The Binary Mathematical Morphology on the Triangular Grid

Mathematical morphology is a part of digital image processing which has strong mathematical foundation and also has several applications. In digital image processing images are understood on (usually, a finite segment of) a grid. Historically, the square grid is the most used one, theory on the square grid has been developed first and it is applied in the most cases. However, it is also known that other grids have some advantages over the square grid. There are two other regular tessellations of the plane, the hexagonal and the triangular grids. In this thesis, we considered mathematical morphology on the triangular grid. The two basic operations of mathematical morphology are the dilation and the erosion. The input image is changed by the help of structural elements with these operations. Since the triangular grid is not a point lattice we have needed to face to some difficulties when defining dilations and erosions, namely, the triangular grid is not closed under vector addition. We have proposed four possible solutions, namely the”strict”, the”weak”, the”strong” and the”independent” approaches. Definitions, examples and properties of the operations are investigated in each case. Keywords: Mathematical morphology, dilation, erosion, non-traditional grids, triangular grid, adjunction relation, digital image processing, binary images