Locality preserving homogeneous Hilbert curves by use of arbitrary kernels

Abstract: Homogeneous Hilbert curves (HHC) in two dimensions are generalized by introducing the construction of the space filling curves from the same affine transformations but using an arbitrary kernel, we call such curves HHCK. The new curves are still space filling that comply with the nesting condition but violates the adjacency property. The freedom of building new HHC curves with arbitrary kernels, is only limited by the constrain that the chosen kernel must allow a well behaved connectivity between quadrants. Two examples of such curves are discussed. The important property of locality preservation in space filling curve mapping is discussed. Besides the common used dilation factor, the paper introduces and discuss difference map as a site locality measure, allowing to describe locality preservation in a more detailed way than dilation factors. The strength of such analysis is proven and global descriptors from the difference map are derived. Locality of all HHC curves and the two discussed HHCK curves are studied by difference map.