Optimized gradient filters for hexagonal matrices

Digital images are represented nowadays as square lattices. Everyday items, such as digital cameras, displays, as well as many systems for vision or image processing use square lattices to represent an image. However, as the distance between adjacent pixels is not constant, any filter based on square lattices presents inherent anisotropy. Ando introduced consistent gradient filters to cope with this problem, with filters derived in order to get the minimum inconsistency. Square lattices are not, however, the only way to order pixels. Another placement method can be found, for example, in the human retina, where receptors adopt an hexagonal structure. In contrast to square lattices, the distance between adjacent pixels is a constant for such structures. The principal advantage of filters based on hexagonal matrices is, then, their isotropy. In this paper, we derive consistent gradient filters of hexagonal matrices following Ando's method to derive consistent gradient filters of square matrices. The resultant hexagonal consistent gradient filters are compared with square ones. The results indicate that the hexagonal filters derived in this paper are superior to square ones in consistency, in proportion of consistency to output power, and in localization.