A new estimator (approximation) for the Euler-Poincaré characteristic of a planar set $K$ in the extended convex ring is suggested. As input, it uses only the digital image of $K$, which is modeled as the set of all points of a regular lattice falling in $K$. The key idea is to estimate the two planar Betti numbers of $K$ (number of connected components and number of holes) by approximating $K$ and its complement by polygonal sets derived from the digitization. In contrast to earlier methods, only certain connected components of these approximations are counted. The estimator of the Euler characteristic is then defined as the difference of the estimators for the two Betti numbers. Under rather weak regularity assumptions on $K$, it is shown that all three estimators yield the correct result, whenever the resolution of the image is sufficiently high.