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

Rigid 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.

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