FRI Sensing: Retrieving the Trajectory of a Mobile Sensor From Its Temporal Samples

In this article, contrary to current research trend which consists of fusing (big) data from many different sensors, we focus on one-dimensional samples collected by a unique mobile sensor (e.g., temperature, pressure, magnetic field, etc.), without explicit positioning information (such as GPS). We demonstrate that this stream of 1D data contains valuable 2D geometric information that can be unveiled by adequate processing-using a high-accuracy Finite Rate of Innovation (FRI) algorithm: “FRI Sensing”. Our key finding is that, despite the absence of any position information, the basic sequence of 1D sensor samples makes it possible to reconstruct the sampling trajectory (up to an affine transformation), and then the image that represents the physical field that has been sampled. We state the FRI Sensing sampling theorem and the hypotheses needed for this trajectory and image reconstruction to be successful. The proof of our theorem is constructive and leads to a very efficient and robust algorithm, which we validate in various conditions. Moreover, although we essentially model the images as finite sums of 2D sinusoids, we also observe that our algorithm works accurately for real textured images.