Image formation in image scanning microscopy, including the case of two-photon excitation

The effect of combining the image scanning microscopy (ISM) technique with two-photon fluorescence microscopy is analyzed. The effective spatial frequency cutoff can be doubled, as compared with conventional two-photon fluorescence microscopy, and the magnitude of the optical transfer function near the cutoff of conventional two-photon microscopy is increased by orders of magnitude. For the two-photon case, it is found that the optimum pixel reassignment factor in ISM is not equal to one half, as is often assumed in single-photon fluoresence image scanning microscopy, because the excitation and detection point spread functions are different. The optimum reassignment factor depends on the noise level, and in general the useful cutoff spatial frequency is about 1.8 times that for conventional two-photon microscopy. The effect of altering the reassignment factor in single-photon fluorescence ISM with a Stokes shift is also investigated. Illumination using pupil filters, such as by a Bessel beam, is considered. Using a ring detector array is found to result in good imaging behavior, exhibiting a sharpening of the point spread function by a factor of 1.7 compared with conventional fluorescence. Image formation in ISM can be considered in a four-dimensional spatial frequency space, giving new insight into the imaging properties. This approach is related to phase space representations such as the Wigner distribution function and the ambiguity function. A noniterative algorithm for image restoration is proposed.