Area-Preserving Parameterization

Mesh parameterization maps a complicated triangular mesh to a simple domain, enabling easier processing of irregular three dimenstional shapes.

Most existing literatures focus on either angle or area preservation: angle preservation maintains local shape (see feet), while area preservation maintains relative size (see ears).

Area preservation maintains geometry of surface uniform on the parameter domain, which is advantageous for applications that require uniformly sampling.

For geometry images, area preservation is crucial for faithfully storing mesh geometry. Area distortion can lose fine details in the reconstruction (see ears).

Spherical parameterization supports shape analysis via spherical harmonics, and area preservation is essential for computing accurate coefficients of spherical harmonic.

As more spherical-harmonic basis functions are used, reconstructions from area-preserving coefficients become increasingly faithful, while those from angle-preserving coefficients tend to distort geometry.

Distortion-Balancing Parameterization

However, strict area preservation can introduce large angle distortion. In some applications, it is important to keep both distortions low without overemphasizing either.

To balance them, we minimize and equalize global angle and area distortion, improving texture of geometry-image reconstructions over strict area preservation.

Another apprach is to directly truncate the excessive angle distortion after computing an area-preserving parameterization using quasi-conformal theory.

This is especially useful for geometry-image reconstruction of sliced closed surfaces, improving geometric integrity compared with strict area preservation.

Volume-Preserving Parameterization

Area preservation in 2-dimension corresponds to volume preservation in 3-dimension.

We can extend our method on triangular meshes to tetrahedral meshes to compute volume-preserving parameterizations.

Volume-preserving maps can serve as a preprocessing step for brain tumor segmentation by neural network.

We map the brain to a cube volume-preservingly, reducing raw image size while preserving geometric information (e.g., tumor size).