Length estimation in 3-D using cube quantization

The estimation of the length of a continuous three dimensional curve from its digital image is considered. This requires a definition of the digitization process used for converting the continuous curve to the discrete representation. The two dimensional case has been extensively studied in the literature. The few available estimators for the 3-D case are based on 26- directional chain code representation of the digital curve. That representation provides natural classification of the chain code links which is necessary for accurate length estimation. Three- dimensional curve quantization methods are first considered. Desirable properties of curve representation schemes are identified and quantitative comparison of the various methods is carried out. It is shown that grid intersect quantization and other discretization schemes that lead to 26-directional chain code representations are inferior methods for curve quantization in 3-D and that cube quantization, leading to 6-directional chain codes, should be preferred. Accurate length estimation based on cube quantization has not been so far attempted due to the lack of obvious link classification criteria. In this paper simple but powerful link classification criteria for 6-directional digital curves are suggested. They are used to obtain unbiased length estimators, with rms errors as low as 0.57% for equally distributed straight lines, about five times better than in previous estimators that are based on 26-directional chain code representations.