Rotations in 2D and 3D discrete spaces

This thesis presents a study on rotation in 2 dimensional and 3 dimensional discrete spaces. In computer science, using floating numbers is problematic due to computation errors. Thus we chose during this thesis to work only in discrete space. In the field of computer vision, the rotation is a transformation required for many applications. Using discretized Euclidean rotation gives bad results. Then, it is necessary to develop new rotation methods adapted to the discrete spaces. We mainly studied the hinge angles that represent the discontinuity of the rotation in the discrete space. Indeed, it is possible to perform two rotations of the same digital image with two angles that are slightly different and obtain the same result. This is captured by hinge angles. Using these angles allow us to describe a discrete rotation that gives the same results than the discretized Euclidean rotation without using floating numbers. They also allow describing an incremental rotation that performs all possible rotations of a given digital image. Using hinge angles can also be extended to the rotations in 3 dimensional discrete spaces. The extension requires the multi-grids that are rotation planes containing three sets of parallel lines. These parallel lines represent the discontinuities of the rotation in 3D discrete space. Thus they are useful to describe the hinge angles in rotation planes. Multi-grids allow obtaining the same results in 3D discrete rotations than the results obtained in 2D discrete rotations. This thesis presents a study on rotation in 2 dimensional and 3 dimensional discrete spaces. In computer science, using floating numbers is problematic due to computation errors. Thus we chose during this thesis to work only in discrete space. In the field of computer vision, the rotation is a transformation required for many applications. Using discretized Euclidean rotation gives bad results. Then, it is necessary to develop new rotation methods adapted to the discrete spaces. We mainly studied the hinge angles that represent the discontinuity of the rotation in the discrete space. Indeed, it is possible to perform two rotations of the same digital image with two angles that are slightly different and obtain the same result. This is captured by hinge angles. Using these angles allow us to describe a discrete rotation that gives the same results than the discretized Euclidean rotation without using floating numbers. They also allow describing an incremental rotation that performs all possible rotations of a given digital image. Using hinge angles can also be extended to the rotations in 3 dimensional discrete spaces. The extension requires the multi-grids that are rotation planes containing three sets of parallel lines. These parallel lines represent the discontinuities of the rotation in 3D discrete space. Thus they are useful to describe the hinge angles in rotation planes. Multi-grids allow obtaining the same results in 3D discrete rotations than the results obtained in 2D discrete rotations