A computational method for semantic image segmentation with distributional uncertainty estimation is proposed based on representing the prediction as a Dirichlet distribution field. Unlike approaches that require multiple stochastic inference runs (MC dropout) or averaging over an ensemble of independent models, the method computes uncertainty maps in closed form based on the Dirichlet field parameters predicted in a single forward pass of the neural network. The method is formulated as the minimization of a composite functional including the expected logarithmic loss function (expected log-loss), KL regularization for controlling the distribution concentration, and spatial smoothing that takes into account local image intensity variations (edge-aware). For fixed smooth fields, the asymptotic discretization accuracy of the spatial regularizers used is established: the discrete Dirichlet energy approximates the corresponding continuous integral with a first-order error over the grid step. Additionally, a formal decomposition of the overall uncertainty into epistemic and data-supported components was introduced, which can be used in further analysis of the method's behavior and the development of extensions. Computational experiments were performed on three medical image datasets (ACDC, Synapse, CHAOS) with 10 independent initializations. In the main comparison with the baseline model trained using cross-entropy, the differences are statistically significant across initializations on all datasets; for ACDC, significance at the patient level was further confirmed. The method improves segmentation quality and improves the calibration of probability estimates with an overhead of approximately 17 %. In the task of detecting pixel-level segmentation errors, the uncertainty map achieves an AUROC of 0.891.
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