  We study a generalization of additive bases into a planar setting. A planar
additive basis is a set of non-negative integer pairs whose vector sumset
covers a given rectangle. Such bases find applications in active sensor arrays
used in, for example, radar and medical imaging. The problem of minimizing the
basis cardinality has not been addressed before.
  We propose two algorithms for finding the minimal bases of small rectangles:
one in the setting where the basis elements can be anywhere in the rectangle,
and another in the restricted setting, where the elements are confined to the
lower left quadrant. We present numerical results from such searches, including
the minimal cardinalities for all rectangles up to $[0,11]\times[0,11]$, and up
to $[0,46]\times[0,46]$ in the restricted setting. We also prove asymptotic
upper and lower bounds on the minimal basis cardinality for large rectangles.
