION

In a manner akin to [15], we propose a self-supervised conformal prediction strategy that removes the need for ground-truth data by making use of a statistical estimator of (5). Namely, we leverage a Poisson Unbiased Risk Estimator (PURE) [18], [19], which provides accurate and unbiased estimates of (5). We apply this estimator individually to a collection of $M$ statistically exchangeable problems of the form $(Y | X = x_i^*) \sim \mathcal{P}(\gamma A x_i^*)$, where $\gamma$ and $A$ are known.

To define the non-conformity measure (5), we set $\Lambda(y) = A^{\top}A$ to approximate the error inverse-covariance, as we expect estimates of $x^*$ to be more accurate in the dominant eigenvector directions of $A^{\top}A$ while the estimation error is likely to concentrate along its weaker eigenvectors. The resulting score is given by:

$$s(x, y) = \frac{1}{m} \|Ax - A\hat{x}(y)\|_2^2, \quad (6)$$

where we recall that $m = \text{dim}(y)$ and that $\hat{x}(y)$ is an estimator of $x^*$. Note that $A$ being full rank is crucial, as otherwise, $\Lambda(y)$ would be only positive semidefinite leading to unbounded prediction sets.

To perform calibration without relying on ground-truth data, we replace the direct sample quantiles of $\{s(x_i, y_i)\}_{i=1}^M$ with PURE-based estimates derived solely from the observed measurements $\{y_i\}_{i=1}^M$. Assuming that the estimator $\hat{x}$ is differentiable almost everywhere, and let $\partial(A\hat{x}(y))/\partial y$ denote the diagonal of the Jacobian of $y \mapsto A\hat{x}(y)$, PURE for (6) takes the form

$$\text{PURE}(y) = \frac{1}{m} \left\| \frac{y}{\gamma} - A\hat{x}(y) \right\|_2^2 - \frac{1}{\gamma^2 m} \mathbf{1}^{\top} y + \frac{2}{\gamma m} y^{\top} \frac{\partial}{\partial y}(A\hat{x}(y)). \quad (7)$$

Note that (7) is evaluated directly from $y$, unlike (6).

Following a law of large numbers argument, in settings with large $m = \text{dim}(y)$, PURE not only remains unbiased but also exhibits a low variance (this is analysed analytically for the Gaussian case

<footnote n="2">²Note that condition (4) holds w.r.t. the joint distribution of $(X, Y)$, in contrast with Bayesian credibility regions which are derived from a similar condition but w.r.t. to the posterior probability of $(X|Y=y)$ for all $y \in \mathbb{R}^m$. Condition (4) is also weaker than frequentist confidence regions, which seek to satisfy the condition w.r.t. $(Y|X=x)$ for all $x \in \mathbb{R}^n$.</footnote>
in [20] and verified empirically in Poisson problems [21]). Consequently, we expect that calibrating conformal prediction by using PURE quantiles will yield prediction sets in close agreement with the true quantiles of $s(X, Y)$, thereby preserving the intended coverage properties. Using PURE as opposed to exact scores introduces a small bias in the resulting quantiles, but the effect is minor (the unbiased errors from PURE act as mild smoothing on the sample quantiles).

Adopting a leave-one out strategy, we compute $\hat{x}(y_i)$ and $\text{PURE}(y_i)$ for $i \in \{1, \dots, M\}$. Then, for any $\alpha \in (0, 1)$, we set

$$ \hat{C}(y_i) = \{ x \in \mathbb{R}^n : \|Ax - A\hat{x}(y_i)\|_2^2/m \le \hat{Q}_{\alpha}^{(i)} \}, $$

where $\hat{Q}_{\alpha}^{(i)}$ is the $\frac{[M(1-\alpha)]}{(M-1)}$-quantile of $\{\text{PURE}(y_j)\}_{j=1}^M$ with the $i$th term removed.

For most estimators $\hat{x}(Y)$ of practical interest, computing the PURE estimate typically involves a numerical approximation of $\frac{\partial}{\partial y}(A\hat{x}(y))$. Monte Carlo PURE (MC-PURE) [22] is perhaps the most widely used strategy to implement PURE, but we find that it requires careful hyper-parameter tuning. Instead, we recommend adopting a more robust alternative based on Hutchinson's stochastic trace approximation [16], which in the considered setting yields

$$ \frac{1}{m} y^{\top} \frac{\partial}{\partial y}(A\hat{x}(y)) = \mathbb{E}_{\tilde{n} \sim \mathcal{N}(0, \text{diag}(y))} \left[ \tilde{n}^{\top} \mathbf{J}_{h(y)} \tilde{n} \right] (h(y) = A\hat{x}(y)) \quad (8) $$

$$ \approx \frac{1}{m} \sum_{i=1}^{m} \tilde{n}_{i}^{\top} \mathbf{J}_{h(y)} \tilde{n}_{i} \quad (\text{Hutchinson's estimator}) \quad (9) $$

$$ \approx \frac{1}{m} \tilde{n}^{\top} \mathbf{J}_{\tilde{n}^{\top} h(y)}, \quad (10) $$

where the last step is efficiently computed via automatic differentiation [16]. In our experience, this approach provides accurate and computationally efficient estimates, even in large-scale settings.

