Stationary behavior of locally normalized asymmetric kernels

Determine the stationary behavior of locally normalized asymmetric kernels used for adaptive blue-noise generation, which lack a symmetric pairwise Hamiltonian and fall outside the analyzed Gibbs model.

Background

The paper’s adaptive blue-noise model uses symmetric pairwise interaction weights so that the energy can be represented by a Hamiltonian and analyzed as a Gibbs distribution. The authors report that locally normalized asymmetric kernels have occasionally produced visually sharper adaptive results, but they observe no consistent pattern and provide no measurements.

Because these asymmetric kernels do not admit the symmetric pairwise Hamiltonian used in the paper’s analysis, their long-run or stationary behavior is not characterized by the presented theory. Establishing that behavior would clarify whether such kernels define meaningful sampling distributions and how their empirical visual advantages relate to convergence and equilibrium statistics.