Publications

Papers & preprints

Research centers on mapping surfaces and volumes onto simple domains, such as a disk, a sphere, or a ball, while preserving area or volume. It combines variational formulations, efficient algorithms, and convergence analysis. Each entry below gives a brief summary, with links to the full text, code, and citation.

Figures interpolate numerical results; explore iterations and distortion in the interactive demonstrations.

Preprint

Area-Preserving Parameterization: Variational Principle, Gradient Flow, and Discrete Approximation

arXiv:2608.17873, 2026

S.-Y. Liu, Kento Sakai, M.-H. Yueh

We characterize area preservation through the variational principle of stretch energy and establish the discrete-to continuum convergence of its minimal value.

00.50

A 1,350-triangle mesh with fixed boundary. Color: |J_f − 1|.

Full text

@misc{LiKY26,
  author        = {Shu-Yung Liu and Kento Sakai and Mei-Heng Yueh},
  title         = {Area-Preserving Parameterization: Variational Principle, Gradient Flow, and Discrete Approximation},
  year          = {2026},
  archiveprefix = {arXiv},
  eprint        = {2608.17873},
  primaryclass  = {math.NA},
}

2026

Volume-Preserving Parameterizations via Preconditioned Nonlinear Conjugate Gradient Method

Journal of Optimization Theory and Applications, 209(3)

S.-Y. Liu, T.-M. Huang, W.-W. Lin, M.-H. Yueh

We compute volume-preserving maps of tetrahedral meshes by minimizing an isovolumetric energy, with boundary vertices free to move on the unit sphere. A preconditioned nonlinear conjugate gradient method solves the problem, with global convergence under strong Wolfe line searches.

00.15

Brain mesh · 19,313 tetrahedra · Color: local volume distortion.

Full text arXiv Demo GitHub

@article{LiHL26,
  author  = {Liu, S.-Y. and Huang, T.-M. and Lin, W.-W. and Yueh, M.-H.},
  title   = {Volume-Preserving Parameterizations via Preconditioned Nonlinear Conjugate Gradient Method},
  journal = {J. Optim. Theory Appl.},
  volume  = {209},
  number  = {3},
  year    = {2026},
  issn    = {1573-2878},
  doi     = {10.1007/s10957-026-03002-5},
}

2026

Spherical Area-Preserving Parameterization via Energy Minimization

SIAM Journal on Imaging Sciences, 19(1):207–235

S.-Y. Liu, M.-H. Yueh

We map genus-zero closed surfaces onto the sphere by minimizing the spherical authalic energy. The algorithm converges globally, and a Riemannian correction step addresses bijectivity.

00.50

Right hand · 8,808 triangles · Color: |log₂ area ratio|.

Full text arXiv Demo GitHub

@article{LiYu26,
  author  = {Liu, S.-Y. and Yueh, M.-H.},
  title   = {Spherical Area-Preserving Parameterization via Energy Minimization},
  journal = {SIAM J. Imaging Sci.},
  volume  = {19},
  number  = {1},
  pages   = {207--235},
  year    = {2026},
  doi     = {10.1137/25M1736979},
}

2025

Energy-Based Distortion-Balancing Parameterization for Open Surfaces

SIAM Journal on Imaging Sciences, 18(4):2059–2093

S.-Y. Liu, M.-H. Yueh

We map simply connected open surfaces to the plane by balancing angle preservation (conformal) and area preservation (authalic). The resulting iterative algorithm comes with a global convergence guarantee.

01.50

Max Planck · 4,999 triangles. Color: the larger of angle (blue) and area (orange) distortion.

Full text Demo GitHub

@article{LiYu25,
  author  = {Liu, S.-Y. and Yueh, M.-H.},
  title   = {Energy-Based Distortion-Balancing Parameterization for Open Surfaces},
  journal = {SIAM J. Imaging Sci.},
  volume  = {18},
  number  = {4},
  pages   = {2059--2093},
  year    = {2025},
  doi     = {10.1137/24M1708437},
}

2024

Convergent Authalic Energy Minimization for Disk Area-Preserving Parameterizations

Journal of Scientific Computing, 100(2):43

S.-Y. Liu, M.-H. Yueh

We compute area-preserving maps from a surface to the unit disk by minimizing the authalic energy. The algorithm is globally convergent and shows improved performance than existing methods.

00.40

Stanford Bunny · 5,000 triangles · Color: local area distortion.

Full text Demo GitHub

@article{LiYu24,
  author  = {Liu, S.-Y. and Yueh, M.-H.},
  title   = {Convergent Authalic Energy Minimization for Disk Area-Preserving Parameterizations},
  journal = {J. Sci. Comput.},
  volume  = {100},
  number  = {2},
  pages   = {43},
  year    = {2024},
  issn    = {1573-7691},
  doi     = {10.1007/s10915-024-02594-2},
}