Uncertainty principles for images defined on the square

This paper discusses uncertainty principles of images defined on the square, or, equivalently, uncertainty principles of signals on the 2‐torus. Means and variances of time and frequency for signals on the 2‐torus are defined. A set of phase and amplitude derivatives are introduced. Based on the derivatives, we obtain three comparable lower bounds of the product of variances of time and frequency, of which the largest lower bound corresponds to the strongest uncertainty principles known for periodic signals. Examples, including simulations, are provided to illustrate the obtained results. To the authors' knowledge, it is in the present paper, and for the first time, that uncertainty principles on the torus are studied. Copyright © 2016 John Wiley & Sons, Ltd.