# On the metric properties of discrete space-filling curves

@article{Gotsman1994OnTM, title={On the metric properties of discrete space-filling curves}, author={Craig Gotsman and Michael Lindenbaum}, journal={Proceedings of the 12th IAPR International Conference on Pattern Recognition, Vol. 2 - Conference B: Computer Vision \& Image Processing. (Cat. No.94CH3440-5)}, year={1994}, pages={98-102 vol.3} }

Discrete space-filling curves are commonly used to reduce a multidimensional problem to a one-dimensional problem, and, as such, are exploited in a variety of image processing applications. The space-filling curve is essentially a linear traversal of the discrete multidimensional space. In order that this traversal be effective, the curve should preserve "locality", namely, that points close in the original multidimensional space be close in their ordering along the curve, and vice versa. We…

## 150 Citations

### Norm-Based Locality Measures of Two-Dimensional Hilbert Curves

- Computer Science
- 2019

The analytical results identify all candidate representative grid-point pairs (realizing the locality-measure values) for all real norm-parameters in the unit interval [1, 2] and grid-orders.

### Norm-Based Locality Measures of Two-Dimensional Hilbert Curves

- Computer Science
- 2020

The analytical results identify all candidate representative grid-point pairs (realizing the locality-measure values) for all real norm-parameters in the unit interval [1, 2] and grid-orders.

### Norm-Based Locality Measures of Two-Dimensional Hilbert Curves

- Computer ScienceAAIM
- 2016

The analytical results identify all candidate representative grid-point pairs (realizing the locality-measure values) for all real norm-parameters in the unit interval [1, 2] and grid-orders.

### An inventory of three-dimensional Hilbert space-filling curves

- MathematicsArXiv
- 2011

It is argued that the properties that make Hilbert's curve unique in two dimensions, are shared by 10694807 structurally different space-filling curves in three dimensions.

### On the Locality Properties of Space-Filling Curves

- MathematicsISAAC
- 2003

An analytical study of the locality properties of the m-dimensional k-order discrete Hilbert and z-order curve families is presented, which suggests that the z- order curve family performs better than the Hilbert curve family over the considered parameter ranges.

### On Multidimensional Curves with Hilbert Property

- Mathematics, Computer ScienceTheory of Computing Systems
- 2000

It is shown that there are 1536 structurally different simple three-dimensional Hilbert curves, and the formalism lays the groundwork for potential mechanized analysis of locality properties of multidimensional Hilbert curves.

### Space-Filling Curves based on Residue Number System

- Computer ScienceINCoS
- 2017

This paper proposes a novel algorithm for a space-filling curve based on the Residue Number System that is extensively studied during the last thirty years and has specific behavior and may be controlled by several parameters.

### Locality Properties of Space-Filling Curves

- Mathematics
- 2013

The overall properties of space-filling curves, together with the results for the examples shown in Figs. 10.2 and 10.3, indicate that space-filling curves are a good heuristics for an efficient load…

### Space-Filling Curve for Image Dynamical Indexing

- Computer ScienceISCIS
- 2012

A new fast technique for image indexing is developed which enables rapid insertions of new images without changing existing data, and a proposal of fast mapping facilitates the computing of 1-D Hilbert indexes from high dimensional features.

### Mapping high dimensional features onto Hilbert curve: Applying to fast image retrieval

- Computer ScienceProceedings of the 21st International Conference on Pattern Recognition (ICPR2012)
- 2012

Hilbert curve, the most neighborhood preserving space-filling curve, is used in shape retrieval and a proposal of fast mapping facilitate the computing of 1-D Hilbert indexes from high dimensional features.

## References

SHOWING 1-10 OF 10 REFERENCES

### Peano scanning of arbitrary size images

- MathematicsProceedings., 11th IAPR International Conference on Pattern Recognition. Vol. III. Conference C: Image, Speech and Signal Analysis,
- 1992

The authors aim to preserve two-dimensional continuity as much as possible by minimizing the weighted sum of the distances of the points in the curve, where the weights are inversely proportional to the spatial distance between the points.

### Space-filling curves

- Mathematics
- 1994

The subject of space-filling curves has generated a great deal of interest in the 100 years since the first such curve was discovered by Peano. Cantor, Hilbert, Moore, Knopp, Lebesgue and Polya are…

### Manipulation and Presentation of Multidimensional Image Data Using the Peano Scan

- Computer ScienceIEEE Transactions on Pattern Analysis and Machine Intelligence
- 1983

A fractal curve first discussed by Giuseppe Peano has useful statistical properties in image processing and applications include multispectral image display, compression in the color and spa-tial domain, classification, and color display.

### Compression of two-dimensional data

- Computer ScienceIEEE Trans. Inf. Theory
- 1986

The proposed picture compressibility is shown to possess the properties that one would expect and require of a suitably defined concept of two-dimensional entropy for arbitrary probabilistic ensembles of infinite pictures.

### Space diffusion: an improved parallel halftoning technique using space-filling curves

- Materials ScienceSIGGRAPH
- 1993

It is shown that by combining dot diffusion with a space-filling curve traversal technique, a parallelizable halftoning technique results that does not pay a cost in image quality.

### Optimal numberings of an N N array

- Mathematics
- 1986

Given a numbering of the vertices of a graph, one can define the edgesum [6] as the sum of differences between adjacent vertices. The problem of finding numberings which are optimal in the sense of…

### Edge-isoperimetric inequalities in the grid

- MathematicsComb.
- 1991

The result is apparently the first example of an isoperimetric inequality for which the extremal sets do not form a nested family and there are at leastkn−1 edges between a subset of [k]n of given cardinality and its complement.

### Über die stetige Abbildung einer Linie auf ein Flächenstück

- Philosophy
- 1935

Peano hat kurzlich in den Mathematischen Annalen2 durch eine arithmetische Betrachtung gezeigt, wie die Punkte einer Linie stetig auf die Punkte eines Flachenstuckes abgebildet werden konnen. Die fur…

### Ueber die stetige Abbildung einer Line auf ein Flächenstück

- Mathematics
- 1891

DigiZeitschriften e.V. gewährt ein nicht exklusives, nicht übertragbares, persönliches und beschränktes Recht auf Nutzung dieses Dokuments. Dieses Dokument ist ausschließlich für den persönlichen,…