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“Estimation of Image Signals with Poisson Noise”
by Chun Moo Lo
June 1979
An optimal filter in the sense of maximum a posteriori probability (MAP) is derived for image signals detected at low light levels. These signals suffer from Poisson noise and blurring degradations.
The low level photon resolved image signal is modeled as an inhomogeneous Poisson point process. The photon noise is inherent in any detected image, and is particularly serious at low light levels. At these low light levels, the emission of photons is described by a Poisson point process, with the average rate of emission proportional to the integrated intensity. The blurring degradation model in the system includes space-variant and space-invariant effects such as atmospheric turbulence, linear motion, diffraction, and aberrations. The estimation is performed assuming that the photon events counted in each detector are independent, Poisson distributed random processes for the large time-bandwidth product case. Since the variance of the Poisson distribution is identical to its mean, the Poisson noise is neither multiplicative noise nor a linear additive Gaussian noise, and is generally signal-dependent. It has been demonstrated that MAP estimation with the Poisson noise model has improved performance because the MAP filter can be generalized to linear or nonlinear image models and to noise models different from additive Gaussian noise. In addition, the MAP filter can be a local adaptive processing filter and extended to the case of space-variant blurring. It also has been shown that image models with a nonstationary mean and stationary variance give useful a priori information for the MAP filter. The MAP estimation equations are nonlinear and have large dimensionality. A sectioning method with a Newton-Raphson solution has been adapted to cope with these problems. It has been shown that the strategy is an effective and fast way to solve nonlinear MAP estimation equations.
The Cramer-Rao lower bound (CRLB) on the mean-square estimation error of the MAP unbiased estimate is derived for the Poisson noise model. It is shown to be a very useful bound for finding the best suboptimal sectioning filter. Finally, a comparison between the performance of the MAP filter and that of the linear minimum mean-square error (LMMSE) filter is made for Poisson noise models. The performance of the MAP filter is much better than that of the LMMSE filter. The LMMSE filter works very well for higher signal-to-noise ratios, but the MAP filter works better for low signal-to-noise ratios where Poisson noise dominates.