Exploring the Geometry of One-Dimensional Signals

The wide availability of inexpensive sensors of all kinds (inertia, magnetic field, light, temperature, pressure, chemicals etc.) makes it possible to empower a host of novel applications. We have shown in a previous paper that, if the field sensed can be expressed as a finite sum of 2D sinusoids, it is possible to reconstruct the sampling curve from the 1D sequence of image samples alone (up to a linear transformation)—without extra positioning information. Here, we explore the validity of this result if, instead, we assume the image to be directional or, as an extreme case, laminar and we simplify our previous approach to the single sinusoid fitting of segments of the 1D samples. We obtain predictive results that quantify the accuracy with which the frequencies found can be used to estimate the slope of the sampling trajectory. We also develop a robust algorithm to retrieve the sampling trajectory and estimate the laminar image that underlies the 1D samples. We finally demonstrate the validity of our approach on synthetic and well-chosen real images.