Sampling theorem for geometric moment determination and its application to a laser beam position detector

Area of Science:

• Signal Processing
• Applied Mathematics
• Optics

Background:

• Accurate calculation of distribution moments is crucial in various scientific and engineering applications.
• The Whittaker-Shannon sampling theorem provides a basis for signal reconstruction but may not be optimal for moment determination.
• Existing methods for moment calculation from sampled data often require a large number of samples.

Purpose of the Study:

• To derive a specific sampling requirement for accurately calculating geometric moments of a distribution.
• To develop a formula for calculating moments from discrete samples.
• To assess the practical applicability and accuracy of the derived method in real-world scenarios.

Main Methods:

• Fourier analysis was employed to derive the sampling requirement.
• The derivation builds upon the Whittaker-Shannon sampling theorem, identifying a coarser interval for moment determination.
• Numerical analysis was conducted to evaluate the accuracy of the first moment calculation under non-ideal sampling conditions.

Main Results:

• A theoretical sampling requirement was established, ensuring sufficient information for accurate geometric moment calculation.
• A formula for computing moments from samples was derived.
• Numerical analysis quantified the accuracy of the first moment calculation, considering factors like sampling aperture, quantization, and noise.

Conclusions:

• The derived sampling requirement is less stringent than that for full distribution reconstruction, enabling efficient moment determination.
• The developed method and formula are validated through numerical analysis and practical application.
• The theory is successfully applied to a laser beam position detector for precise measurement of laser printer raster line accuracy.