Unexpected structure in images of astronomical sources often presents itself upon
visual inspection of the image, but such apparent structure may either correspond to
true features in the source or be due to noise in the data. This paper presents a
method for testing whether inferred structure in an image with Poisson noise represents a
significant departure from a baseline (null) model of the image. To infer image structure,
we conduct a Bayesian analysis of a full model that uses a multiscale component to
allow flexible departures from the posited null model. As a test statistic, we use a
tail probability of the posterior distribution under the full model. This choice of test
statistic allows us to estimate a computationally efficient upper bound on a p-value
that enables us to draw strong conclusions even when there are limited computational
resources that can be devoted to simulations under the null model. We demonstrate
the statistical performance of our method on simulated images. Applying our method
to an X-ray image of the quasar 0730+257, we find significant evidence against the null
model of a single point source and uniform background, lending support to the claim of
an X-ray jet.