Geometric transformation of the RBF implicit surface

When we use a Radial Basis Function (RBF) implicit surface, we may need to transform it. The conventional algorithm to transform an RBF implicit surface is to apply inverse transformation to the given point and to evaluate the original function at the inversely transformed point. The algorithm keeps the initial RBF centers. Sometimes, we need the transformed RBF centers. In these cases, if we still use the conventional algorithm, we need to keep both the initial and the transformed RBF centers. Obviously, this is a problem that wastes the memory. We have derived the relationship between the initial and the transformed RBF coefficients, which can solve the previous problem. Our method only needs to keep the transformed RBF centers, and save much memory. Our method works for both globally and compactly supported RBF. We also compare our algorithm with the conventional algorithm about the time efficiency in details. The theoretical analysis and experiment results show that our algorithm is faster than the conventional algorithm in many cases. We also applied our method on RBF-based CSG operations and improved the RBF-based Boolean operation algorithm to be more efficient. Moreover, we present a solution to relieve the bumps in CSRBF Boolean operations.