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Distance Bounds for High Dimensional Consistent Digital Rays and 2-D Partially-Consistent Digital Rays.

Man-Kwun Chiu1, Matias Korman2, Martin Suderland3

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Summary
This summary is machine-generated.

Researchers explored digitalizing Euclidean segments, finding that existing lower bounds on construction error do not apply in higher dimensions. This opens new possibilities for more accurate digital representations and introduces the bichromatic discrepancy concept.

Keywords:
Computer visionConsistent digital line segmentsDigital geometryDiscrepancy

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Area of Science:

  • Computational Geometry
  • Digitalization Techniques
  • Geometric Algorithms

Background:

  • Digitalizing Euclidean segments requires consistent constructions satisfying geometric axioms.
  • Previous 2D constructions had asymptotically tight error bounds due to a lower bound.
  • This lower bound was previously thought to extend to higher dimensions.

Purpose of the Study:

  • To investigate constructive methods for digitalizing Euclidean segments in higher dimensions.
  • To re-evaluate the lower bounds on construction error in d-dimensional space.
  • To explore the relationship between axiom consistency, error, and construction techniques.

Main Methods:

  • Developed an alternative argument to establish a new lower bound for d-dimensional consistent constructions.
  • Linked the error of high-dimensional constructions to weak constructions in 2D.
  • Introduced and defined the bichromatic discrepancy concept.

Main Results:

  • Established that the 2D lower bound does not directly extend to higher dimensions.
  • Showed that any consistent d-dimensional construction must have a specific error bound.
  • Demonstrated a connection between high-dimensional construction error and 2D weak constructions.

Conclusions:

  • The findings challenge previous assumptions about error limitations in digitalizing Euclidean segments.
  • Opens avenues for developing more accurate high-dimensional digital constructions.
  • Highlights potential research in the trade-offs between axiom violations and construction error.