How many modes can a mixture of Gaussians with uniformly bounded means have?

We show, by an explicit construction, that a mixture of univariate Gaussian densities with variance $1$ and means in $[-A,A]$ can have $\varOmega (A^2)$ modes. This disproves a recent conjecture of Dytso et al. (2020, IEEE Trans. Inf. Theory, 66, 2006–2022) who showed that such a mixture can have at most $O(A^{2})$ modes and surmised that the upper bound could be improved to $O(A)$. Our result holds even if an additional variance constraint is imposed on the mixing distribution. Extending the result to higher dimensions, we exhibit a mixture of Gaussians in ${\mathbb{R}}^{d}$, with identity covariances and means inside ${[-A,A]}^{d}$, that has $\varOmega (A^{2d})$ modes.

Infrared Small-Faint Target Detection Using Non-i.i.d. Mixture of Gaussians and Flux Density

The robustness of infrared small-faint target detection methods to noisy situations has been a challenging and meaningful research spot. The targets are usually spatially small due to the far observation distance. Considering the underlying assumption of noise distribution in the existing methods is impractical; a state-of-the-art method has been developed to dig out valuable information in the temporal domain and separate small-faint targets from background noise. However, there are still two drawbacks: (1) The mixture of Gaussians (MoG) model assumes that noise of different frames satisfies independent and identical distribution (i.i.d.); (2) the assumption of Markov random field (MRF) would fail in more complex noise scenarios. In real scenarios, the noise is actually more complicated than the MoG model. To address this problem, a method using the non-i.i.d. mixture of Gaussians (NMoG) with modified flux density (MFD) is proposed in this paper. We firstly construct a novel data structure containing spatial and temporal information with an infrared image sequence. Then, we use an NMoG model to describe the noise, which can be separated with the background via the variational Bayes algorithm. Finally, we can select the component containing true targets through the obvious difference of target and noise in an MFD maple. Extensive experiments demonstrate that the proposed method performs better in complicated noisy scenarios than the competitive approaches.