Length Estimation in 3

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 suuer 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 classiication 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 classiication for length estimation. In this paper length estimators for 3-D curves digitized using cube quantization are developed. Simple but powerful link classiication 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, about ve times better than in the best estimators that have been so far available.