Encoding and Decoding the Hilbert Order

Abstract

Explicit formulas are presented to generate the coordinates of a point on the two-dimensional Hilbert curve from its location code and vice versa. Execution-time assessments suggest that the proposed algorithms are faster than the ones published previously.

All-nearest-neighbors finding based on the Hilbert curve

“…4(a) shows the Hilbert curve of order n = 1 which linearly orders four blocks by four sequence numbers ranged from 0 to 3. The total ordering of the Hilbert curve is that the adjacent blocks in the two-dimensional space always correspond to the adjacent line intervals in the curve Liu & Schrack, 1996). Fig.…”

“…The number of disk accesses for the neighboring objects could be reduced by sequentially accessing adjacent blocks. The Hilbert curve has been widely used in a variety of fields including spatial databases (Chen & Chang, 2005;Jagadish, 1997;Lawder & King, 2001;Mokbel et al, 2003;Yiu, Tao, & Mamoulis, 2008), geographic information systems (Guo & Gahegan, 2006), image processing and compression (Chung, Huang, & Liu, 2007;Liang, Chen, Huang, & Liu, 2008), genome visualization (Deng et al, 2008) and scientific computing (Castro, Georgiopoulos, Demara, & Gonzalez, 2005;Liu & Schrack, 1996). Some neighbor finding strategies have been proposed based on the Hilbert curve (Bartholdi & Goldsman, 2001;Koudas, 2000;Liao, Lopez, & Leutenegger, 2001).…”

Efficient algorithms for coding Hilbert curve of arbitrary-sized image and application to window query

“…Considering a sub-image of size 2 r · 2 r , according to the efficient encoding scheme by Liu and Schrack [27], the pixel at location ðx; yÞ ¼ ððx rÀ1 . .…”

“…12.003 Hilbert curve is the most well-known. Previously, Liu and Schrack [27,28] presented efficient encoding and decoding algorithms to map 2-D/3-D images to Hilbert curves. Based on the Hilbert curve representation, many applications [3,4,6,8,10,18,[23][24][25]35,36,39,43,11,9,15,5,29] have been developed.…”

“…Section 2 presents the previous efficient coding scheme proposed by Liu and Schrack [27] for coding the Hilbert curve of a 2 r · 2 r image. Based on the proposed snake scan-based approach, Section 3 presents an efficient algorithm for coding the Hilbert curve of an arbitrary-sized image.…”