Smoothing of three dimensional models by convolution

Any 3D shape can be described as a function g(x,y,z) where g>0 inside the shape and g<0 outside. The convolution of g with a suitable filter describes a smoothed shape where sharp edges and corners have been rounded. This idea provides a simple and uniform method to create blends and fillets for engineering objects and a way to build more organic shapes by smoothing idealised geometrical shapes. Convolution in three dimensions requires too much computation to use this idea directly but we can make a useful approximation by representing the convolved function at points on a three dimensional grid and interpolating between these points. The grid can be regular or adaptive (octree). Using this approach, we have successfully modelled a variety of objects including engineering parts and animal forms