Fast, near-optimal, multiresolution estimation of poisson signals and images

The nonparametric multiscale methods presented here are powerful tools for Poisson signal and image denoising and reconstruction. These methods offer near minimax convergence rates for broad classes of functions. However, the computational burden these methods impose makes them impractical for many applications which involve iterative algorithms, such as deblurring and tomographic reconstruction. The techniques described in this paper allow multiscale Poisson signal and image reconstruction methods to be implemented with significantly less computational complexity than previously possible. Fast translation-invariant Haar denoising for Poisson data is accomplished by deriving the relationship between maximum penalized likelihood tree pruning decisions and the undecimated wavelet transform coefficients. Fast wedgelet and platelet methods are accomplished with a coarse-to-fine technique which detects possible boundary locations before performing wedgelet or platelet fits.