  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.
