Incremental and Transitive Discrete Rotations

Abstract

A discrete rotation algorithm can be apprehended as a parametric map f α from Z[i] to Z[i], whose resulting permutation "looks like" the map induced by an Euclidean rotation. For this kind of algorithm, to be incremental means to compute successively all the intermediate rotated copies of an image for angles in-between 0 and a destination angle. The discretized rotation consists in the composition of an Euclidean rotation with a discretization; the aim of this article is to describe an algorithm which computes incrementally a discretized rotation. The suggested method uses only integer arithmetic and does not compute any sine nor any cosine. More precisely, its design relies on the analysis of the discretized rotation as a step function: the precise description of the discontinuities turns to be the key ingredient that will make the resulting procedure optimally fast and exact. A complete description of the incremental rotation process is provided, also this result may be useful in the specification of a consistent set of definitions for discrete geometry.

Computing upper and lower bounds of rotation angles from digital images

“…On the other hand, Nouvel and Rémila proposed in [7] another discrete rotation based on hinge angles, which gives the same results as DER. It is known that a sequence of hinge angles is a set of particular angles determined by a digital image in the sense that any angle between two consecutive hinge angles gives the identical rotated digital image.…”

“…Nouvel and Rémila presented a method for computing all possible hinge angles for a grid point or a pixel in a digital image [7]. Their method can be used for finding our interesting hinge angle which is the lower bound of the admissible rotation angles.…”

“…⌋ different hinge angles in each quadrant [7]. We can compare in magnitude any pair of hinge angles.…”

“…For such a set of admissible rotation angles, there exist two angles that achieve the lower and the upper bounds. To find those lower and upper bounds, we use hinge angles as used in [7]. A sequence of hinge angles is a set of particular angles determined by a digital image in the sense that any angle between two consecutive hinge angles gives the identical rotation of the digital image.…”

Efficient Neighbourhood Computing for Discrete Rigid Transformation Graph Search

“…This paradigm relies on a combinatorial structure, called discrete rigid transformation graph (DRT graph, for short) [12]. This structure describes the quantification of the parameter space of rigid transformations, in the framework of hinge angles, pioneered in [13][14][15].…”