Function spaces defined by scaling functions are used to generate bandlimited noise octaves and other attribute data sets. Scaling functions are basis functions that admit multiresolution analysis and include piecewise constant scaling functions, piecewise polynomial scaling functions, bandlimited scaling functions, Daubeschies scaling functions, as well as other multiresolution analysis scaling basis functions known to those of skill in the art. Scaling basis functions can be locally supported or have infinite support. The properties of the scaling basis functions used to construct bandlimited noise octaves may ensure that any bandlimited noise octave at resolution level N is orthogonal to bandlimited noise octaves and their associated scaling basis functions at all resolution levels less than N. Bandlimited noise octaves can be scaled to any resolution level and guaranteed to have no effect on images at any lower resolution level.