Offline handwritten mathematical expression recognition is often considered much harder than its online counterpart due to the absence of temporal information and the presence of background noise. In order to take advantage of the more developed techniques on online recognition and save resources, an oversegmentation approach is proposed to recover strokes from a textual bitmap image automatically. The proposed algorithm first break down the skeleton of a binarized image into junctions and segments, then segments are merged to form strokes, finally the ordering is determined by recursive projection and topological sort.Given a state-of-art online handwritten mathematical expression recognition system, the proposed procedure correctly recognized 58.22%, 65.65% and 65.05% of the offline formulas rendered from CROHME 2014, 2016 and 2019 respectively. Therefore, the effectiveness of stroke extraction to offline recognition is justified.

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In this paper, we consider a vector-valued one-phase Bernoullitype free boundary problem on a metric measure space (X, d, µ) with Riemannian curvature-dimension condition RCD(K, N ). We first prove the existence and the local Lipschitz regularity of the solutions, provided that the space X is non-collapsed, i.e. µ is the N -dimensional Hausdorff measure of X. And then we show that the free boundary of the solutions is an (N −1)-dimensional topological manifold away from a relatively closed subset of Hausdorff dimension N − 3. Contents 1. Introduction 1.1. The Bernoulli-type free boundary problems on Euclidean spaces 1.2. Free boundary problems in RCD-spaces and the main results 2. Preliminaries 2.1. RCD(K, N ) metric measure spaces and their calculus 2.2. Non-collapsed RCD(K, N ) metric measure spaces 2.3. Sets of finite perimeter and the reduced boundary 3. Existence of a minimizer 4. Hölder continuity of local minimizers 5. Lipschitz continuity of local minimizers 5.1. Mean value inequality 5.2. Lipschitz continuity of local minimizers of J Q 6. Local finiteness of perimeter for the free boundary 6.1. Nondegeneracy 6.2. Density estimates near the free boundary 6.3. Local finiteness of perimeter 7. Compactness and the Euler-Lagrange equation 8. Regularity of the free boundary Appendix A. Weiss-type monotonicity on cones References