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 extensive ly 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 curv es. The selection and thorough understanding of the digital curve repr esentation scheme is crucial to the design of 3-D length estimators. A comprehensive study on the digitization of 3-D curves was recently ca rried 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 quantiz ation, that leads to 6-directional chain codes, should be preferred. T he few 3-D length estimators that have been suggested are based on 26- directional chain coding that naturally provides a classification of t he chain links, which is necessary for accurate length estimation. Cub e quantization is mathematically well-behaved but the symmetry and uni formity of the 6-directional digital chain elements create a challenge in their classification for length estimation. In this paper length e stimators for 3-D curves digitized using cube quantization are develop ed. 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 str aight lines.