A new relative chain code in 3D

• We defined new rotations to be used to propose a 3D code.
• Equivalent simple paths allow to save information to be labeled.
• We found that a set of 25 symbols codify any face, edge and vertex connected 3D path.
• We applied the new code to a set of skeletons and digital elevation model data.
• We found invariance under transformations and high compression level of the new 3D code.

A new chain code to represent 3D discrete curves is proposed. The method is based on a search for relative changes in the 3D Euclidean space, composed of three main vectors: a reference vector, a support vector, and a change direction vector, utilized to obtain a directed simple path in a grid of 26 connected components. A set of rotation transformations is defined in the 3D Euclidean space, and an alphabet of only 25 symbols is required to represent any face, edge or vertex-connected discrete curve. Important properties of this code are found: independence under translation, rotation and mirror transformations, as well as high compression levels. A set of 3D curve-skeletons and digital elevation model data to study the terrain were utilized to prove the proposed code. Compared with the state-of-the-art, our method has more advantages: at first, it represents voxelized paths independently of vicinity, also it gives better representation for the tested objects and detects better the redundant parts. This fact is shown in the entropy calculated for 3D curve-skeletons: our method gives 3.03 bits/symbol, whereas the state-of-the-art method gives 4.35 bits/symbol. On the other hand, our proposed chain code uses 23% less memory than the well known Freeman code of 26 directions. In case of digital elevation models, our method improves memory for 36.1% regarding Freeman code and 10.7% regarding the well known relative code called orthogonal direction change chain code. Finally, average length of the chain code proposed is 14% shorter than the relative code of the state-of-the-art.