|M.Sc Student||Farchi Nahum|
|Subject||Integer-Only Cross Field Computation|
|Department||Department of Computer Science||Supervisor||Professor Mirela Ben-Chen|
Directional fields are important objects in geometry processing with applications
ranging from texture synthesis to non-photorealistic rendering, quadrangular
remeshing, and architectural design. In this thesis, we focus our attention on cross
fields - a direction field in which four unit vectors with π/2 symmetry are defined at
each point on the surface.
Computing smooth cross fields on triangle meshes is challenging, as the problem
formulation inherently depends on integer variables to encode the invariance of the
crosses to rotations by integer multiples of π/2. Furthermore, finding the optimal
placement for the cone singularities is essentially a hard combinatorial problem.
We propose a new iterative algorithm for computing smooth cross fields on triangle
meshes that is simple, easily parallelizable on the GPU, and finds solutions with lower
energy and fewer cone singularities than state-of-the-art methods. Furthermore, the
output cross fields are such that there is no relocation of a single ±π/2 singularity that
will reduce the energy.
Our approach is based on a formal equivalence, which we prove, between two
formulations of the optimization problem. This equivalence allows us to eliminate the
real variables and design an efficient grid search algorithm for the cone singularities.
We make use of a recent graph-theoretical approximation of the resistance distance
matrix of the triangle mesh to speed up the computation and enable a trade-off
between the computation time and the smoothness of the output.