^{1}

^{*}

^{1}

^{*}

^{1}

^{*}

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 development of information technologies has inspired also the development of the information compression, the most famous part of which is the image and video compression. The compression is based on the information structure in order to optimize compression speed, compression rate and the possible losses of information during the compression. Development of the theory of orbit functions opens a space for their use in the processing of the information sampled on grids in simplexes and polyhedra in n-dimensional space. These functions can be used for decomposition of any discrete values on the grids to orthogonal series. The density of grid points is controlled by a suitable choice of parameter. Moreover, we can glue together more simplexes and study the information carried in the grid in this ensemble. In this paper, we focus on the simplest non-trivial case of utilization of orbit functions in two dimensions. It corresponds to a two-dimensional digital image processing. In comparison with the most widespread method for image processing—Fourier analysis, i.e., the decomposition into exponential series in two perpendicular directions, we decompose discrete functions on points of the grid in a number of orbit functions without the division into several directions. Our approach is a generalization of discrete Fourier and cosine transform.

In this paper, we summarize the properties of Cand E-orbit functions connected with Weyl groups of simple Lie algebras

The paper is organized as follows. Section 2 summarizes some known facts about the spatial filtering using a convolution. In Section 3, we remind basic notations from the theory of Weyl group orbit functions. In particular, we describe the discrete transforms based on finite families of orbit functions in SubSection 3.3. In Section 4, we define Cand E-orbit convolution and formulate the orbit convolutions theorems. Finally, in SubSection 4.2, we provide examples of image processing using C-orbit functions. We include two appendices with technical details for the orbit transforms.

A variety of filters play an important role in image processing, in image improving and in detail recognition. For example, the spatial filtering uses convolution of functions which is performed via Fourier transform as a multiplication of the Fourier images. Fourier analysis is based on the decomposition of brightness values in each digitized image points along the rows and columns into Fourier series. The Fourier transform is then processed. The inverse discrete Fourier transform shows processing of digital images. This way we can highlight some features of the image—remove the noise or enhance blur edges. The whole process is described in several papers, for an overview see for example [1,2]. For image compression JPEG the discrete cosine transforms are used. They are of four types and the convolution via multiplications in these cases is more complicated, it combines cosine and sine discrete transform except the discrete cosine transform of type II. The simplest filtering technique is the averaging the light intensities at points. Intensity of each new pixel is the mean value of the intensities of the 8 neighboring pixels and the pixel itself in the original image. Other filters use the intensities of neighboring pixels multiplied by different relative weights and the pixel is assigned by a mean value of 9 intensities. Other filters take into account a number of other surrounding pixels, 25 pixels together with the center. Intensities in 9 or 25 pixel can be expressed as

Filters mentioned above are called linear spatial filters. Their application to a digital image creates a new image using a linear combination of brightness values in the surrounding pixels. The intensities of the digital image in each pixel are defined by the matrix

New digital image has the intensity in each pixel given by a matrix

This corresponds to the sum of all the values of the

For defining the orbit convolutions we proceed in a similar way as for the discrete cosine transform DCT II, where for two functions

and for cosine transform

We consider the simple Lie algebras of rank two, namely

We denote the reflections with respect to the hyperplanes orthogonal to the simple roots by

Let

The affine Weyl group

and

The even Weyl group

Three families of Weyl group orbit functions, so-called C-, Sand E-functions, are defined in the context of any Weyl group. Their complete description can be found in the series of papers [6-8]. The family of C-functions is defined as follows: For every

The functions are invariant with respect to the affine Weyl group, therefore, we can consider

The family of S-functions is defined for every

They are antiinvariant with respect to

Finally, the E-orbit functions are defined for every

They are invariant with respect to the even affine Weyl group, we restrict them on

For Weyl groups with two different lengths of root in their root system other families of orbit functions can be defined. For more details see [9,10]. In this paper, we consider convolution based on the Cand E-functions, S-functions do not differ significantly from the C-functions case.

The method of discretization of orbit functions was described in detail in the papers [4,5]. The general idea is the following: In the fundamental domain we define a finite grid of points

We consider a space of discrete functions sampled on the points of

The weight function

For every

where the coefficient

The discrete orthogonality allows us to perform a Fourier like transform, called C-orbit transform. We consider a function

where we require

In the case of E-orbit functions we consider the grids

The weight function

For every

where the coefficient

The E-orbit transform is provided as follows. We consider a function

where we require

The main aim of this work is to define a discrete orbit functions convolution, i.e., a mapping of two functions sampled on

The

Such a convolution is well defined, the shifts in the convolution kernel

Theorem 1 Let

where

Its proof is straightforward, it uses the relations (4) and the following formula for the product of an orbit function with the complex conjugate of an orbit function with the same label but different argument:

Analogously, the

The E-orbit convolution theorem is then the following.

Theorem 2 Let

where

For the purpose of demonstrating the differences between the orbit convolution and convolution on

In

The filters are constructed to be as similar to the filters used for orbit convolution as possible. There are some restrictions for the orbit convolution coming from its definition, the most significant is the summation over all reflections of the convolution kernel. This property is unpleasant, since it does not give us the possibility to apply changes in a single direction, i.e., detecting only horizontal edges. For this reason we cannot use all convolution kernels we can use for image filtering in

When developing a spatial filter for orbit convolution from kernel for filtering in

There are two major restrictions for the orbit convolution kernels: the reflection of the kernel, which disables filtering in a single direction, and the placement of the kernel center. For the convolution on

Filters for orbit convolution are defined in the following way:

For the orbit convolution demonstration we used the hexagon image, see

The differences between the convolution on

1) In the case of

orbit convolution theorem can be formulated for each of them. This gives us bigger choice of the shape of the fundamental domain suitable for the image.

2) The method described here can be generalized to Weyl group of any rank. Therefore, it can be used for more general problems than the image processing.

3) The orbit convolution takes an advantage from the symmetry of the underlying Weyl group. On the other hand, as there is no fast algorithm yet, the computation takes more time than standard Fourier or cosine transform. One of our future projects is finding such a fast algorithm.

This work is supported by the European Union under the project Support of inter-sectoral mobility and quality enhancement of research teams at Czech Technical University in Prague CZ.1.07/2.3.00/30.0034.

In this Section we describe in detail the grids of points and grids of parameters used in the discretization of orbit functions [4,5].

We consider four lattices in

Two finite lattice grids depending on an integer parameter

invariant group

The explicit formulas are then obtain by using the marks

For the grid of parameters we take the

where the duals marks

The grids for the

where

We summarize values of all the constants and functions needed in formulas (2), (3), (6), (7).

The orders of the corresponding Weyl groups and even Weyl groups are:

The determinants of the corresponding Cartan matrix are:

The values of

Let

Let