Likelihood ratio Haar variance stabilization and normalization for Poisson and other non-Gaussian noise removal

Abstract: We propose a new methodology for denoising, variance-stabilizing and normalizing signals whose both mean and variance are parameterized by a single unknown varying parameter, such as Poisson or scaled chi-squared. Key to our methodology is the observation that the signed and square-rooted generalized log-likelihood ratio test for the equality of the local means is approximately and asymptotically distributed as standard normal under the null. We use these test statistics within the Haar wavelet transform at each scale and location, referring to them as the likelihood ratio Haar (LRH) coefficients of the data. In the denoising algorithm, the LRH coefficients are used as thresholding decision statistics, which enables the use of thresholds suitable for i.i.d. Gaussian noise, despite the standard Haar coefficients of the signal being heteroscedastic. In the variance-stabilizing and normalizing algorithm, the LRH coefficients replace the standard Haar coefficients in the Haar basis expansion. To the best of our knowledge, the variance-stabilizing and normalizing properties of the generalized likelihood ratio test have not been interpreted or exploited in this manner before. We prove the consistency of our LRH smoother for Poisson counts with a near-parametric rate, and various numerical experiments demonstrate the good practical performance of our methodology.