Fast multiresolution photon-limited image reconstruction

The techniques described in this paper allow multiscale photon-limited image reconstruction methods to be implemented with significantly less computational complexity than previously possible. Methods such as multiscale Haar estimation, wedgelets, and platelets are all promising techniques in the context of Poisson data, but the computational burden they impose makes them impractical for many applications which involve iterative algorithms, such as deblurring and tomographic reconstruction. With the advent of the proposed implementation techniques, hereditary translation-invariant Haar wavelet-based estimates can be calculated in O (N log N) operations and wedgelet and platelet estimates can be computed in O (N-7/6) operations, where N is the number of pixels; these complexities are comparable to those of standard wavelet denoising (O (N)) and translation-invariant wavelet denoising (O (N log N)). 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.