Measurement of length in a finite picture
B. Roy Frieden · Annual Meeting Optical Society of America · 1985
The finite size of a picture should affect any length determination within the picture. This follows because the histogram of all possible lengths within a picture is not flat but is peaked at a length 0.46b and cuts off at length 1.414b (the diagonal), where b is the width of the square picture. With such a preference for length 0.46b, any Bayesian estimate of length (such as the posterior mean) will be biased toward it. The following scenario is presumed: (a) Prior to taking the picture, the two end points defining length can be equally anywhere within the picture. Only after viewing the picture are two end points of interest recognized, (b) The X and Y positions of each end point are located with additive, Gaussian error, (c) The end point measurements lie within the picture. A Monte Carlo approach was used to create true lengths and measured lengths in accordance with the preceding, forming a likelihood histogram whose mean is the required estimate. Results are shown as plots of estimated length vs measured length for a given sigma of noise in the end point location. If sigma is a decent fraction of picture size b, the estimated length departs significantly from the measurement.