The study of the evolution process of our visual system indicates the existence of variational spatial arrangement; from densely hexagonal in the fovea to a sparse circular structure in the peripheral retina. Today’s sensor spatial arrangement is inspired by our visual system. However, we have not come further than rigid rectangular and, on a minor scale, hexagonal sensor arrangements. Even in this situation, there is a need for directly assessing differences between the rectangular and hexagonal sensor arrangements, i.e., without the conversion of one arrangement to another. In this paper, we propose a method to create a common space for addressing any spatial arrangements and assessing the differences among them, e.g., between the rectangular and hexagonal. Such a space is created by implementing a continuous extension of discrete Weyl Group orbit function transform which extends a discrete arrangement to a continuous one. The implementation of the space is demonstrated by comparing two types of generated hexagonal images from each rectangular image with two different methods of the half-pixel shifting method and virtual hexagonal method. In the experiment, a group of ten texture images were generated with variational curviness content using ten different Perlin noise patterns, adding to an initial 2D Gaussian distribution pattern image. Then, the common space was obtained from each of the discrete images to assess the differences between the original rectangular image and its corresponding hexagonal image. The results show that the space facilitates a usage friendly tool to address an arrangement and assess the changes between different spatial arrangements by which, in the experiment, the hexagonal images show richer intensity variation, nonlinear behavior, and larger dynamic range in comparison to the rectangular images.

We deal with the Fourier-like analysis of functions on discrete grids in two-dimensional simplexes using C-and E-Weyl group orbit functions. For these cases, we present the convolution theorem. We provide an example of application of image processing using the C-functions and the convolutions for spatial filtering of the treated image.

No abstract

We deal with the Fourier-like analysis of functions on discrete grids in two-dimensional simplexes using C− and E− Weyl group orbit functions. For these cases we present the convolution theorem. We provide an example of application of image processing using the C− functions and the convolutions for spatial filtering of the treated image.

The study of the evolution process of our visual system indicates the existence of variational spatial arrangement; from densely hexagonal in the fovea to a sparse circular structure in the peripheral retina. Today's sensor spatial arrangement is inspired by our visual system. However, we have not come further than rigid rectangular and, in a minor scale, hexagonal sensor arrangement. Even in this situation, there is a need to be able to directly assess differences between the rectangular and hexagonal sensor arrangements, i.e., without the conversion of one arrangement to another. In this paper, we propose a method to create a common space for directly assessing the differences among any spatial arrangements, e.g., between the rectangular and hexagonal. Such a space is created by implementing a continuous extension of discrete Weyl Group orbit function transform which extends a discrete arrangement to a continuous one. The implementation of the space is demonstrated by comparing two types of generated hexagonal images from each rectangular image with two different methods of the half-pixel shifting method and virtual hexagonal method. In the experiment, a group of ten texture images are generated with variational curviness content using ten different Perlin noise patterns adding to an initial 2D Gaussian distribution pattern image. Then, the common space is obtained from each of the discrete images to assess the differences between the original rectangular image and its corresponding hexagonal image. The results show that the space facilitates an easy tool to assess the changes between different spatial arrangements by which, in the experiment, the hexagonal images show richer intensity variation, nonlinear behavior, and larger dynamic range in comparison to the rectangular images.