Fast Sampling from Wiener Posteriors for Image Data with Dataflow Engines

We use Dataflow Engines (DFE) to construct an efficient Wiener filter of noisy and incomplete image data, and to quickly draw probabilistic samples of the compatible true underlying images from the Wiener posterior. Dataflow computing is a powerful approach using reconfigurable hardware, which can be deeply pipelined and is intrinsically parallel. The unique Wiener-filtered image is the minimum-variance linear estimate of the true image (if the signal and noise covariances are known) and the most probable true image (if the signal and noise are Gaussian distributed). However, many images are compatible with the data with different probabilities, given by the analytic posterior probability distribution referred to as the Wiener posterior. The DFE code also draws large numbers of samples of true images from this posterior, which allows for further statistical analysis. Naive computation of the Wiener-filtered image is impractical for large datasets, as it scales as $n^3$, where $n$ is the number of pixels. We use a messenger field algorithm, which is well suited to a DFE implementation, to draw samples from the Wiener posterior, that is, with the correct probability we draw samples of noiseless images that are compatible with the observed noisy image. The Wiener-filtered image can be obtained by a trivial modification of the algorithm. We demonstrate a lower bound on the speed-up, from drawing 10$^5$ samples of a 128$^2$ image, of 11.3 ${\pm}$ 0.8 with 8 DFEs in a 1U MPC-X box when compared with a 1U server presenting 32 CPU threads. We also discuss a potential application in astronomy, to provide better dark matter maps and improved determination of the parameters of the Universe.