A proof on the invariance of the Hirschman Uncertainty to the Rényi entropy parameter and an observation on its relevance in the image texture classification problem

In [1] we developed a new uncertainty measure which incorporates Rényi entropy instead of Shannon entropy. This new uncertainty measure was conjectured to be invariant to the Rényi order α > 0, whereas for discrete signals other than picket fence signal the uncertainty measure decreases for α > 0. In this paper, we prove this invariance, and test whether this invariance is predictive in the problem of texture classification for digital images. In the preliminary results, we find that it does, in that the recognition performance does not depend significantly on the Rényi parameter α. We hope that these results will be extended to other problems where Rényi entropy is used.