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

Abstract

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

Bijective Digitized Rigid Motions on Subsets of the Plane

“…The main difficulty, with respect to the study of 2D digitized rigid motions, lies in the lack of a natural order of critical planes and high dimensionality of the parameter space. We recently proposed preliminary results on these topics in [10]. Neighborhood motion maps of G U 1 , as label maps, for θ ∈ π 6 , π 4 that differ from these for θ ∈ 0, π 6 .…”

Homotopic Digital Rigid Motion: An Optimization Approach on Cellular Complexes

“…Translations [5,19], rotations [1,2,6,13,27,28,31,37] and more generally rigid motions [22-26, 29, 32] in the Cartesian grids have been studied with various purposes: describing the combinatorial structure of these transformations with respect to R n vs. Z n [5,6,19,22,30,38], guaranteeing their bijectivity [1,2,13,27,31,32,37] or transitivity [28] in Z n , preserving geometrical properties [24,25] and, less frequently, ensuring their topological invariance [23,26] in Z n . These are non-trivial questions, and their difficulty increases with the dimension of the Cartesian grid [29].…”