A Spectral Analysis of Function Composition and its Implications for Sampling in Direct Volume Visualization

In this paper we investigate the effects of function composition in the form g( f (x)) = h(x) by means of a spectral analysis of h. We decompose the spectral description of h(x) into a scalar product of the spectral description of g(x) and a term that solely depends on f (x) and that is independent of g(x). We then use the method of stationary phase to derive the essential maximum frequency of g( f (x)) bounding the main portion of the energy of its spectrum. This limit is the product of the maximum frequency of g(x) and the maximum derivative of f (x). This leads to a proper sampling of the composition h of the two functions g and f. We apply our theoretical results to a fundamental open problem in volume rendering—the proper sampling of the rendering integral after the application of a transfer function. In particular, we demonstrate how the sampling criterion can be incorporated in adaptive ray integration, visualization with multi-dimensional transfer functions, and pre-integrated volume rendering.