The Wide Field Imager (WFI) flying on Athena will usher in the next era of studying the hot and energetic Universe. Among Athena's ambitious science programs are observations of faint, diffuse sources limited by statistical and systematic uncertainty in the background produced by high-energy cosmic ray particles. These particles produce easily identified "cosmic-ray tracks" along with less easily identified signals produced by secondary photons or Xrays generated by particle interactions with the instrument. Such secondaries produce identical signals to the X-rays focused by the optics, and cannot be filtered without also eliminating these precious photons. As part of a larger effort to estimate the level of unrejected background and mitigate its effects, we here present results from a study of background-reduction techniques that exploit the spatial correlation between cosmic-ray particle tracks and secondary events. We use Geant4 simulations to generate a realistic particle background signal, sort this into simulated WFI frames, and process those frames in a similar way to the expected flight and ground software to produce a realistic WFI observation containing only particle background. The technique under study, Self Anti-Coincidence or SAC, then selectively filters regions of the detector around particle tracks, turning the WFI into its own anti-coincidence detector. We show that SAC is effective at improving the systematic uncertainty for observations of faint, diffuse sources, but at the cost of statistical uncertainty due to a reduction in signal. If sufficient pixel pulse-height information is telemetered to the ground for each frame, then this technique can be applied selectively based on the science goals, providing flexibility without affecting the data quality for other science. The results presented here are relevant for any future silicon-based pixelated X-ray imaging detector, and could allow the WFI and similar instruments to probe to truly faint X-ray surface brightness.