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The area of hexagonal image representation has been explored for several decades based on the hexagonal structure's many advantages over conventional rectangular image grids. Due to existing image capture and display hardware being produced for rectangular structures only, most research to date has focused on developing an efficient software approach to mimic hexagonal grid structures. Only recently has the area of hexagonal image processing been investigated, where techniques such as rotation, scaling and edge detection have been implemented. In this paper, we present an approach to the development of multi-scale hexagonal gradient operators using the finite element framework. We apply these operators in a sub-pixel virtual hexagonal environment and provide both visual and quantitative results when comparing with standard rectangular image processing techniques.