Lastly, although PURE introduces some estimation bias, it is minor compared to likely distribution shifts in real deployment, which are a serious issue for supervised methods (being fully self-supervised provides robustness to distribution shifts). Furthermore, the required $\text{PURE}(y_i)$ values can be computed in parallel. The resulting algorithm is summarized in Algorithm 1.

### Algorithm 1 PURE-based conformal prediction

**Require:** Forward operator $A$, noise variance $\sigma^2$, estimator $\hat{x}$, measurement $y$, samples $\{y_i\}_{i=1}^M$, precision level $1 - \alpha \in (0, 1)$.

1: **for** $i \leftarrow 1$ to $M$ **do**

2: &nbsp;&nbsp;&nbsp;&nbsp; $S_i \leftarrow \text{PURE}(y_i)$ using (7) and (10)

3: **end for**

4: &nbsp;&nbsp;&nbsp;&nbsp; $\hat{Q}_{\alpha}^{(i)} \leftarrow \text{top} \frac{[M(1-\alpha)]}{(M-1)}$-quantile of $\{S_j\}_{\substack{j=1 \\ j \neq i}}^M$

5: &nbsp;&nbsp;&nbsp;&nbsp; $m \leftarrow \text{dim}(y)$

6: &nbsp;&nbsp;&nbsp;&nbsp; $\hat{C}(y) \leftarrow \{x \in \mathbb{R}^n \mid \|Ax - A\hat{x}(y)\|_2^2/m \le \hat{Q}_{\alpha}\}$

**Ensure:** Prediction set $\hat{C}(y)$

## IV. POISSON IMAGING EXPERIMENTS

### A. Experimental setup

We illustrate our proposed self-supervised conformal prediction framework on two canonical Poisson image denoising tasks: Poisson image denoising and non-blind Poisson image blurring. In each case, we employ Algorithm 1 to construct conformal prediction

<figure><img src="image_2.png" /><figcaption>Fig. 1: Image reconstruction results for image denoising (top) and non-blind image blurring (bottom), by using a self-supervised neural-network estimator $\hat{x}(y)$ and images from DIV2K.</figcaption></figure>

sets over a range of confidence levels $\alpha \in (0, 1)$, sampled from 0% to 100%. By design, each prediction set should contain the ground-truth solution with a probability close to $1 - \alpha$. To verify this in practice, we compute empirical coverage probabilities on a separate test set. Specifically, for each $\alpha$, we measure the proportion of images in the test set that lie within their corresponding prediction sets. In both experiments, the calibration sets are of size $M = 900$, which is sufficiently large to ensure that comparisons with supervised methods are fair; the test set is of size 200. (We chose this split between calibration and test set to facilitate comparisons with supervised alternatives; however, our self-supervised method can also be naturally implemented by using a leave-one-out strategy). With regards to the choice of $\hat{x}$, for both experiments we use a deep neural network estimator trained end-to-end in a self-supervised manner by using the PyTorch Deep Inverse library<fnref n="3" />. More precisely, we train $\hat{x}$ by using the PURE-based self-supervised training approach described in [22]. As a baseline for comparison, we also report results using a fully supervised conformal prediction technique implemented by using the same estimator $\hat{x}$ but trained in a supervised manner, so where ground-truth data are available both for training the estimator as well as for calibrating the conformal prediction sets.

### B. Image Poisson Denoising

We consider colour images of size $d = 256 \times 256$ pixels taking values in the hypercube $[0, 1]^{3d}$, obtained by cropping images from the DIV2K dataset [23], which we artificially corrupt by applying the considered Poisson observation model with $A = \mathbb{I}$ and $\gamma = 4$. As image restoration method, we use an unrolled proximal gradient network with 4 iterations and no weight-tying across iterations with a backbone denoiser set as the U-Net architecture with 2 scales. The training is done in a self-supervised manner by using the PURE loss [22]. The training data consists of 900 noisy measurements, we do not use any form of ground truth data neither for training nor for conformal prediction calibration. Fig. 1 (top) shows an example of a clean image, its noisy measurement, and the estimated reconstruction.

