Computational geometry: an introduction
Computational geometry: an introduction
Theory of linear and integer programming
Theory of linear and integer programming
Polynomial time algorithms for finding integer relations among real numbers
SIAM Journal on Computing
Handbook of theoretical computer science (vol. A)
On the recognition of digital planes in three-dimensional space
Pattern Recognition Letters
A subexponential randomized simplex algorithm (extended abstract)
STOC '92 Proceedings of the twenty-fourth annual ACM symposium on Theory of computing
The real number model in numerical analysis
Journal of Complexity
A parametrization of digital planes by least-squares fits and generalizations
Graphical Models and Image Processing
Real data—integer solution problems with the Blum-Shub-Smale computational model
Journal of Complexity
Complexity and real computation
Complexity and real computation
Discrete analytical hyperplanes
Graphical Models and Image Processing
A Fast Algorithm for the Two-Variable Integer Programming Problem
Journal of the ACM (JACM)
Linear Programming in Linear Time When the Dimension Is Fixed
Journal of the ACM (JACM)
Lattice computers for approximating Euclidean space
Journal of the ACM (JACM)
Object discretizations in higher dimensions
Pattern Recognition Letters
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
Digital Planarity of Rectangular Surface Segments
IEEE Transactions on Pattern Analysis and Machine Intelligence
DGCI '02 Proceedings of the 10th International Conference on Discrete Geometry for Computer Imagery
Recognizing arithmetic straight lines and planes
DCGA '96 Proceedings of the 6th International Workshop on Discrete Geometry for Computer Imagery
A linear incremental algorithm for naive and standard digital lines and planes recognition
Graphical Models - Special issue: Discrete topology and geometry for image and object representation
Finding a small root of a bivariate integer equation; factoring with high bits known
EUROCRYPT'96 Proceedings of the 15th annual international conference on Theory and application of cryptographic techniques
EUROCRYPT'97 Proceedings of the 16th annual international conference on Theory and application of cryptographic techniques
Three-Dimensional Digital Planes
IEEE Transactions on Pattern Analysis and Machine Intelligence
Complexity analysis for digital hyperplane recognition in arbitrary fixed dimension
DGCI'05 Proceedings of the 12th international conference on Discrete Geometry for Computer Imagery
Computational aspects of digital plane and hyperplane recognition
IWCIA'06 Proceedings of the 11th international conference on Combinatorial Image Analysis
A generalized preimage for the digital analytical hyperplane recognition
Discrete Applied Mathematics
Recognition of digital hyperplanes and level layers with forbidden points
IWCIA'11 Proceedings of the 14th international conference on Combinatorial image analysis
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In this paper we present an algorithm for the integer linear programming (ILP) problem within an algebraic model of computation and use it to solve the following digital plane segment recognition problem: Given a set of points M={p^1,p^2,...,p^m}@?R^n, decide whether M is a portion of a digital hyperplane and, if so, determine its analytical representation. In our setting p^1, p^2, ...,p^m may be arbitrary points (possibly, with rational and/or irrational coefficients) and the dimension n may be any arbitrary fixed integer. We reduce this last problem to an ILP to which our general integer programming algorithm applies. It performs O(mlogD) arithmetic operations, where D is a bound on the norm of the domain elements. For the special case of problem dimension two, we propose an elementary algorithm that takes advantage of the specific geometry of the problem and appears to be optimal. It implies an efficient algorithm for digital line segment recognition.