  We investigate algorithms for testing whether an image is connected. Given a
proximity parameter $\epsilon\in(0,1)$ and query access to a black-and-white
image represented by an $n\times n$ matrix of Boolean pixel values, a (1-sided
error) connectedness tester accepts if the image is connected and rejects with
probability at least 2/3 if the image is $\epsilon$-far from connected. We show
that connectedness can be tested nonadaptively with $O(\frac 1{\epsilon^2})$
queries and adaptively with $O(\frac{1}{\epsilon^{3/2}}
\sqrt{\log\frac{1}{\epsilon}})$ queries. The best connectedness tester to date,
by Berman, Raskhodnikova, and Yaroslavtsev (STOC 2014) had query complexity
$O(\frac 1{\epsilon^2}\log \frac 1{\epsilon})$ and was adaptive. We also prove
that every nonadaptive, 1-sided error tester for connectedness must make
$\Omega(\frac 1\epsilon\log \frac 1\epsilon)$ queries.