We then use these same noisy measurements to compute our proposed self-supervised conformal prediction sets, and assess their accuracy empirically by using 200 measurement-truth pairs from the test dataset. The results are reported in Fig. 2 above, together with

<footnote n="3">³https://deepinv.github.io/deepinv/</footnote>
<figure><img src="image_3.png" /><figcaption>Fig. 2: **Poisson image denoising experiment:** desired confidence level vs. empirical coverage; the proposed self-supervised conformal prediction method delivers prediction sets with near perfect coverage.</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Fig. 3: **Poisson image deblurring experiment:** desired confidence level vs. empirical coverage; the proposed self-supervised conformal prediction method delivers prediction sets with near perfect coverage.</figcaption></figure>

the results obtained by using the equivalent supervised conformal prediction approach that relies on ground truth data for calibration and the supervised version of the estimator. We observe that our method delivers prediction sets that are remarkably accurate and in close agreement with the results obtained by using supervised conformal prediction, demonstrating that the bias stemming from using PURE instead of ground truth data is negligible.

### C. Image Poisson Deblurring

We now consider a non-blind Poisson image deblurring experiment with grayscale images of size 256 × 256 pixels taking values in the hypercube $[0, 1]^d$, also derived from the DIV2K dataset [23] and artificially degraded with an isotropic Gaussian blur of standard deviation $\sigma = 2$ pixels, and Poisson noise with $\gamma = 60$. As our reconstruction network, we use the state-of-the-art SwinIR architecture [24], trained with the loss function proposed [25]. This loss leverages both scale-invariance and an unbiased risk estimator (PURE) to form a self-supervised training objective. Fig. 1 (bottom) depicts an example of a clean image, its noisy measurement, and its estimate $\hat{x}(y)$.

Again, we use 900 blurred and noisy images for training $y \mapsto \hat{x}(y)$ and for computing our proposed self-supervised conformal prediction sets. To evaluate their accuracy, we use 200 measurement–truth pairs from the test dataset. The corresponding results are presented in Figure 3, alongside those obtained using an analogous supervised conformal prediction approach that relies on ground truth data for training and calibration. As in the previous experiment, our method produces prediction sets that closely match those from the fully supervised approach, indicating that the bias introduced by using PURE in place of ground truth data is again negligible.

## V. DISCUSSION AND CONCLUSION

Leveraging the Poisson Unbiased Risk Estimator, a novel self-supervised conformal prediction method has been proposed. The method is useful for uncertainty quantification in Poisson imaging problems, a key class of imaging problems that were beyond the scope of existing self-supervised conformal prediction methodology [15]. Unlike standard split conformal prediction, the proposed method does not require any ground truth data for calibration, thus significantly extending the application scope of conformal uncertainty

quantification. In addition to bypassing the efforts associated with acquiring reliable ground truth data, self-calibration also provides robustness to distribution shift, a key issue when deploying conformal methods in populations that might not be well represented by the training data available. The proposed approach is particularly powerful when combined with modern self-supervised image reconstruction techniques, trained directly from measurements without the need for ground truth data. The effectiveness of the proposed self-supervised conformal prediction method was assessed through Poisson image denoising and Poisson image deblurring experiments with the DIV2K dataset, where it delivered highly accurate conformal prediction regions and performed on par with fully supervised alternatives.

Future work will seek to establish mathematical guarantees for the proposed method, with particular attention to the interplay between the number of considered imaging problems $M$, the dimensionality of $X$ and $Y$, the regularity of the estimator $\hat{x}$, and the number of projections used in the stochastic trace estimator within PURE. Furthermore, future work could also consider extensions of the proposed framework to other noise models from the exponential family, as well as to problems in which the parameters of the noise model are unknown [19]. For example, it would be interesting to tackle blind Poisson-Gaussian noise models, which are commonly encountered in microscopy [4].

Lastly, it would also be interesting to modify the proposed method for uncertainty quantification in time series analysis. For example in the context of epidemiology, when monitoring an ongoing epidemic such as COVID-19 [26]. State-of-the-art models for viral epidemics describe the number of new infections at a given time $t$ as a Poisson random variable whose parameter depends on previous counts and on a key unknown quantity, the so-called *reproduction number* $R_t$, which reflects the strength of the epidemic [27]. A major challenge in the estimation of $R_t$ during an ongoing pandemic is that available data are highly corrupted by administrative noise, making the estimation of $R_t$ an ill-conditioned inverse problem. A ambitious perspective for future research is to extend the proposed self-supervised PURE-based conformal prediction framework to the estimation of $R_t$ by leveraging the recently proposed Autoregressive Poisson Unbiased Risk Estimator [28]. A key difficulty in that case will be to obtain accurate PURE estimates from short epidemiological time series data.
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