\section{Related Work}\label{sec:sota}


\paragraph{Cell instance segmentation:}  
HoVer-Net~\cite{graham2019hover} established the dominant multi-decoder paradigm for nuclear instance segmentation by jointly predicting semantic masks, horizontal/vertical offset maps, and cell-type labels. Transformer-based adaptations such as CellViT~\cite{cellvit} and HistoNext~\cite{chen2025histonext} retain this multi-head structure while incorporating long-range contextual modelling to refine boundaries and improve classification accuracy, highlighting the effectiveness of combining semantic and detection cues for reliable cell delineation. In our previous work, DualU-Net~\cite{anglada2025dualunet} streamlines this design to only two decoders: a semantic segmentation head and a centroid regression head. The centroid head predicts a continuous Gaussian density map centred at each nucleus, constructed during training using a fixed standard deviation~\(\sigma\) that reflects the expected nucleus scale in the dataset \cite{xie2018microscopy}. At inference, instance segmentation is obtained by combining both decoder outputs through a marker-controlled watershed procedure. Local maxima are first extracted from the predicted Gaussian centroid map and used as instance markers. These markers are then propagated over the semantic segmentation mask using the watershed algorithm, yielding a partition of the foreground into individual cell instances.


\paragraph{Uncertainty estimation and calibration:}  
Predictive uncertainty in deep learning usually decomposes into \emph{aleatoric} uncertainty, arising from intrinsic ambiguity in the data, and \emph{epistemic} uncertainty, reflecting limited model knowledge or out-of-distribution behavior ~\cite{gal2017uncertainties}. Estimating both components simultaneously remains difficult in many tasks. Multi-pass methods such as MC Dropout (MCD) ~\cite{gal2016dropout} or deep ensembles (DE) ~\cite{lakshminarayanan2017deep} provide good approximations of epistemic uncertainty, with the latter shown to remain robust under distribution shift~\cite{ovadia2020can}, but they are computationally expensive for day-to-day diagnostic workflows and do not yield explicit aleatoric estimates. Probabilistic segmentation frameworks such as Probabilistic U-Net~\cite{kohl2018probunet} or PhiSeg~\cite{baumgartner2019phiseg} introduce latent sampling or generative priors and can capture ambiguity, yet they require multiple stochastic passes and are not well suited to densely packed nuclei. None of these approaches provide simple, closed-form estimates of both uncertainty types. Uncertainty has also been investigated for error prediction and active learning in biomedical imaging~\cite{tan2025uncert,Anglada-Rotger_2024_CVPR}, though most efforts remain in semantic or single-task settings.

Calibration is equally important, as cross-entropy-trained models often produce overconfident predictions. Post-hoc techniques such as temperature scaling~\cite{guo2017calibration} adjust confidence after training, while train-time strategies (e.g., MMCE~\cite{kumar2018trainable}, focal-loss variants~\cite{mukhoti2019focal} or BSCE-GRA~\cite{lin2025bscegra}) aim to regularize confidence throughout optimization. Despite these advances, calibrated and instance-aware uncertainty estimation for multi-task cell segmentation remains under-explored.

\paragraph{Evidential Deep Learning (EDL):}
EDL introduces a probabilistic view of classification in which the network does not output a single categorical distribution, but instead predicts the parameters of a istribution over categorical distributions. In a standard setting, a categorical likelihood for an input $x$ with class probabilities $\mathbf{p} = (p_1,\dots,p_K)$ is
$
p(y=k \mid \mathbf{p}) = p_k,
$
with $\mathbf{p}$ typically produced by a softmax layer. EDL generalizes this by placing a Dirichlet prior over $\mathbf{p}$. Following Sensoy et al.~\cite{sensoy2018evidential}, the network outputs non-negative evidence values $e_k$, which define concentration parameters $\alpha_k = e_k + 1$ of a Dirichlet distribution $D(\mathbf{p}\mid\boldsymbol{\alpha})$. The predictive probabilities are given by the Dirichlet mean (see Section \ref{sec:methods}). The Dirichlet formulation allows uncertainty to be read directly from the predicted parameters $\boldsymbol{\alpha}$. The total evidence $S=\sum_k \alpha_k$ reflects how strongly the model supports its prediction: when $S$ is small, the Dirichlet distribution is broad, indicating that the model has not accumulated enough evidence to commit to any class. This behaviour is captured by vacuity, which represents uncertainty due purely to a lack of support in the data. In contrast, the spread of the Dirichlet around its mean captures the remaining uncertainty and gives rise to analytic measures of aleatoric and epistemic uncertainty. All these quantities are obtained in closed form, allowing EDL to produce calibrated uncertainty estimates from a single forward pass without sampling or ensembles. Training encourages the model to increase evidence when predictions are correct and suppress it when they are wrong, preventing unwarranted confidence.

EDL has also been explored in semantic segmentation. In ~\cite{ancha2024icra} evidential models are applied to pixelwise OOD-aware segmentation. EDL has been also used in several biomedical tasks, such as semantic segmentation~\cite{tan2024edlbiomed}, uncertainty-guided 3D mitochondria segmentation~\cite{jiang2024aaai}, interpretable evidential uncertainty supervision~\cite{li2025miccai}, or semi-supervised segmentation via mutual evidential learning~\cite{he2024bibm}. These works demonstrate growing interest in evidential segmentation, but they remain limited to single-task semantic settings: none provide interpretable uncertainty at the instance level, nor do they extend evidential modeling to multi-task formulations. Recent works have also critically examined the theoretical foundations of evidential deep learning, questioning whether Dirichlet-based uncertainty measures should be interpreted as faithful Bayesian epistemic and aleatoric uncertainty estimates~\cite{shen2024uncertainty}. In line with these findings, we treat the evidential outputs in this work as practically useful uncertainty proxies rather than strictly probabilistic quantities, and focus on their empirical ability to correlate with model errors at pixel and instance level.

