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We propose a new methodology for denoising, variance-stabilizing andnormalizing signals whose both mean and variance are parameterized by a singleunknown varying parameter, such as Poisson or scaled chi-squared. Key to ourmethodology is the observation that the signed and square-rooted generalizedlog-likelihood ratio test for the equality of the local means is approximatelyand asymptotically distributed as standard normal under the null. We use thesetest 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 thresholdingdecision statistics, which enables the use of thresholds suitable for i.i.d.Gaussian noise, despite the standard Haar coefficients of the signal beingheteroscedastic. In the variance-stabilizing and normalizing algorithm, the LRHcoefficients replace the standard Haar coefficients in the Haar basisexpansion. To the best of our knowledge, the variance-stabilizing andnormalizing properties of the generalized likelihood ratio test have not beeninterpreted or exploited in this manner before. We prove the consistency of ourLRH smoother for Poisson counts with a near-parametric rate, and variousnumerical experiments demonstrate the good practical performance of ourmethodology.

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