"Interactive Texture mapping", J. Maillot, H. Yahia and
A. Verroust, Proceedings of Siggraph '93. This was one of the first
papers to use the concept of an atlas, or collection of
parameterizations. They "cut up" the mesh into regions based on
curvature. Each of these regions was flattened out using an
optimization procedure, with a parameter that trades distortion for
- Optimization used: They measure the deformation energy in the
first fundamental form. Specifically, let J be the Jacobian of S. Let
IS be the dot product of J and J transpose, which is a 2X2
matrix. Take the length squared of || IS - I ||, where I is the
identity matrix. This is the equivalent of the integral over u and v
of (dS^2/du - 1)^2 + 2 (dS/du dS/dv) + (dS^2/dv - 1)^2 du dv. Mechanical engineers know this as the Green-Lagrange deformation tensor.
- The above optimization, if used directly on a mesh, results in a
sum of fractions, with numerator of degree 8 and denominator of degree
9. This is too nasty to send to an optimizer, so they do the following
simplification using just the length of the edges (the three edge
lengths of a triangle completely determine the angles).
- They use a modified version of the spring energy to measure edge length distortion. For each edge, let el be the length of the edge in parameter space, and EL be the length on the mesh. Then minimize the sum of (el^2 - EL^2)^2/EL^2.
- May cause face flipping. To reduce flipping, they add an energy
term that is big for flipped faces, zero otherwise. Essentially, take
the cross product of the edges in the plane and make sure it points in
the correct direction.
- Cutting up the mesh into an atlas. They use the Gauss map and
segmentation to divide up the surface. Essentially, grow a region on
the mesh until the surface normals vary too much.
- Equations in the paper:
- Energy measure based entirely on edge lengths (similar to the spring energy term)
- Energy measure for "flipped" triangles.
- A measure of similarity of surface normals.
- A discrete curvature calculation based on the local points, similar to Taubin's integral eigenvalue calculation.