New theory in wavelet equation and its applications in astronomy

Methods based on hypothesis tests (HTs) in the Haar domain are widely used to denoise Poisson count data. Facing large datasets or real-time applications, Haar-based denoisers have to use the decimated transform to meet limited-memory or computation-time constraints. Unfortunately, for regular underlying intensities, decimation yields discontinuous estimates and strong "staircase" artifacts. In this paper, we propose to combine the HT framework with the decimated biorthogonal Haar (Bi-Haar) transform instead of the classical Haar, The Bi-Haar filter bank is normalized such that the p-values of Bi-Haar coefficients (PBH) provide good approximation to those of Haar (PH) for high-intensity settings or large scales; for low-intensity settings and small scales, we show that PBH are essentially upper-bounded by PH. Thus. we may apply the Haar-based HTs to Bi-Haar coefficients to control a prefixed false positive rate. By doing so, we benefit from the regular Bi-Haar filter bank to gain a smooth estimate while always maintaining a low computational complexity, A Fisher-approximation-based threshold implementing the HTs is also established. The efficiency of this method is illustrated on an example of hyperspectral-source-flux estimation.