A quantitative optical microscopy arrangement is described. Specifically, a digital filter derived from linear discriminant analysis is described for recovering impulse responses in applications that may include photon counting from a high speed photodetector and applied to remove ringing distortions from impedance mismatch in multiphoton fluorescence microscopy. Training of the digital filter is achieved by defining temporally coincident and non-coincident transients and identifying the projection within filter-space that best separates the two classes. The training allows rapid data analysis by digital filtering. The LDA filter is also capable of recovering deconvolved impulses for single photon counting from highly distorted ringing waveforms from an impedance mismatched photomultiplier tube. The LDA filter is also successful in removing these ringing distortions from two-photon excited fluorescence micrographs and may extend the dynamic range of photon counting by about three orders of magnitude through minimization of detector paralysis.