Computing upper and lower bounds of rotation angles from digital images

Abstract

Rotations in the discrete plane are important for many applications such as image matching or construction of mosaic images. We suppose that a digital image A is transformed to another digital image B by a rotation. In the discrete plane, there are many angles giving the rotation from A to B, which we called admissible rotation angles from A to B. 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. We propose a method for obtaining the lower and the upper bounds of admissible rotation angles using hinge angles from a given Euclidean angle or from a pair of corresponding digital images.

Sufficient Conditions for Topological Invariance of 2D Images under Rigid Transformations

“…Similar notions for the case of discrete rotations can be found in [5,6]. In particular, a discrete rigid transformation T behaves like a bijection for single pixels.…”

“…While rigid transformations are topology-preserving operations in R 2 , this property is generally lost in Z 2 , due to the discontinuities induced by the mandatory digitization from R to Z. In particular, discrete rigid transformations (DRTs) -that include discrete rotations [3,4,5,6]-are not guaranteed to preserve the homotopy type of digital images, as exemplified in Fig. 1.…”

Decomposition of nD-rotations: Classification, properties and algorithm

“…As an application of this work, we plan to use it in order to describe nD discrete rotations by extending the hinge angle notion developed for 3D rotations in [6].…”

“…Since, our framework is experimental, the method must be somewhat robust to noise. This work is supposed to be a basis for the description of nD discrete rotations by extension of hinge angles for 3D discrete rotations [6]. In this paper, we suppose without loss of generality that the rotations are centred on the origin of the frame and that the angles are in [0; π].…”

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].…”