In this contribution, the extent to which the Nyquist criterion can be violated in optical imaging systems with a digital sensor, e.g., a digital microscope, is investigated. In detail, we analyze the subpixel uncertainty of the detected position of a step edge, the edge of a stripe with a varying width, and that of a periodic rectangular pattern for varying pixel pitches of the sensor, thus also in aliased conditions. The analysis includes the investigation of different algorithms of edge localization based on direct fitting or based on the derivative of the edge profile, such as the common centroid method. In addition to the systematic error of these algorithms, the influence of the photon noise (PN) is included in the investigation. A simplified closed form solution for the uncertainty of the edge position caused by the PN is derived. The presented results show that, in the vast majority of cases, the pixel pitch can exceed the Nyquist sampling distance by about 50% without an increase of the uncertainty of edge localization. This allows one to increase the field-of-view without increasing the resolution of the sensor and to decrease the size of the setup by reducing the magnification. Experimental results confirm the simulation results.
The influence of photon noise to the signal evaluation of digital microscopy using a sinusoidal fringe pattern illumination with incoherent light is shown. The signal is evaluated by calculating the contrast for every charge coupled device (CCD) pixel when the object is defocused and the fringe pattern illumination shifted by a defined phase over the sample for every z -position. Every CCD pixel gets a certain number of irradiance values for every z -position which allows calculating the contrast. The result is the focal depth response (FDR) for every pixel. The FDR is Gaussian shaped and contains the height information of the specimen in the maximum. To accelerate the signal evaluation it is common to compute a fit curve to detect the maximum of the FDR. Due to the statistical photon noise, every measured irradiance value and every computed contrast value contains an error and thus also the maximum of the three-point-Gauss-curve-fit. The error of the maximum of the three-point-Gauss-curve-fit is the uncertainty of the measured height information. A general and a simplified analytical closed form solution are derived to calculate this uncertainty. An easily manageable equation allows calculating the optimal spatial frequency for an incoherent sinusoidal fringe pattern illumination and the corresponding sampling distance.
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