3D behaviour of Frieden filters in confocal imaging

The three-dimensional (3D) focal behaviour of the super-resolving Frieden filters is investigated numerically. It is shown that, as the central bright spot is sharpened, super-giant secondary maximums are formed on the optic axis. These lobes are much higher that the well-known side-lobes inherent to spatial filtering that surround the restricted, utilisable field, whose characteristics in the meridional plane are depicted for various values of the space-bandwidth parameter and for various numbers of terms that compose the window function. The two-terms filter is found to present, for the first time to my knowledge, some axial apodizing properties. To be compatible with practical realisation, the use of this class of filters in a single- and two-photon confocally scanned system is discussed in terms of 3D super-resolution with an intentionally limited light-power loss. It is shown that these filters match particularly well with recently designed axial apodizers for the transmission-mode confocal scanning microscope and provide a 3D intensity point-spread volume reduction of variable amount as high as 37 percent. The filtering process is shown to vary significantly with the mode of operation.