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Yanwen Luo

Publications and source records attributed to Yanwen Luo.

At least 19 recordsLinked to original sources
arXiv2026

Discrete uniformization of polyhedral surfaces

The main result of the paper shows that each connected polyhedral surface with a hyperbolic background metric, or a Euclidean background metric with uniform boundedness of radii of circumdisks, is discrete conformal to a complete constant-curvature Riemannian surface equipped with a non-empty closed discrete subset. We also prove a discrete Riemann mapping theorem. The proofs are based on the recent work on the discrete Schwarz lemma, the discrete Liouville theorem for polyhedral surfaces, and a Weyl-type realization theorem for hyperbolic surfaces.

math.GT↗
arXiv2026

Morphing Graphs on Hyperbolic Surfaces

We propose the first algorithm to morph geometric graphs on hyperbolic surfaces. It is based on a generalization of Tutte's spring embedding theorem on essentially 3-vertex-connected graphs. We describe the algorithms in detail and show experiments with triangulations and graphs on a hyperbolic surface of genus two, the Bolza surface, and a hyperbolic surface of genus three, the Klein quartic.

math.GT↗
arXiv2026

A Rigidity Theorem for Convex Sets in Hyperbolic 3-Space

Pogorelov's rigidity theorem states that a compact convex body in the hyperbolic 3-space is determined up to isometry by the intrinsic path metric on its boundary. The main result of this paper addresses a rigidity problem for non-compact closed convex 3-dimensional subsets in hyperbolic 3-space. We show that the intrinsic path metric on the boundary determines a closed convex set up to isometry, provided that the set of limit points of the convex set at infinity of the hyperbolic 3-space has vanishing 1-dimensional Hausdorff measure, i.e., zero length. Furthermore, this zero-length condition is optimal. This can be considered as an analogue of the Painlev\'e removability theorem in complex analysis, which states that sets of zero length are removable for bounded holomorphic functions. As a corollary, we show that if the underlying complex structure of a connected polyhedral surface is of parabolic type, then it is discrete conformal, unique up to scaling, to a complete flat surface marked with a discrete subset. The proof uses Pogorelov's rigidity theorem for compact convex bodies in $\mathbb R^3$, the Pogorelov map, and the Tabor--Tabor theorem on the extension of locally convex functions.

math.GT↗
arXiv2025

Infinite rigidity of inversive distance circle packings in the Poincar\'{e} disk

The maximum principle for hyperbolic inversive distance circle packings on polyhedral surfaces is established,which unifies and generalizes existing maximum principles for various types of circle packings in the literature.As an application of this principle, a discrete Schwarz-Ahlfors lemma is established.Furthermore, an infinite rigidity theorem for weighted Delaunay triangulations of the Poincar\'{e} disk is proved,which generalizes He's hyperbolic rigidity result \cite{He2}.

math.DG↗
arXiv2025

A Foundational Generative Model for Breast Ultrasound Image Analysis

Foundational models have emerged as powerful tools for addressing various tasks in clinical settings. However, their potential development to breast ultrasound analysis remains untapped. In this paper, we present BUSGen, the first foundational generative model specifically designed for breast ultrasound image analysis. Pretrained on over 3.5 million breast ultrasound images, BUSGen has acquired extensive knowledge of breast structures, pathological features, and clinical variations. With few-shot adaptation, BUSGen can generate repositories of realistic and informative task-specific data, facilitating the development of models for a wide range of downstream tasks. Extensive experiments highlight BUSGen's exceptional adaptability, significantly exceeding real-data-trained foundational models in breast cancer screening, diagnosis, and prognosis. In breast cancer early diagnosis, our approach outperformed all board-certified radiologists (n=9), achieving an average sensitivity improvement of 16.5% (P-value<0.0001). Additionally, we characterized the scaling effect of using generated data which was as effective as the collected real-world data for training diagnostic models. Moreover, extensive experiments demonstrated that our approach improved the generalization ability of downstream models. Importantly, BUSGen protected patient privacy by enabling fully de-identified data sharing, making progress forward in secure medical data utilization. An online demo of BUSGen is available at https://aibus.bio.

cs.AI↗
arXiv2024

Knowledge-driven AI-generated data for accurate and interpretable breast ultrasound diagnoses

Data-driven deep learning models have shown great capabilities to assist radiologists in breast ultrasound (US) diagnoses. However, their effectiveness is limited by the long-tail distribution of training data, which leads to inaccuracies in rare cases. In this study, we address a long-standing challenge of improving the diagnostic model performance on rare cases using long-tailed data. Specifically, we introduce a pipeline, TAILOR, that builds a knowledge-driven generative model to produce tailored synthetic data. The generative model, using 3,749 lesions as source data, can generate millions of breast-US images, especially for error-prone rare cases. The generated data can be further used to build a diagnostic model for accurate and interpretable diagnoses. In the prospective external evaluation, our diagnostic model outperforms the average performance of nine radiologists by 33.5% in specificity with the same sensitivity, improving their performance by providing predictions with an interpretable decision-making process. Moreover, on ductal carcinoma in situ (DCIS), our diagnostic model outperforms all radiologists by a large margin, with only 34 DCIS lesions in the source data. We believe that TAILOR can potentially be extended to various diseases and imaging modalities.

eess.IV↗
arXiv2022

Rigidity of infinite inversive distance circle packings in the plane

In 2004, Bowers-Stephenson [2] introduced the inversive distance circle packings as a natural generalization of Thurston's circle packings. They further conjectured the rigidity of infinite inversive distance circle packings in the plane. Motivated by the recent work of Luo-Sun-Wu [22] on Luo's vertex scaling, we prove Bower-Stephenson's conjecture for inversive distance circle packings in the hexagonal triangulated plane. This generalizes Rodin-Sullivan's famous result [13] on the rigidity of infinite tangential circle packings in the hexagonal triangulated plane. The key tools include a maximal principle for generic weighted Delaunay inversive distance circle packings and a ring lemma for the inversive distance circle packings in the hexagonal triangulated plane.

