The Hexagonal Efficient Coordinate System (HECS), formerly known as Array Set Addressing (ASA), is a coordinate system for hexagonal grids that allows hexagonally sampled images to be efficiently stored and processed on digital systems. HECS represents the hexagonal grid as a set of two interleaved rectangular sub-arrays, which can be addressed by normal integer row and column coordinates and are distinguished with a single binary coordinate. Hexagonal sampling is the optimal approach for isotropically band-limited two-dimensional signals and its use provides a sampling efficiency improvement of 13.4% over rectangular sampling. The HECS system enables the use of hexagonal sampling for digital imaging applications without requiring significant additional processing to address the hexagonal array.
Background The advantages of sampling on a hexagonal grid instead of the standard rectangular grid for digital imaging applications include: more efficient sampling, consistent connectivity, equidistant neighboring pixels, greater angular resolution, and higher circular symmetry. Sometimes, more than one of these advantages compound together, thereby increasing the efficiency by 50% in terms of computation and storage when compared to rectangular sampling. Researchers have shown that the hexagonal grid is the optimal sampling lattice and its use provides a sampling efficiency improvement of 13.4% over rectangular sampling for isotropically band-limited two-dimensional signals. Despite all of these advantages of hexagonal sampling over rectangular sampling, application prior to the introduction of HECS was limited because of the lack of an efficient coordinate system.
Hexagonal Efficient Coordinate System
Description
The Hexagonal Efficient Coordinate System (HECS) is based on the idea of representing the hexagonal grid as a set of two rectangular arrays which can be individually indexed using familiar integer-valued row and column indices. The arrays are distinguished using a single binary coordinate so that a full address for any point on the hexagonal grid is uniquely represented by three coordinates.
( a , r , c ) ∈ { 0 , 1 } × Z × Z {\displaystyle (a,r,c)\in \{0,1\}\times \mathbb {Z} \times \mathbb {Z} }
where the coordinates represent the array, row, and column, respectively. The hexagonal grid is separated into rectangular arrays by taking every other row as one array and the remaining rows as the other array, as shown in the figure.
Nearest neighbors
The addresses of the nearest neighbors of a pixel (or grid point) are easily determined by simple expressions which are functions of the pixel's coordinates, as shown.
Convert to Cartesian Converting coordinates in HECS to their Cartesian counterparts is done with a simple matrix multiplication
[ x y ] = [ 1 2 0 1 3 2 3 0 ] [ a r c ] = [ a 2 + c ( 3 ) ( a 2 + r ) ] . {\displaystyle {\begin{bmatrix}x\\y\end{bmatrix}}={\begin{bmatrix}{\frac {1}{2}}&0&1\\{\frac {\sqrt {3}}{2}}&{\sqrt {3}}&0\end{bmatrix}}{\begin{bmatrix}a\\r\\c\end{bmatrix}}={\begin{bmatrix}{\frac {a}{2}}+c\\({\sqrt {3}})({\frac {a}{2}}+r)\end{bmatrix}}.}
Operators
Preliminaries Let the set of all possible HECS addresses be HECS = { 0 , 1 } × Z × Z . {\displaystyle {\text{HECS}}=\{0,1\}\times \mathbb {Z} \times \mathbb {Z} .}
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