This looks like the image you are using is shaped like the tile that you are attempted to fill so you are getting the surrounding area when the angled edge is 'travelling' along the outside corners of each pixel, 'grabbing' a pixel from the outside part.
A couple of possible solutions are:
- Expand your diamond-shaped texture to fill the pixel around it as well
- Contract your texture co-ordinate so that it no longer passes outside
The second suggestion above can be achieved by adding only a half to each co-ordinate (remembered to subtract from the highest values) if the final display will be the same size pixel-wise as the texture, otherwise it'd need to be a full one that gets added and subtracted to be sure.