Since their introduction in the 1980s, wavelets have played a major role in the analysis of signals, revealing underlying structures on different scales. When applied to point processes, they have been successfully used to estimate their intensity in a non-parametric fashion. While both linear and non-linear wavelet estimators have been proposed, the theory resulting from the intersection of point processes and wavelet analysis has not been fully explored, in particular a sequential multiresolution analysis for point processes is lacking for further applications. We address this with newly defined multiscale properties named Jth-level homogeneity and Lth-level innovation, which together form a first-order multiresolution analysis of point processes. Using Haar wavelets, we propose likelihood ratio tests for these properties under the Poisson model, which in turn allows us to implement statistical thresholding procedures for estimating the intensity. Simulation studies reveal these procedures to have excellent performance on a range of intensity models.