What is a mathematically rigorous definition of "blue noise"?

Let $d\in\mathbb N$ , $I$ be a finite nonempty set, $(x_i)_{i\in I}\subseteq[0,1)^d$ , $(w_i)_{i\in I}\subseteq[0,\infty)$ with $\sum_{i\in I}w_i=1$ and $$\sigma:=\sum_{i\in I}w_i\delta_{x_i}.$$ I really have trouble understanding what's meant by the property of the point pattern $(x_i)_{i\in I}$ to "admit blue noise characteristics". I think the term "blue noise" is used quite vaguely in the literature. One thing I've often read is that $(x_i)_{i\in I}$ has "blue noise characteristics" if the corresponding power spectrum is near $0$ for low frequencies. This should mean that the absolute value of the Fourier transform $$\hat\sigma(x)=\sum_{i\in I}w_ie^{-{\rm i}2\pi\langle x,\:x_i\rangle}\;\;\;\text{for }x\in\mathbb R^d$$ is small (or near $0$ ) whenever $\left\|x\right\|$ is small. There are also papers (like Blue Noise through Optimal Transport and Wasserstein Blue Noise Sampling ) which seem to relate the blue noise property to a suitable minimization of a Wasserstein distance, but I actually don't understand why and in which sense that minimizer has blue noise characteristics. What I actually would like to investigate is whether a specific Markov chain (e.g. generated by the Metropolis-Hastings algorithm) has blue noise characteristics.