Optimal sampling strategies for surface roughness measurement.

This dissertation re-evaluates the two-dimensional sampling problem for measuring surface roughness and establishes that an optimal sampling strategy (rather than uniform) can result by utilizing the least discrepancy point sequences developed in Number theory. By modeling the random surface as Wiener process, the root-mean-square (RMS) error of measurement is shown to be equivalent to the $L\sb2$-discrepancy of the compliment of the sampling points. It is further shown that this relationship holds approximately for more general surfaces, thus linking the minimum RMS error of the measurement to the celebrated lower bound on $L\sb2$-discrepancy asserted by Roth, a 1958 Fields medalist. As a result of this dissertation, optimal sampling strategies for measuring surface roughness are obtained based on the Hammersley and Zaremba sequences, for they approach (for Hammersley sequence) or reach (for Zaremba sequence) the Roth's bound. As an evidence of improvement, it is shown that for the same order of RMS error in measurement, the number of sample points required by the optimal sampling sequences is reduced almost quadratically from the number of points required by the uniform sampling sequence. The theoretical developments are also demonstrated by simulations.PhDIndustrial and Operations EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/103102/1/9303771.pdfDescription of 9303771.pdf : Restricted to UM users only