A Measure of Shape Dissimilarity for 3D Curves

We present a quantitative approach to the measurement of shape dissimilarity between two 3D (three-dimensional) curves. Any 3D continuous curve can be digitalized and represented as a 3D discrete curve. Thus, a 3D discrete curve is composed of constant orthogonal straightline segments. In order to represent 3D discrete curves, we use the orthogonal direction change chain code. The chain elements represent the orthogonal direction changes of the contiguous straight-line segments of the discrete curve. This chain code only considers relative direction changes, which allows us to have a curve descriptor invariant under translation and rotation. Also, this curve descriptor may be starting point normalized and mirroring curves may be obtained with ease. Thus, using the above-mentioned chain code it is possible to have a unique 3D-curve descriptor. To find out how close in shape two 3D curves are, a measure of shapeof-curve dissimilarity between them is introduced; analogous curves will have a low measure of shape dissimilarity, while different curves will have a high measure of shape dissimilarity. When this measure of shape dissimilarity is normalized, its values vary continuously from 0 to 1. If Author to whom correspondence should be addressed 728 E. Bribiesca and W. Aguilar two curves are identical, the value of the measure of shape dissimilarity is equal to 0. The computation of this measure for two curves is based on the analysis of their common and different subcurves represented by their chain elements. Finally, we present some results of the computation of the proposed measure for 15 curves. Mathematics Subject Classification: 65D17