Length Estimation in 3-D Using Cube Quantization

Amnon Jonas, Nahum Kiryati

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
Pages (from-to)215-238
Number of pages24
JournalJournal of Mathematical Imaging and Vision
Issue number3
StatePublished - 1998


  • 3-D length estimation
  • Chain code
  • Cube quantization
  • Shape analysis
  • Three-dimensional digital geometry


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