A comparison of normal estimation schemes

The task of reconstructing the derivative of a discrete function is essential for its shading and rendering as well as being widely used in image processing and analysis. We survey the possible methods for normal estimation in volume rendering and divide them into two classes based on the delivered numerical accuracy. The three members of the first class determine the normal in two steps by employing both interpolation and derivative filters. Among these is a new method which has never been realized. The members of the first class are all equally accurate. The second class has only one member and employs a continuous derivative filter obtained through the analytic derivation of an interpolation filter. We use the new method to analytically compare the accuracy of the first class with that of the second. As a result of our analysis we show that even inexpensive schemes can in fact be more accurate than high order methods. We describe the theoretical computational cost of applying the schemes in a volume rendering application and provide guidelines for helping one choose a scheme for estimating derivatives. In particular we find that the new method can be very inexpensive and can compete with the normal estimations which pre-shade and pre-classify the volume (M. Levoy, 1988).