Hierarchical representations of surfaces have many advantages for digital geometry processing applications. Normal meshes are particularly attractive, since their level-to-level displacements are in the local normal direction only. Consequently, they can be specified with only scalar coefficients.

In this paper, the authors propose a novel method to approximate a given mesh with a normal mesh. Instead of building an associated parameterization on the fly, they assume a globally smooth parameterization at the beginning, and cast the problem as one of perturbing this parameterization. Controlling the magnitude of this perturbation gives them explicit control over the range, between fully constrained (only scalar coefficients) and unconstrained (3-vector coefficients) approximations. With the unconstrained problem giving the lowest approximation error, they characterize the error cost of normal meshes as a function of the number of nonnormal offsets, and find a significant gain for little (error) cost.

Because the normal mesh construction creates a geometry-driven approximation, the authors replace the difficult geometric distance minimization problem with a much simpler least squares problem. This variational approach reduces the magnitude and structure (aliasing) of the error further. Their method separates the parameterization construction into an initial setup followed only by subsequent perturbations, resulting in an algorithm that is far simpler to implement, more robust, and significantly faster.

The basis functions in the authors’ construction are still interpolating (though not the approximation itself). Perhaps even better approximations can be built when using, for example, cubic B-splines. In that case, all coefficients would have to be solved simultaneously in a nonlinear minimization problem. Another interesting avenue for future work is the construction of normal meshes fine-to-coarse. These offer the potential for fast transform methods.