[Paper Review] Computing Quasiconformal Maps on Riemann surfaces using Discrete Curvature Flow

Wei Zeng, Lok Ming Lui+5 more|arXiv (Cornell University)|May 25, 2010

Advanced Numerical Analysis Techniques29 references3 citations

TL;DR

This paper proposes a novel numerical method to compute quasiconformal maps on general Riemann surfaces by solving the Beltrami equation via a discrete curvature flow approach. It introduces an auxiliary metric that transforms the quasiconformal map into a conformal one under the new metric, enabling solution via discrete Yamabe flow; the method converges to the continuous solution as mesh resolution increases, demonstrating high accuracy and generality on real-world scanned surfaces of various topologies.

ABSTRACT

Surface mapping plays an important role in geometric processing. They induce both area and angular distortions. If the angular distortion is bounded, the mapping is called a {\it quasi-conformal} map. Many surface maps in our physical world are quasi-conformal. The angular distortion of a quasi-conformal map can be represented by Beltrami differentials. According to quasi-conformal Teichmüller theory, there is an 1-1 correspondence between the set of Beltrami differentials and the set of quasi-conformal surface maps. Therefore, every quasi-conformal surface map can be fully determined by the Beltrami differential and can be reconstructed by solving the so-called Beltrami equation. In this work, we propose an effective method to solve the Beltrami equation on general Riemann surfaces. The solution is a quasi-conformal map associated with the prescribed Beltrami differential. We firstly formulate a discrete analog of quasi-conformal maps on triangular meshes. Then, we propose an algorithm to compute discrete quasi-conformal maps. The main strategy is to define a discrete auxiliary metric of the source surface, such that the original quasi-conformal map becomes conformal under the newly defined discrete metric. The associated map can then be obtained by using the discrete Yamabe flow method. Numerically, the discrete quasi-conformal map converges to the continuous real solution as the mesh size approaches to 0. We tested our algorithm on surfaces scanned from real life with different topologies. Experimental results demonstrate the generality and accuracy of our auxiliary metric method.

Motivation & Objective

• To develop an effective numerical algorithm for computing quasiconformal maps on general Riemann surfaces, including those of arbitrary genus.
• To address the long-standing challenge of numerically solving the Beltrami equation on surfaces beyond simple domains in the complex plane.
• To establish a discrete analog of quasiconformal maps on triangular meshes for computational geometric processing.
• To enable the reconstruction of quasiconformal maps from prescribed Beltrami differentials using a metric-based transformation approach.

Proposed method

• Formulate a discrete quasiconformal map on triangular meshes by defining an auxiliary metric that renders the map conformal under the new metric.
• Construct the auxiliary metric based on the given Beltrami differential, ensuring the original quasiconformal map becomes conformal under this new metric.
• Apply the discrete Yamabe flow to evolve the metric toward constant curvature, thereby computing the conformal map under the auxiliary metric.
• Use the solution of the discrete Yamabe flow to recover the quasiconformal map on the original metric.
• Leverage the 1-1 correspondence between Beltrami differentials and quasiconformal maps via Teichmüller theory to ensure uniqueness and correctness.
• Prove local convexity and convergence of the discrete Yamabe flow by analyzing the Hessian of the energy functional and showing it is positive definite under the hyperbolic setting.

Experimental results

Research questions

• RQ1 Can the Beltrami equation be effectively solved on general Riemann surfaces using a discrete curvature flow method?
• RQ2 How can a quasiconformal map be transformed into a conformal map via a discrete auxiliary metric construction?
• RQ3 What is the convergence behavior of the discrete quasiconformal map to the continuous solution as mesh resolution increases?
• RQ4 Is the proposed method robust and accurate across surfaces of different topologies and real-world geometries?
• RQ5 Can the discrete Yamabe flow be proven to converge to a solution that satisfies the quasiconformal mapping condition under the auxiliary metric?

Key findings

• The proposed method successfully computes quasiconformal maps on Riemann surfaces of arbitrary genus, including complex topologies from real scanned data.
• The discrete quasiconformal map converges to the continuous solution as the mesh size approaches zero, demonstrating numerical consistency.
• The auxiliary metric transformation ensures that the quasiconformal map becomes conformal under the new metric, enabling use of the discrete Yamabe flow.
• The Hessian of the discrete Yamabe energy is proven to be positive definite on the relevant domain, guaranteeing local convexity and convergence of the flow.
• The discrete Yamabe flow solution satisfies exponential decay of curvature deviation, with |K_i(t) - K̄_i| ≤ c₁e^(-c₂t), indicating stable and fast convergence.
• Experimental results on real-world scanned surfaces confirm the method's generality, accuracy, and robustness across diverse topologies and geometric complexities.