PDE based level set methods in Radar remote sensing, shape reconstruction, and reflectivity analysis.

The present post concern the research topic I am investigating since May 2017, when I started working at the Georgia Institute of Technology (GaTech), with Prof. Anthony J. Yezzi.

The mathematical method employed have close connection with the research I conducted in 2014 at GaTech, except that in 2014 I investigated the use of active surfaces using PDE2 based level set methods for the detection and geometrical description of inclusions or defects in solids based on elastic wave measurements, while since May 2017 I have been investigating applications of active surfaces and their use in connection with radar signals for remote sensing and shape reconstruction as well as reflectivity analysis.

Nowadays radar finds application in many different fields, spanning from localization, tracking, to imaging. It has been proven a powerful tool to remotely sense the complex reflectivity of a target (or scene), and after signal processing, higly detailed images can be delivered. In this context, a wide literature is available on the use of a plurality of antennas, or alternatively, the use of a single moving antenna (e.g. as in SAR1). Similarly, when echos are recorded from a sufficient number of locations distributed in space, a reflectivity volume can be generated. The latter approach is better known as VolSAR. IIn both SAR and VolSAR, the result, after the signals have processed, is a distribution of pixels. As such, shape recovery from these distributions of pixels is usually achieved through segmentation software.

It is worth mentioning that some approaches have been proposed to use the information from multiple images to retrieve the actual shape of the illuminated scene.

The basic idea of the present research effor is to to infer the shape (or at least, a good approximation) of an illuminated object using PDE methods (such as the active surfaces) in connection with radar echos recorded at multiple locations.

The following videos illustrate our early result.

Among the possible choices, we decided to experiment the concepts assuming the target (1Km squared) to be illuminated using a X-band chirp pulse. Ninety antenna locations are used, all distributed around the scene, with different heights. All locations are 20 Km far from the center of the scene.

Video-1 shows an optimization test using a coarse mesh.

Video-2 shows the same test using the “accelerated” gradient descent method.

Video-3 shows a coarse-to-fine approach.

I believe this is a very promising research, especially because practical applications at different scales can be conceived.



1. SAR: Synthetic Aperture Radar

2. PDE: Partial Differential Equation