Length Estimation in 3-D Using Cube Quantization

Estimators for the original length of a continuous 3-D curve given its digital representation are developed. The 2-D case has been extensively studied. The few estimators that have been suggested for 3-D curves suffer from serious drawbacks, partly due to incomplete understanding of the characteristics of digital representation schemes for 3-D curves.The selection and thorough understanding of the digital curve representation scheme is crucial to the design of 3-D length estimators. A comprehensive study on the digitization of 3-D curves was recently carried out. It was shown that grid intersect quantization and other 3-D curve discretization schemes that lead to 26-directional chain codes do not satisfy several fundamental requirements, and that cube quantization, that leads to 6-directional chain codes, should be preferred.The few 3-D length estimators that have been suggested are based on 26-directional chain coding that naturally provides a classification of the chain links, which is necessary for accurate length estimation. Cube quantization is mathematically well-behaved but the symmetry and uniformity of the 6-directional digital chain elements create a challenge in their classification for length estimation.In this paper length estimators for 3-D curves digitized using cube quantization are developed. Simple but powerful link classification criteria for 6-directional digital curves are presented. They are used to obtain unbiased length estimators, with RMS errors as low as 0.57% for randomly oriented straight lines.