Area-Preserving Parameterization

In computer graphics, complex 3-dimensional surfaces are typically represented as triangular meshes. Mesh parameterization is the process of mapping these complex surfaces onto simpler 2D domains, such as a unit disk or a square.

Most existing methods focus on one of two properties:

  Angle preservation (conformal) : Maintains the local shape and features of the surface.

  Area preservation (authalic) : Maintains the relative size of each region across the map.

While angle preservation is common, area preservation is crucial for applications like geometry Images, which store vertex positions as pixels of images.

If the parameterization distorts the area, some regions of the mesh get squeezed into too few pixels, leading to a loss of fine detail during reconstruction.

We introduce a novel numerical method to compute these area-preserving maps with significantly higher accuracy and efficiency than previous state-of-the-art methods.

We extend our method to surfaces topologically equivalent to a sphere, computing a spherical area-preserving parameterization.

Spherical parameterizations supports shape analysis via spherical harmonics. Because area-preserving maps provide a uniform distribution of the surface, they are essential for computing accurate harmonic coefficients.

As the number of basis functions increases, reconstructions using area-preserving coefficients become increasingly precise, whereas those using angle-preserving maps often suffer from geometric distortion.

For more details, please refer to the following papers:

Distortion-Balancing Parameterization

However, strict area preservation can often lead to extreme angle distortion. In many practical applications, the best results come from finding a trade-off between minimizing both angle and area distortion without overemphasizing one over the other.

We propose a numerical method that minimizes global angle and area distortions simultaneously while keeping them in balance. This approach further enhances the quality of geometry images, recovering finer details than strict area-preserving one.

Another approach is to truncate excessive angular distortion, which may prevent the creation of overly thin triangles that can occur during area preservation.

We truncate angular distortion using quasi-conformal theory for sliced closed surfaces, significantly improving the geometric integrity of reconstructions by geometry images.

For more details, please refer to the following papers:

Volume-Preserving Parameterization

Area preservation in two dimensions is directly analogous to volume preservation in three dimensions.

We have extended our framework to tetrahedral meshes, allowing for the volume-preserving parameterization of complex 3D volumetric objects (3-manifolds).

These volume-preserving maps serve as a preprocessing step for brain tumor segmentation via neural networks.

By mapping the brain to a cube while strictly maintaining its volume, we can effectively remove redundant space outside the brain.

Replacing raw imaging data with this cubic representation offers significant advantages:

  Reduces memory storage requirements by nearly 70%, decreasing training times.

  Enhances the accuracy of tumor region identification.

For more details, please refer to the following papers: