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A set of probabilities along with corresponding quantiles are often used to define predictive distributions or probabilistic forecasts. These quantile predictions offer easily interpreted uncertainty of an event, and quantiles are generally straightforward to estimate using standard statistical and machine learning methods. However, compared to a distribution defined by a probability density or cumulative distribution function, a set of quantiles has less distributional information. When given estimated quantiles, it may be desirable to estimate a fully defined continuous distribution function. Many researchers do so to make evaluation or ensemble modeling simpler. Most existing methods for fitting a distribution to quantiles lack accurate representation of the inherent uncertainty from quantile estimation or are limited in their applications. In this manuscript, we present a Gaussian process model, the quantile Gaussian process –based on established asymptotic results of quantile functions and sample quantiles– to construct a probability distribution given estimated quantiles. In some applications, the form of an unknown distribution function from which sample quantiles are drawn must be estimated, for which case we propose the use of a latent truncated Dirichlet process mixture model for estimation. A Bayesian application of the quantile Gaussian process is evaluated for parameter inference and distribution approximation in simulation studies as well as in a real data analysis of quantile forecasts from the 2023-24 US Centers for Disease Control collaborative flu forecasting initiative. The simulation studies and data analysis show that compared to other existing methods, the quantile Gaussian process leads to accurate inference on model parameters, estimation of a continuous distribution, and uncertainty quantification of sample quantiles. 

The numerical computation of shortest paths or geodesics on surfaces, along with the associated geodesic distance, has a wide range of applications. Compared to Euclidean distance computation, these tasks are more complex due to the influence of surface geometry on the behavior of shortest paths. This paper introduces a primal-dual level set method for computing geodesic distances. A key insight is that the underlying surface can be implicitly represented as a zero level set, allowing us to formulate a constraint minimization problem. We employ the primal-dual methodology, along with regularization and acceleration techniques, to develop our algorithm. This approach is robust, efficient, and easy to implement. We establish a convergence result for the high resolution PDE system, and numerical evidence suggests that the method converges to a geodesic in the limit of refinement. 

Abstract—Generating realistic synthetic microscopy images is critical for training deep learning models in label-scarce environments, such as cell counting with many cells per image. However, traditional domain adaptation methods often struggle to bridge the domain gap when synthetic images lack the complex textures and visual patterns of real samples. In this work, we adapt the Inversion-Based Style Transfer (InST) framework originally designed for artistic style transfer to biomedical microscopy images. Our method combines latent-space Adaptive Instance Normalization with stochastic inversion in a diffusion model to transfer the style from real fuorescence microscopy images to synthetic ones, while weakly preserving content structure. We evaluate the effectiveness of our InST-based synthetic dataset for downstream cell counting by pre-training and fne tuning EffcientNet-B0 models on various data sources, including real data, hard-coded synthetic data, and the public Cell200-s dataset. Models trained with our InST-synthesized images achieve up to 37% lower Mean Absolute Error (MAE) compared to models trained on hard-coded synthetic data, and a 52% reduction in MAE compared to models trained on Cell200-s (from 53.70 to 25.95 MAE). Notably, our approach also outperforms models trained on real data alone (25.95 vs. 27.74 MAE). Further improvements are achieved when combining InST-synthesized data with lightweight domain adaptation techniques such as DACS with CutMix. These findings demonstrate that InST-based style transfer most effectively reduces the domain gap between synthetic and real microscopy data. Our approach offers a scalable path for enhancing cell counting performance while minimizing manual labeling effort. The source code and resources are publicly available at: https://github.com/MohammadDehghan/ InST-Microscopy 

Abstract—Accurate cell counting is essential in various biomedical research and clinical applications, including cancer diagnosis, stem cell research, and immunology. Manual counting is labor-intensive and error-prone, motivating automation through deep learning techniques. However, training reliable deep learning models requires large amounts of high-quality annotated data, which is difficult and time-consuming to produce manually. Consequently, existing cell-counting datasets are often limited, frequently containing fewer than 500 images. In this work, we introduce a large-scale annotated dataset comprising 3,023 images from immunocytochemistry experiments related to cellular differentiation, containing over 430,000 manually annotated cell locations. The dataset presents significant challenges: high cell density, overlapping and morphologically diverse cells, a long-tailed distribution of cell count per image, and variation in staining protocols. We benchmark three categories of existing methods: regression-based, crowd-counting, and cell-counting techniques on a test set with cell counts ranging from 10 to 2,126 cells per image. We also evaluate how the Segment Anything Model (SAM) can be adapted for microscopy cell counting using only dot-annotated datasets. As a case study, we implement a density-map-based adaptation of SAM (SAM-Counter) and report a mean absolute error (MAE) of 22.12, which outperforms existing approaches (second-best MAE of 27.46). Our results underscore the value of the dataset and the benchmarking framework for driving progress in automated cell counting and provide a robust foundation for future research and development. 

Cell counting in biomedical imaging is pivotal for various clinical applications, yet the interpretability of deep learning models in this domain remains a signifcant challenge. We propose a novel prototype-based method for interpretable cell counting via density map estimation. Our approach integrates a prototype layer into the density estimation network, enabling the model to learn representative visual patterns for both cells and background artifacts. The learned prototypes were evaluated through a survey of biologists, who confirmed the relevance of the visual patterns identified, further validating the interpretability of the model. By generating interpretations that highlight regions in the input image most similar to each prototype, our method offers a clear understanding of how the model identifies and counts cells. Extensive experiments on two public datasets demonstrate that the tour method achieves interpretability without compromising counting effectiveness. This work provides researchers and clinicians with a transparent and reliable tool for cell counting, potentially increasing trust and accelerating the adoption of deep learning in critical biomedical applications. Code is available at https://github.com/NRT-D4/CountXplain. Keywords: Cell Counting, Biomedical Imaging, Deep Learning, Interpretability, Density, Map Estimation