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Image translation and rotation are becoming essential operations in many application areas such as image processing, computer graphics and pattern recognition. Conventional translation moves image from pixels to pixels and conventional rotation usually comprises of computation-intensive CORDIC operations. Traditionally, images are represented on a...
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... This implies that there will be less ambiguity in detecting symmetry of images. In general, the hexagonal structure provides a more flexible and efficient way to perform image translation and rotation without losing image information [6] and demonstrate the ability to better represent curved structures [7]. A considerable amount of research in hexagonally sampled images processing is taking place now despite the fact that there are no hardware resources that currently produce or display hexagonal images. ...
... Indeed, in the continuous complex plane, for any fixed 0 w = re iα ∈ C it is just composition of the rotation by an angle α about the origin O, with the r-scaling. Unfortunately, in discrete cases rotations and scalings are much more hard to define (see, for example, [19, p. 377] for square images and [1, p. 97], [6,7] for hexagonal images). Nevertheless, based on the analogy between C and E(p), one may expect that some properties of the transform f (z) → f (wz) could be similar to rotations even in the discrete case. ...
... Then the Eisenstein rotation R w [ f ] of f consists of three separate fragments wU 0 , wU 1 and wU −1 , localized near the points 0, w and −w respectively. 6 A. Karkishchenko, V. Mnukhin / Procedia Engineering 00 (2017) 000-000 ...
A new algebraic method for analysis and processing of hexagonally sampled images is presented. The method is based on the interpretation of such images as functions on “Eisenstein fields”. These are finite fields GF(p^2) of special characteristics p = 12k+5, where k > 0 is an integer. Some properties of such fields are studied; in particular, it is shown that its elements may be considered as ”discrete Eisenstein numbers” and are in natural correspondence with hexagons in a (p � p)-diamond-shaped fragment of a regular plane tiling. We show that in some cases multiplications in Eisenstein fields correspond to rotations combined with appropriate scalings, and use this fact for hexagonal images sharpening, smoothering and segmentation. The proposed algorithms have complexity O(p2) and can be used also for processing of square-sampled digital images over finite Gaussian fields.
This paper considers an algebraic method for symmetry analysis of hexagonally sampled images, based on the interpretation of such images as functions on “Eisenstein fields”. These are finite fields \(\mathbb {GF}(p^2)\) of special characteristics \(p=12k+5\), where \(k>0\) is an integer. Some properties of such fields are studied; in particular, it is shown that its elements may be considered as “discrete Eisenstein numbers” and are in natural correspondence with hexagons in a \((p\times p)\)-diamond-shaped fragment of a regular plane tiling. The concept of logarithm in Eisenstein fields is introduced and used to define a “log-polar”-representation of hexagonal images. Next, an algorithm for threefold symmetry detection in gray-level images is proposed.
