Approximate Euclidean shortest path in 3-space
SCG '94 Proceedings of the tenth annual symposium on Computational geometry
Length Estimation in 3-D Using Cube Quantization
Journal of Mathematical Imaging and Vision
Static and kinetic geometric spanners with applications
SODA '01 Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms
Digital Curves in 3D Space and a Linear-Time Length Estimation Algorithm
IEEE Transactions on Pattern Analysis and Machine Intelligence
Segmentation and Length Estimation of 3D Discrete Curves
Digital and Image Geometry, Advanced Lectures [based on a winter school held at Dagstuhl Castle, Germany in December 2000]
Minimum-Length Polygons in Simple Cube-Curves
DGCI '00 Proceedings of the 9th International Conference on Discrete Geometry for Computer Imagery
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Proceedings of the thirty-fifth annual ACM symposium on Theory of computing
Digital Geometry: Geometric Methods for Digital Picture Analysis
Digital Geometry: Geometric Methods for Digital Picture Analysis
New lower bound techniques for robot motion planning problems
SFCS '87 Proceedings of the 28th Annual Symposium on Foundations of Computer Science
Minimum-length polygons of first-class simple cube-curves
CAIP'05 Proceedings of the 11th international conference on Computer Analysis of Images and Patterns
Minimum-Length polygon of a simple cube-curve in 3d space
IWCIA'04 Proceedings of the 10th international conference on Combinatorial Image Analysis
The class of simple cube-curves whose MLPs cannot have vertices at grid points
DGCI'05 Proceedings of the 12th international conference on Discrete Geometry for Computer Imagery
Automated DNA fragments recognition and sizing through AFM image processing
IEEE Transactions on Information Technology in Biomedicine
An approximation algorithm for computing minimum-length polygons in 3D images
ACCV'10 Proceedings of the 10th Asian conference on Computer vision - Volume Part IV
Finding the shortest path between two points in a simple polygon by applying a rubberband algorithm
PSIVT'06 Proceedings of the First Pacific Rim conference on Advances in Image and Video Technology
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Since 1987 it is known that the Euclidean shortest path problem is NP-hard. However, if the 3D world is subdivided into cubes, all of the same size, defining obstacles or possible spaces to move in, then the Euclidean shortest path problem has a linear-time solution, if all spaces to move in form a simple cube-curve. The shortest path through a simple cube-curve in the orthogonal 3D grid is a minimum-length polygonal curve (MLP for short). So far only one general and linear (only with respect to measured run times) algorithm, called the rubberband algorithm, was known for an approximative calculation of an MLP. The algorithm is basically defined by moves of vertices along critical edges (i.e., edges in three cubes of the given cube-curve). A proof, that this algorithm always converges to the correct MLP, and if so, then always (provable) in linear time, was still an open problem so far (the authors had successfully treated only a very special case of simple cube-curves before). In a previous paper, the authors also showed that the original rubberband algorithm required a (minor) correction. This paper finally answers the open problem: by a further modification of the corrected rubberband algorithm, it turns into a provable linear-time algorithm for calculating the MLP of any simple cube-curve. The paper also presents an alternative provable linear-time algorithm for the same task, which is based on moving vertices within faces of cubes. For a disticntion, we call the modified original algorithm now the edge-based rubberband algorithm, and the second algorithm is the face-based rubberband algorithm; the time complexity of both is in ${\cal O}(m)$, where m is the number of critical edges of the given simple cube-curve.