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}
}
  • C. GotsmanM. Lindenbaum
  • Published 9 October 1994
  • Computer Science
  • Proceedings of the 12th IAPR International Conference on Pattern Recognition, Vol. 2 - Conference B: Computer Vision & Image Processing. (Cat. No.94CH3440-5)
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… 

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References

SHOWING 1-10 OF 10 REFERENCES

Peano scanning of arbitrary size images

  • A. PerezS. KamataE. Kawaguchi
  • Mathematics
    Proceedings., 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 AND A MEASURE OF COHERENCE

Space-filling curves

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

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

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

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

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

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

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

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