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Digital Curves in 3D Space and a Linear-Time Length Estimation Algorithm
, 1999
"... We consider simple digital curves in a 3D orthogonal grid as special polyhedrally bounded sets. These digital curves model digitized curves or arcs in three-dimensional euclidean space. The length of such a simple digital curve is defined to be the length of the minimum-length polygonal curve ful ..."
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Cited by 10 (10 self)
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We consider simple digital curves in a 3D orthogonal grid as special polyhedrally bounded sets. These digital curves model digitized curves or arcs in three-dimensional euclidean space. The length of such a simple digital curve is defined to be the length of the minimum-length polygonal curve fully contained and complete in the tube of this digital curve. So far no algorithm was known for the calculation of such a shortest polygonal curve. This paper provides an iterative algorithmic solution, including a presentation of its foundations and of experimental results.
Exact and approximate algorithms for the calculation of shortest paths
- IMA Minneapolis, Report 2141 on www.ima.umn.edu/preprints/oct2006
, 2006
"... Ptolemy once asked Euclid whether there was any shorter way to a knowledge of geometry than by a study of the Elements, whereupon Euclid answered that there was no royal road to geometry. iii Proclus Diadochus ..."
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Cited by 5 (5 self)
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Ptolemy once asked Euclid whether there was any shorter way to a knowledge of geometry than by a study of the Elements, whereupon Euclid answered that there was no royal road to geometry. iii Proclus Diadochus
Minimum-length polygon of a simple cube-curve in 3D space
- In Proc. IWCIA, LNCS 3322: 502–511
, 2004
"... Abstract. We consider simple cube-curves in the orthogonal 3D grid of cells. The union of all cells contained in such a curve (also called the tube of this curve) is a polyhedrally bounded set. The curve’s length is defined to be that of the minimum-length polygonal curve (MLP) fully contained and c ..."
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Cited by 3 (3 self)
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Abstract. We consider simple cube-curves in the orthogonal 3D grid of cells. The union of all cells contained in such a curve (also called the tube of this curve) is a polyhedrally bounded set. The curve’s length is defined to be that of the minimum-length polygonal curve (MLP) fully contained and complete in the tube of the curve. So far, only a ”rubber-band algorithm” is known to compute such a curve approximately. We provide an alternative iterative algorithm for the approximative calculation of the MLP for curves contained in a special class of simple cube-curves (for which we prove the correctness of our alternative algorithm), and the obtained results coincide with those calculated by the rubber-band algorithm. 1
R.: Rubber Band Algorithm for Estimating the Length of Digitized Space-Curves, 15th Int
- Con. Pattern Recog., ICPR
, 2000
"... We consider simple digital curves in a 3D orthogonal grid as special polyhedrally bounded sets. These digital curves model digitized curves or arcs in three-dimensional euclidean space. The length of such a simple digital curve is defined to be the length of the minimum-length polygonal curve fully ..."
Abstract
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Cited by 3 (2 self)
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We consider simple digital curves in a 3D orthogonal grid as special polyhedrally bounded sets. These digital curves model digitized curves or arcs in three-dimensional euclidean space. The length of such a simple digital curve is defined to be the length of the minimum-length polygonal curve fully contained and complete in the tube of this digital curve. So far no algorithm was known for the calculation of such a shortest polygonal curve. This paper provides an iterative algorithmic solution, including a presentation of its foundations and of experimental results. 1.
Minimum-Length Polygons of First-Class Simple Cube-Curve
- In Proc. Computer Analysis Images Patterns, LNCS 3691
, 2005
"... Abstract. We consider simple cube-curves in the orthogonal 3D grid. The union of all cells contained in such a curve (also called the tube of this curve) is a polyhedrally bounded set. The curve’s length is defined to be that of the minimum-length polygonal curve (MLP) fully contained and complete i ..."
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Cited by 2 (2 self)
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Abstract. We consider simple cube-curves in the orthogonal 3D grid. The union of all cells contained in such a curve (also called the tube of this curve) is a polyhedrally bounded set. The curve’s length is defined to be that of the minimum-length polygonal curve (MLP) fully contained and complete in the tube of the curve. So far only one general algorithm called rubber-band algorithm was known for the approximative calculation of such an MLP. A proof that this algorithm always converges to the correct curve, is still an open problem. This paper proves that the rubber-band algorithm is correct for the family of first-class simple cube-curves. 1
Analysis of the rubberband algorithm
, 2006
"... Abstract. We consider simple cube-curves in the orthogonal 3D grid of cells. The union of all cells contained in such a curve (also called the tube of this curve) is a polyhedrally bounded set. The curve’s length is defined to be that of the minimum-length polygonal curve (MLP) contained and complet ..."
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Cited by 1 (1 self)
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Abstract. We consider simple cube-curves in the orthogonal 3D grid of cells. The union of all cells contained in such a curve (also called the tube of this curve) is a polyhedrally bounded set. The curve’s length is defined to be that of the minimum-length polygonal curve (MLP) contained and complete in the tube of the curve. Only one general algorithm, called rubberband algorithm, was known for the approximative calculation of such an MLP so far. An open problem in [7] is related to the design of algorithms for the calculation of the MLP of a simple cube-curve: Is there a simple cubecurve such that none of the nodes of its MLP is a grid vertex? This paper constructs an example of such a simple cube-curve, and we also characterize the class of all of such cube-curves. This study leads to a correction in Option 3 of the rubberband algorithm (by adding one missing test). We also prove that the rubberband algorithm has linear time complexity O(m) where m is the number of critical edges of a given simple cube curve, which solves another open problem in the context of this algorithm. 1
Approximation of 3D Shortest Polygons in Simple Cube Curves
, 2001
"... One possible definition of the length of a digitized curve in 3D is the length of the shortest polygonal curve lying entirely in a cube curve. In earlier work the authors proposed an iterative algorithm for the calculation of this minimal length polygonal curve (MLP). This paper reviews the algorith ..."
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Cited by 1 (1 self)
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One possible definition of the length of a digitized curve in 3D is the length of the shortest polygonal curve lying entirely in a cube curve. In earlier work the authors proposed an iterative algorithm for the calculation of this minimal length polygonal curve (MLP). This paper reviews the algorithm and suggests methods to speed it up by reducing the set of possible locations of vertices of the MLP, or by directly calculating MLP-vertices in specific situations. Altogether, the paper suggests an in-depth analysis of cube curves.
Two Approximative Algorithms for Calculating Minimum-Length Polygons in 3D Space
"... We consider simple cube-curves in the orthogonal 3D grid. The union of all cells contained in such a curve (also called the tube of this curve) is a polyhedrally bounded set. The curve’s length is defined to be that of the minimum-length polygonal curve (MLP) fully contained and complete in the tube ..."
Abstract
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We consider simple cube-curves in the orthogonal 3D grid. The union of all cells contained in such a curve (also called the tube of this curve) is a polyhedrally bounded set. The curve’s length is defined to be that of the minimum-length polygonal curve (MLP) fully contained and complete in the tube of the curve. So far no provable general algorithm is known for the approximative calculation of such an MLP. This paper presents two approximative algorithms for computing the MLP of a general simple cube-curve in O(n 4) time, where n is the total number of critical edges of the given simple cube-curve.

