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1. Automatic and Robust Skull Registration Based on Discrete Uniformization

   https://doi.org/10.1109/ICCV.2019.00052  

   Zhao, Junli; Qi, Xin; Wen, Chengfeng; Lei, Na; Gu, Xianfeng. (January 2019, IEEE International Conference on Computer Vision workshops)

   Skull registration plays a fundamental role in forensic science and is crucial for craniofacial reconstruction. The complicated topology, lack of anatomical features, and low quality reconstructed mesh make skull registration challenging. In this work, we propose an automatic skull registration method based on the discrete uniformization theory, which can handle complicated topologies and is robust to low quality meshes. We apply dynamic Yamabe flow to realize discrete uniformization, which modifies the mesh combinatorial structure during the flow and conformally maps the multiply connected skull surface onto a planar disk with circular holes. The 3D surfaces can be registered by matching their planar images using harmonic maps. This method is rigorous with theoretic guarantee, automatic without user intervention, and robust to low mesh quality. Our experimental results demonstrate the efficiency and efficacy of the method. 

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2. Barcode entropy of geodesic flows

   https://doi.org/10.4171/JEMS/1572  

   Ginzburg, Viktor L; Gürel, Başak Z; Mazzucchelli, Marco (December 2024, Journal of the European Mathematical Society)

   We introduce and study the barcode entropy for geodesic flows of closed Riemannian manifolds, which measures the exponential growth rate of the number of not-too-short bars in the Morse-theoretic barcode of the energy functional. We prove that the barcode entropy bounds from below the topological entropy of the geodesic flow and, conversely, bounds from above the topological entropy of any hyperbolic compact invariant set. As a consequence, for Riemannian metrics on surfaces, the barcode entropy is equal to the topological entropy. A key to the proofs and of independent interest is a crossing energy theorem for gradient flow lines of the energy functional. 

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3. ariaDNE: A robustly implemented algorithm for Dirichlet energy of the normal

   https://doi.org/10.1111/2041-210X.13148  

   Shan, Shan; Kovalsky, Shahar Z.; Winchester, Julie M.; Boyer, Doug M.; Daubechies, Ingrid; Cooper, ed., Natalie (February 2019, Methods in Ecology and Evolution)

   Abstract Shape characterizers are metrics that quantify aspects of the overall geometry of a three‐dimensional (3D) digital surface. When computed for biological objects, the values of a shape characterizer are largely independent of homology interpretations and often contain a strong ecological and functional signal. Thus, shape characterizers are useful for understanding evolutionary processes. Dirichlet normal energy (DNE) is a widely used shape characterizer in morphological studies.Recent studies found that DNE is sensitive to various procedures for preparing 3D mesh from raw scan data, raising concerns regarding comparability and objectivity when utilizing DNE in morphological research. We providearobustlyimplementedalgorithm for computing the Dirichlet energy of the normal (ariaDNE) on 3D meshes.We show through simulation that the effects of preparation‐related mesh surface attributes, such as triangle count, mesh representation, noise, smoothing and boundary triangles, are much more limited on ariaDNE than DNE. Furthermore, ariaDNE retains the potential of DNE for biological studies, illustrated by its effectiveness in differentiating species by dietary preferences.Use of ariaDNE can dramatically enhance the assessment of the ecological aspects of morphological variation by its stability under different 3D model acquisition methods and preparation procedure. Towards this goal, we provide scripts for computing ariaDNE and ariaDNE values for specimens used in previously published DNE analyses. 

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4. Quadrilateral Mesh Generation III : Optimizing Singularity Configuration Based on Abel-Jacobi Theory

   Zheng, Xiaopeng; Zhu, Yiming; Chen, Wei; Lei, Na; Luo, Zhongxuan; Gu, Xianfeng (April 2022, Computer methods in applied mechanics and engineering)

   null (Ed.)

   This work proposes a rigorous and practical algorithm for quad-mesh generation based the Abel-Jacobi theory of algebraic \textcolor{red}{curves}. We prove sufficient and necessary conditions for a flat metric with cone singularities to be compatible with a quad-mesh, in terms of the deck-transformation, then develop an algorithm based on the theorem. The algorithm has two stages: first, a meromorphic quartic differential is generated to induce a T-mesh; second, the edge lengths of the T-mesh are adjusted by solving a linear system to satisfy the deck transformation condition, which produces a quad-mesh. In the first stage, the algorithm pipeline can be summarized as follows: calculate the homology group; compute the holomorphic differential group; construct the period matrix of the surface and Jacobi variety; calculate the Abel-Jacobi map for a given divisor; optimize the divisor to satisfy the Abel-Jacobi condition by integer programming; compute \textcolor{red}{a} flat Riemannian metric with cone singularities at the divisor by Ricci flow; \textcolor{red}{isometrically} immerse the surface punctured at the divisor onto the complex plane and pull back the canonical holomorphic differential to the surface to obtain the meromorphic quartic differential; construct a motorcycle graph to generate a T-Mesh. In the second stage, the deck transformation constraints are formulated as a linear equation system of the edge lengths of the T-mesh. The solution provides a flat metric with integral deck transformations, which leads to the final quad-mesh. The proposed method is rigorous and practical. The T-mesh and quad-mesh results can be applied for constructing Splines directly. The efficiency and efficacy of the proposed algorithm are demonstrated by experimental results on surfaces with complicated topologies and geometries. 

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5. Multiscale Textured Mesh Substrates that Glide Alcohol Droplets and Impede Ice Nucleation

   https://doi.org/10.1002/adem.202101524  

   Bajpayee, Aayushi; Rivera-Gonzalez, Natalia; Braham, Erick_J; Alivio, Theodore_E_G; Anita; Alvi, Scheherzad; Li, Chenxuan; Cool, Nicholas; Al-Hashimi, Mohammed; Fang, Lei; et al (March 2022, Advanced Engineering Materials)

   Textured surfaces are commonly designed to preclude wetting by water. The design of surfaces that are not wetted by alcohols represents a considerable challenge given the low surface tension, viscosity, and density of these liquids. Herein, a hierarchically textured plastronic architecture that can suspend alcohol droplets in a metastable Cassie–Baxter regime is presented. As a result of microtexturation of the underlying stainless steel mesh, multiscale texturation derived from ZnO tetrapods, and surface functionalization with perfluorinated‐polyhedral oligomeric silsesquioxanes, the surfaces glide aliphatic alcohols, water, andn‐hexadecane. The design of surfaces not wetted by alcohols is particularly relevant to “point‐of‐care” environments. Because of the minimized interfacial contact areas, the textured surfaces further greatly inhibit ice nucleation at solid/liquid interfaces. High‐speed video imaging of the freezing and droplet impact shows that the textured surfaces delay ice nucleation by inhibiting heterogeneous nucleation, more effectively channel kinetic energy upon droplet impact to break up impinging droplets, and greatly limit frost formation. Once ice forms, its adhesion is substantially diminished by about three orders of magnitude as compared with planar substrates. The results demonstrate a scalable spray deposition method to generate surfaces for enabling the deterministic flow of liquids as well as inhibit ice formation. 

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