You are invited to attend a series of** tutorial lecture**s by

**Prof. George Papanicolaou**

Mathematics Department

Stanford University

On the subject:

**Synthetic Aperture Imaging**

**Tuesday, June 11 2019, at 10:00-14:00**

**Mathematical problems in imaging**

I will give an overview of a few methods for forming images from different data sets (measurements), how to determine the resolution of the images based on physical as well as signal processing considerations, and how to denoise them. I will also discuss briefly holographic imaging.**2. Synthetic aperture imaging with intensity-only measurements**

I will consider imaging the reflectivity of scatterers from intensity-only data recorded by a single moving transducer that both emits and receives signals, forming a synthetic aperture. By exploiting frequency illumination diversity, one can obtain multiple intensity measurements at each location, from which it is possible to determine (field) cross-correlations using an appropriate illumination strategy and the inner product polarization identity. The cross-correlations obtained this way do not, however, provide all the missing phase information because they are determined up to a phase that depends on the receiver’s location.

The main result that will be presented is an algorithm with which one can recover the cross-correlations up to a single phase that is common to all the data measured over the synthetic aperture, so all the data are synchronized. Thus, one can image coherently with data over all frequencies and measurement locations as if full phase information was recorded.

**Low rank plus sparse decomposition of synthetic aperture radar data for target imaging and tracking (to be given by Matan Leibovich)**We analyze synthetic aperture radar (SAR) imaging of complex ground scenes that contain both sta- tionary and moving targets. In the usual SAR acquisition scheme, we consider ways to preprocess the data so as to separate the contributions of the moving targets from those due to stationary background reflectors. Both components of the data, that is, reflections from stationary and moving targets, are considered as signal that is needed for target imaging and tracking, respectively. The approach we use is to decompose the data matrix into a low rank and a sparse part. This decomposition enables us to capture the reflections from moving targets into the sparse part and those from stationary targets into the low rank part of the data. The computational tool for this is robust principal component analysis (RPCA) applied to the SAR data matrix. We also introduce a lossless baseband trans- formation of the data, which simplifies the analysis and improves the performance of the RPCA algorithm. Our main contribution is a theoretical analysis that determines an optimal choice of parameters for the RPCA algorithm so as to have an effective and stable separation of SAR data coming from moving and stationary targets. This analysis gives also a lower bound for detectable target velocities. We show in particular that the rank of the sparse matrix is proportional to the square root of the targetâ€™s speed in the direction that connects the SAR platform trajectory to the imaging region. The robustness of the approach is illustrated with numerical simulations in the X-band SAR regime.

**Room 206, Wolfson Building, Faculty of Engineering**