The Discrete Fourier Transform on hexagonal remote sensing image

Global discrete grid system will subdivide the earth recursively to form a multi-resolution grid hierarchy with no Overlap and seamless which help build global uniform spatial reference datum and multi-source data processing mode which takes the position as the object and in the aspect of data structure supports the organization, process and analysis of the remote sensing big data. This paper adopts the base transform to realize the mutual transformation of square pixel and hexagonal pixel. This paper designs the corresponding discrete Fourier transform algorithm for any lattice. Finally, the paper show the result of the DFT of the remote sensing image of the hexagonal pixel.


Introduction
With the rapid development of earth observation technology, Human obtain remote sensing information of big data characteristics, so remote data mining and application becomes the focus of attention. Global discrete grid system will subdivide the earth recursively to form a multi-resolution grid hierarchy with no overlap and seamless which help build global uniform spatial reference datum and multi-source data processing mode which takes the position as the object and in the aspect of data structure supports the organization, process and analysis of the remote sensing big data. Only three basic polygons-triangles, rectangles and hexagons can seamlessly tile plane [2] . Since the adjacent triangles differ in directions, the triangle is not the optimal scheme for plane partition. So far the rectangular grid is the most commonly used . Because square pixel image is symmetry, a small amount of calculation ,easy to store and implement and many existing image processing algorithm. However, the image of square pixels also have many disadvantages, such as aliasing effects, quantization errors, inconsistency in adjacent cells and less relevance with visual process and so on. In the image processing , compared with square pixel ,hexagonal pixel have a higher degree of circular symmetry ,consistent contiguity, greater angular resolution, and require less storage space and computing [1,5] . Above all, this article will research remote-sensing image of hexagonal pixels. Since the Fourier transform has been widely used in feature extraction, spatial frequency and filtering, image restoration and texture analysis. This article firstly describes a clear and simple DFT suitable for n-dimensional lattice L .The space domain of the DFT is a system of coset representatives of the quotient group generated by the lattice L and the sublattice of L. Then we apply this result on the image of hexagonal pixel.

The DFT of any lattice
This paper will introduce DFT which is suitable for any lattice. Mathematical set which can undertake DFT is a N order finite Abelian group D . Abelian group is D additive group. A lattice in n dimensional European space is a linear combination of linearly independent vector. Take these n vectors as base vector of the lattice. The row vectors of base matrix are base vectors of the lattice L.denote that L0 is a n dimensional sublattice of the lattice L.
Because the lattice L and the lattice L0 are both Abelian group, The quotient group L=L 0 is also a finite Abelian group [3] .
is a linear transformation [3,4,6] , as shown in the following type: The inverse Fourier transformation: Theorem If the quotient group L=L 0 have elementary divisor N 1 ; N 2 ; ¢ ¢ ¢ ; N d ; DFT and inverse DFT can be reduced to computing multiple one dimensional DFT. The running time needed for this result is: Proof the result that DFT can be reduced to computing multiple one dimensional DFT, the proof of inverse DFT is similar. Given a n dimensional lattice L and a sublattice L 0 ,the base vector of L with respect to L 0 compose the set fṽ 1 ; Hence M is a system of coset representatives of the quotient group L=L 0 . The base vectors of L with respect to L 0 compose the base matrix W L .The base vectors of L compose the base matrix W . The diagonal elements of the diagonal matrix B are composed of the corresponding L=L 0 elementary divisors. As the base vector of L 0 is integer linear combination of the base vector of L, there is an integer matrix T to make TW a base matrix of L0. P 1 ; P 2 ; ¢ ¢ ¢ ; P a andQ 1 ; Q 2 ; ¢ ¢ ¢ ; Q b is the elementary matrix used to diagonalize matrix T . So you can get the diagonal matrix B: With the relationship between matrix W L and the matrixW : and P = P a ¢ ¢ ¢ P 2 P 1 . As the row vector of T W is the base vector of L 0 and V is a unit matrix, the row vector of the matrix W hx; yi is the inner product of a standard European space. The following three properties is: As the quotient group of L=L 0 is the space domain of DFT, according to the above second properties , the quotient group L 0 ¤ =L ¤ is the frequency domain. The base vector fṽ 1 ;ṽ 2 ; ¢ ¢ ¢ ;ṽ n g of L with respect to L 0 is the row vector of the base matrix W . According to the above third properties ,we get The base vector of L ¤ 0 with respect to L ¤ is the row vector fṽ ¤ 1 ;ṽ ¤ 2 ; ¢ ¢ ¢ ;ṽ ¤ n g of the base matrix (BW) ¡T .According to n . The formula (5) in the above formula (1): Because the run time of one dimension DFT of size Ni is O(N i log N i ).So the complexity of the above formular is: Hence the transition matrix C is: Let the vector x = m 1 a 1 + m 2 a 2 = n 1 v 1 + n 2 v 2 ,Therefore According to the above formulas of the base transformation and coordinated transformation ,it can resample square pixel image to the desired hexagonal pixel remote sensing images. This paper uses bilinear interpolation to resample . The hexagonal images resampled are shown as below: The experimental result is shown as below:

Conclusion
This paper firstly describes DFT suitable for n-dimensional lattice and then introduces base the DFT on a particular subdivision structure of hexagonal pixel remote sensing image .