Triangular coverage on summed area table

Describe how to efficiently compute the sum of intensities covered by an arbitrary triangle over a texture map, using a pre-computed summed area table. You can assume that the texture map stores one intensity value per texel and that the vertices of the triangle locate at texel centers. Your calculation should be based purely on values stored in the summed area table and no intersection tests between triangle edges and grid lines should be computed. Hint: Try to cover only cases that are really distinct and use figures to assist the description of your solution. Make you algorithm as efficient as possible.