Quadric Arrangement in Classifying Rigid Motions of a 3D Digital Image

International audienceRigid motions are fundamental operations in image processing. While bijective and isometric in $\mathbb{R}^3$, they lose these properties when digitized in $\mathbb{Z}^3$. To understand how the digitization of 3D rigid motions affects the topology and geometry of a chosen image patch, we classify the rigid motions according to their effect on the image patch. This classification can be described by an arrangement of hypersurfaces in the parameter space of 3D rigid motions of dimension six. However, its high dimensionality and the existence of degenerate cases make a direct application of classical techniques, such as cylindrical algebraic decomposition or critical point method, difficult. We show that this problem can be first reduced to computing sample points in an arrangement of quadrics in the 3D parameter space of rotations. Then we recover information about remaining three parameters of translation. We implemented an ad-hoc variant of state-of-the-art algorithms and applied it to an image patch of cardinality $7$. This leads to an arrangement of 81 quadrics and we recovered the classification in less than one hour on a machine equipped with 40 cores

3D neighborhood motion maps (6-neighborhood)

The file is a sqlite3 database which contains 3D neighborhood motions maps of 6-neighborhood together with sample points. It was computed while using a variation of the algorithm described in: Pluta K., Moroz G., Kenmochi Y., Romon P. (2016) Quadric Arrangement in Classifying Rigid Motions of a 3D Digital Image. In: Gerdt V., Koepf W., Seiler W., Vorozhtsov E. (eds) Computer Algebra in Scientific Computing. CASC 2016. Lecture Notes in Computer Science, vol 9890. Springer, doi:10.1007/978-3-319-45641-6_27 To refer to a version of the algorithm used to compute the file use DOI:10.5281/zenodo.573013 (https://zenodo.org/badge/latestdoi/53963129)

Sets of second degree polynomials used in the problem of 3D neighborhood motion maps computations

The files are Maple source files which contain sets of second degree polynomials used in the problem of computing 3D neighborhood motion maps. The files contain polynomials related to: 6-neighbrhood (see quadrics_N1.mpl); 18-neighbrhood (see quadrics_N2.mpl) and 26-neighbrhood (see quadrics_N3.mpl). The polynomials are related to the paper: Pluta K., Moroz G., Kenmochi Y., Romon P. (2016) Quadric Arrangement in Classifying Rigid Motions of a 3D Digital Image. In: Gerdt V., Koepf W., Seiler W., Vorozhtsov E. (eds) Computer Algebra in Scientific Computing. CASC 2016. Lecture Notes in Computer Science, vol 9890. Springer, doi:10.1007/978-3-319-45641-6_27 To refer to a version of the algorithm used to compute the file use DOI:10.5281/zenodo.573013 (https://zenodo.org/badge/latestdoi/53963129)

Computer Algebra in Scientific Computing [electronic resource] : 18th International Workshop, CASC 2016, Bucharest, Romania, September 19-23, 2016, Proceedings /

This book constitutes the proceedings of the 18th International Workshop on Computer Algebra in Scientific Computing, CASC 2016, held in Bucharest, Romania, in September 2016. The 32 papers presented in this volume were carefully reviewed and selected from 39 submissions. They deal with cutting-edge research in all major disciplines of Computer Algebra.On the Differential and Full Algebraic Complexities of Operator Matrices Transformations -- Resolving Decompositions for Polynomial Modules -- Setup of Order Conditions for Splitting Methods -- Symbolic Manipulation of Flows of Nonlinear Evolution Equations, with Application in the Analysis of Split-Step Time Integrators -- Improved Computation of Involutive Bases -- Computing all Space Curve Solutions of Polynomial Systems by Polyhedral Methods -- Algorithmic Computation of Polynomial Amoebas -- Sparse Gaussian Elimination Modulo p: an Update -- MathCheck2: A SAT+CAS Verifier for Combinatorial Conjectures -- Incompleteness, Undecidability and Automated Proofs (Invited Talk) -- A Numerical Method for Computing Border Curves of Bi-Parametric Real Polynomial Systems and Applications -- The Complexity of Cylindrical Algebraic Decomposition with Respect to Polynomial Degree -- Efficient Simplification Techniques for Special Real Quantifier Elimination with Applications to the Synthesis of Optimal Numerical Algorithms -- Symbolic-Numeric Algorithms for Solving BVPs for a System of ODEs of the Second Order: Multichannel Scattering and Eigenvalue Problems -- Symbolic Algorithm for Generating Irreducible Rotational-Vibrational Bases of Point Groups -- A Symbolic Investigation of the Influence of Aerodynamic Forces on Satellite Equilibria -- Computer Algebra in High-Energy Physics (Invited Talk) -- A Note on Dynamic Gröbner Bases Computation -- Qualitative Analysis of the Reyman-Semenov-Tian-Shansky Integrable Case of the Generalized Kowalewski Top -- On Multiple Eigenvalues of a Matrix Dependent on a Parameter -- A Generalised Branch-and-Bound Approach and its Application in SAT Modulo Nonlinear Integer Arithmetic -- Computing Characteristic Polynomials of Matrices of Structured Polynomials -- Computing Sparse Representations of Systems of Rational Fractions -- On the General Analytical Solution of the Kinematic Cosserat Equations -- Using Sparse Interpolation in Hensel Lifting -- A Survey of Satisfiability Modulo Theory -- Quadric Arrangement in Classifying Rigid Motions of a 3D Digital Image -- A Lower Bound for Computing Lagrange's Real Root Bound -- Enhancing the Extended Hensel Construction by Using Gröbner Bases -- Symbolic-Numerical Optimization and Realization of the Method of Collocations and Least Residuals for Solving the Navier-Stokes Equations -- Pruning Algorithms for Pretropisms of Newton Polytopes -- Computational Aspects of a Bound of Lagrange.This book constitutes the proceedings of the 18th International Workshop on Computer Algebra in Scientific Computing, CASC 2016, held in Bucharest, Romania, in September 2016. The 32 papers presented in this volume were carefully reviewed and selected from 39 submissions. They deal with cutting-edge research in all major disciplines of Computer Algebra