Imaging radar for automated driving functions

This work presents the implementation of a synthetic aperture radar (SAR) at 77 GHz, for automotive applications. This implementation is unique in the sense that it is a radar-only solution for most use-cases. The set-up consists of two radar sensors, one to calculate the ego trajectory and the second for SAR measurements. Thus the need for expensive GNSS-based dead reckoning systems, which are in any case not accurate enough to fulfill the requirements for SAR, is eliminated. The results presented here have been obtained from a SAR implementation which is able to deliver processed images in a matter of seconds from the point where the targets were measured. This has been accomplished using radar sensors which will be commercially available in the near future. Hence the results are easily reproducible since the deployed radars are not special research prototypes. The successful widespread use of SAR in the automotive industry will be a large step forward toward developing automated parking functions which will be far superior to today's systems based on ultrasound sensors and radar (short range) beam-forming algorithms. The same short-range radar can be used for SAR, and the ultrasound sensors can thus be completely omitted from the vehicle.

Skeleton Ideals of Certain Graphs, Standard Monomials and Spherical Parking Functions

Let $G$ be a graph on the vertex set $V = \{ 0, 1,\ldots,n\}$ with root $0$. Postnikov and Shapiro were the first to consider a monomial ideal $\mathcal{M}_G$, called the $G$-parking function ideal, in the polynomial ring $ R = {\mathbb{K}}[x_1,\ldots,x_n]$ over a field $\mathbb{K}$ and explained its connection to the chip-firing game on graphs. The standard monomials of the Artinian quotient $\frac{R}{\mathcal{M}_G}$ correspond bijectively to $G$-parking functions. Dochtermann introduced and studied skeleton ideals of the graph $G$, which are subideals of the $G$-parking function ideal with an additional parameter $k ~(0\le k \le n-1)$. A $k$-skeleton ideal $\mathcal{M}_G^{(k)}$ of the graph $G$ is generated by monomials corresponding to non-empty subsets of the set of non-root vertices $[n]$ of size at most $k+1$. Dochtermann obtained many interesting homological and combinatorial properties of these skeleton ideals. In this paper, we study the $k$-skeleton ideals of graphs and for certain classes of graphs provide explicit formulas and combinatorial interpretation of standard monomials and the Betti numbers.