Irradiance normal mapping is a method to combine two popular techniques, light mapping and normal mapping, and is used in games such as Half-Life 2 or Halo 3. This combination allows using low-resolution light caching on surfaces with only a few coefficients which are evaluated by normal maps to render spatial high-frequency changes in the lighting. Though there are dedicated bases for this purpose such as the Half-Life 2 basis, higher order basis functions such as quadratic Spherical Harmonics are needed for an accurate representation. However, a full spherical basis is not needed since the irradiance is stored on the surface of a scene. In order to represent the irradiance signals efficiently, we propose a novel polynomial, hemispherically orthonormal basis function set that is specifically designed to carry a directional irradiance signal on the hemisphere and which makes optimal use of the number of coefficients. To compare our results with previous work, we analyze the relations ...