Alternative patterns of the multidimensional Hilbert curve

Locality-preserving (distance preserving-mapping) is a useful property to manage multidimensional data. Close points in space remain -as much as possible- close after mapping on curve. That is why Hilbert space-filling curve is used in many domains and applications. Hilbert curve preserves well locality because from a construction aspect, it is guided by adajacency constraint on points ordering : the curve connects all points of a D-dimensional discrete space, without favoring any direction, under the constrainst that two successive points are separated by an unit distance. Originally defined in 2-D, all existing multidimensional extensions of the Hilbert curve satisfy adjacency by using the RBG pattern (based on Reflected Binary Gray code). The RBG pattern is then duplicated and arranged (geometrical transformations) to build the multidimensional Hilbert curve at a given order. In this paper, we emphasize that there are other patterns that can satisfy the adjacency. A formulation is given, an algorithm to find out solutions is provided and their respective level of locality preservation is estimated through a standard criterion. Results show that some new patterns can carry a comparable levels of locality and sometimes better than RBG. Moreover, selecting the best locality preserving pattern allows one to design, through orders, a new curve with a comparable overall locality preserving refer to Hilbert curve. The contribution of new patterns is experimented through a CBIR (Content-Based Image Retrieval) application. Large-scale image retrieval tests show that exploring the image feature space with an alternative way to the classical Hilbert curve can lead to improved image searching performances.