Scale periodicity and its sampling theorem

The Scale transform is a new representation for signals, offering perspective different from the Fourier transform. In this paper, we introduce the notion of a scale periodic function. These functions are then represented through the discrete scale series. We also define the notion of a strictly scale limited signal. Analogous to the Shannon interpolation formula, we show that such signals can be exactly reconstructed from exponentially spaced samples of the signal in the time domain. As an interesting, practical application, we show how properties unique to the Scale transform make it very useful in computing depth maps of a scene.