Laurent Alonso

Geometry-aware direction field processing

Many algorithms in texture synthesis, nonphotorealistic rendering (hatching), or remeshing require to define the orientation of some features (texture, hatches, or edges) at each point of a surface. In early works, tangent vector (or tensor) fields were used to define the orientation of these features. Extrapolating and smoothing such fields is usually performed by minimizing an energy composed of a smoothness term and of a data fitting term. More recently, dedicated structures (N-RoSy and N-symmetry direction fields ) were introduced in order to unify the manipulation of these fields, and provide control over the fields topology (singularities). On the one hand, controlling the topology makes it possible to have few singularities, even in the presence of high frequencies (fine details) in the surface geometry. On the other hand, the user has to explicitly specify all singularities, which can be a tedious task. It would be better to let them emerge naturally from the direction extrapolation and smoothing. This article introduces an intermediate representation that still allows the intuitive design operations such as smoothing and directional constraints, but restates the objective function in a way that avoids the singularities yielded by smaller geometric details. The resulting design tool is intuitive, simple, and allows to create fields with simple topology, even in the presence of high geometric frequencies. The generated field can be used to steer global parameterization methods (e.g., QuadCover).

Determining the majority

Abstract Given a set of n elements each of which is either red or blue, it is known that n–v(n) pairwise equal/not equal colour comparisons are necessary and sufficient to determine the majority color, where v(n) is the number of 1-bits in the binary representation of n. We present a new, simple proof of this lower bound.

The Average-Case Complexity of Determining the Majority

Given a set of

“Lion and Man”: Upper and Lower Bounds

n

Uniform generation of a Motzkin word

elements each of which is either red or blue, it is known that in the worst case

A linear-time algorithm for the generation of trees

n-\nu(n)

On the Tree Inclusion Problem

pairwise equal/not equal color comparisons are necessary and sufficient to determine the majority color, where

Determining plurality

\nu(n)

On the tree inclusion problem

is the number of 1-bits in the binary representation of

The virtual mesh: a geometric abstraction for efficiently computing radiosity

n