Valiron’s Interpolation Formula and a Derivative Sampling Formula in the Mellin Setting Acquired via Polar-Analytic Functions

In this paper, we first recall some recent results on polar-analytic functions. Then we establish Mellin analogues of a classical interpolation of Valiron and of a derivative sampling formula. As consequences a new differentiation formula and an identity theorem in Mellin–Bernstein spaces are obtained. The main tool in the proofs is a residue theorem for polar-analytic functions.

On Derivative Sampling From Image Blur for Reconstruction of Band-Limited Signals

Image sensors are typically characterized by slow sampling rates, which limit their efficacy in signal reconstruction applications. Their integrative nature though produces image blur when the exposure window is long enough to capture relative motion of the observed object relative to the sensor. Image blur contains more information on the observed dynamics than the typically used centroids, i.e., time averages of the motion within the exposure window. Parameters characterizing the observed motion, such as the signal derivatives at specified sampling instants, can be used for signal reconstruction through the derivative sampling extension of the known sampling theorem. Using slow image based sensors as derivative samplers allows for reconstruction of faster signals, overcoming Nyquist limitations. In this manuscript, we present an algorithm to extract values of a signal and its derivatives from blurred image measurements at specified sampling instants, i.e. the center of the exposure windows, show its application in two signal reconstruction numerical examples and provide a numerical study on the sensitivity of the extracted values to significant problem parameters.