Semantic Segmentation with Uncertainty Estimation Based on the Dirichlet Model and Anisotropic Regularization

   Introduction. In computational mathematics, variational methods for minimizing discrete energies are widely used for the regularization of ill-posed problems. However, standard discrete schemes often suffer from scale inconsistency: upon mesh refinement (h→0), weights depending on unnormalized jumps of the function degenerate, leading to trivialization
of the anisotropic properties of the limiting operator. In this paper, a computational method is proposed that solves this problem by parameterizing the Dirichlet distribution and employing rigorously justified anisotropic spatial regularization.

   Materials and Methods. The mathematical model is formulated as an optimization problem for a composite functional in the space of grid functions. The functional includes an expected logarithmic loss function, Kullback-Leibler regularization, and spatial regularizers of the weighted Dirichlet energy type. To ensure the structural consistency of the discrete operator, edge-aware weight functions are constructed strictly through normalized finite differences. The asymptotic behavior of the discrete energies is investigated using the apparatus of Γ-convergence.

   Results. The main theoretical result of the work is a mathematical proof of the Γ-convergence of a family of discrete anisotropic functionals to a non-trivial continuous limit in the L²(Ω) topology. The equicoercivity of the discrete energies is proven, guaranteeing the convergence of a sequence of almost minimizers to the solution of the continuous problem.

   Discussion. The use of normalized finite differences in constructing the weights restores dimensional homogeneity and ensures strict scale invariance of the discretization of non-local operators.

   Conclusion. The proposed method successfully links continuous variational formulations with discrete predictive models, providing a theoretically justified and computationally efficient tool (additional inference costs amount to 17–18 %) with controlled error.

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