# Measuring Sparseness of Noisy Signals

In this paper sparseness measures are reviewed, extended and compared. Special attention is paid on measuring sparseness of noisy data. We review and extend several definitions and measures for sparseness, including the $\ell^{0}$, $\ell^{p}$ and $\ell^{\epsilon}$ norms. A measure based on order statistics is also proposed. The concept of sparseness is extended to the case where a signal has a dominant value other than zero. The sparseness measures can be easily modified to correspond to this new definition. Eight different measures are compared in three examples. It turns out that different measures may give complete opposite results if the distribution does not have a unique mode at zero. As conclusion, we suggest that the kurtosis should be avoided as a sparseness measure and recommend tanh-functions for measuring noisy sparseness.