math.GT↗
arXiv2022

Maximal principles in discrete conformal geometry with application to the rigidity of infinite triangulations

In this paper, maximum principles for Euclidean and hyperbolic discrete conformal structures on polyhedral surfaces are established. These maximum principles unify and generalize the maximum principles for vertex scalings and different types of circle packings in the literature. As an application of the hyperbolic discrete maximum principle, a discrete Schwarz-Ahlfors lemma is established. As another application, an infinite rigidity theorem for small Delaunay triangulations of the hyperbolic plane is proved.

math.MG↗
arXiv2022

Hexagonal Geometric Triangulations

It is well-known that the Euclidean plane has a standard 6-regular geodesic triangulation , and the unit sphere has a 5-regular geodesic triangulation, which is induced from the regular Dodecahedron, and the hyperbolic plane has an n-regular geodesic triangulation for any n > 6. Here we constructed a 6-regular geodesic triangulation of the hyperbolic plane.

math.GT↗
arXiv2022

A note on graph drawings with star-shaped boundaries in the plane

In this note, we propose a straightforward method to produce an straight-line embedding of a planar graph where one face of a graph is fixed in the plane as a star-shaped polygon. It is based on minimizing discrete Dirichlet energies, following the idea of Tutte's embedding theorem. We will call it a geodesic triangulation of the star-shaped polygon. Moreover, we study the homotopy property of spaces of all straight-line embeddings. We give a simple argument to show that this space is contractible if the boundary is a non-convex quadrilateral. We conjecture that the same statement holds for general star-shaped polygons.

cs.CG↗
arXiv2022

Spaces of Geodesic Triangulations Are Cells

It has been shown that spaces of geodesic triangulations of closed negatively curved surfaces are contractible. Here we prove that these spaces are homeomorphic to Euclidean spaces $\mathbb{R}^n$.

math.GT↗
arXiv2022

Comparing shapes of high genus surfaces

In this paper, we define a new metric structure on the shape space of a high genus surface. We introduce a rigorous definition of a shape of a surface and construct a metric based on two energies measuring the area distortion and the angle distortion of a quasiconformal homeomorphism. We show that the energy minimizer in a fixed homotopy class is achieved by a quasiconformal homeomorphism by the lower semicontinuity property of these two energies.

math.DG↗
arXiv2022

Combinatorial Calabi flows on surfaces with boundary

Motivated by Luo's combinatorial Yamabe flow on closed surfaces \cite{L1} and Guo's combinatorial Yamabe flow on surfaces with boundary \cite{Guo}, we introduce combinatorial Calabi flow on ideally triangulated surfaces with boundary, aiming at finding hyperbolic metrics on surfaces with totally geodesic boundaries of given lengths. Then we prove the long time existence and global convergence of combinatorial Calabi flow on surfaces with boundary. We further introduce fractional combinatorial Calabi flow on surfaces with boundary, which unifies and generalizes the combinatorial Yamabe flow and the combinatorial Calabi flow on surfaces with boundary. The long time existence and global convergence of fractional combinatorial Calabi flow are also proved. These combinatorial curvature flows provide effective algorithms to construct hyperbolic surfaces with totally geodesic boundaries with prescribed lengths.

math.DG↗
arXiv2022

The convergence of inversive distance circle packings to the Riemann mapping

Bowers and Stephenson introduced the notion of inversive distance circle packings as a natural generalization of Thurston's circle packings. They further conjectured that discrete conformal maps induced by inversive distance circle packings converge to the Riemann mapping. In this paper, we prove Bowers-Stephenson's conjecture for Jordan domains by establishing a solvability theorem of certain prescribing combinatorial curvature problems for inversive distance circle packings.

math.DG↗
arXiv2022

The Deformation Space of Delaunay Triangulations of the Sphere

In this paper, we determine the topology of the spaces of convex polyhedra inscribed in the unit $2$-sphere and the spaces of strictly Delaunay geodesic triangulations of the unit $2$-sphere. These spaces can be regarded as discretized groups of diffeomorphisms of the unit $2$-sphere. Hence, it is natural to conjecture that these spaces have the same homotopy types as those of their smooth counterparts. The main result of this paper confirms this conjecture for the unit $2$-sphere. It follows from an observation on the variational principles on triangulated surfaces developed by I. Rivin. On the contrary, the similar conjecture does not hold in the cases of flat tori and convex polygons. We will construct simple examples of flat tori and convex polygons such that the corresponding spaces of Delaunay geodesic triangulations are not connected.

math.GT↗
arXiv2021

The Convergence of Discrete Uniformizations for Genus Zero Surfaces

The notion of discrete conformality proposed by Luo and Bobenko-Pinkall-Springborn on triangle meshes has rich mathematical theories and wide applications. Gu et al. proved that the discrete uniformizations approximate the continuous uniformization for closed surfaces of genus $\geq 1$, given that the approximating triangle meshes are reasonably good. In this paper, we generalize this result to the remaining case of genus-zero surfaces, by reducing it to planar cases via stereographic projections.

math.GT↗
arXiv2021

The Deformation Spaces of Geodesic Triangulations of Flat Tori

We prove that the deformation space of geodesic triangulations of a flat torus is homotopy equivalent to a torus. This solves an open problem proposed by Connelly et al. in 1983, in the case of flat tori. A key tool of the proof is a generalization of Tutte's embedding theorem for flat tori. When this paper is under preparation, Erickson and Lin proved a similar result, which works for all convex drawings.

math.GT↗