Spline Smoothing: A Special Case of Diffusion Smoothing

Diffusion Smoothing (DS) implements the smoothing by directly solving a boundary value problem of the diffusion equation — = b V2u with explicit or implicit numerical schemes, it provides a uniform theoretical base for some other smoothing methods. It has shown that the elegant Gaussian smoothing (GS) is equivalent to the initial value problem of DS, and the widely-used Repeated Averaging (RA) is a special case of the explicit DS. This paper further proves that Spline smoothing (SS) is a special case of the explicit DS with a "convex corner cling" boundary condition. This result coincides with Poggio's conclusion However, our proof starts from the diffusion smoothing theory instead of regularisatio n theory and is given in the mask form, thus is simpler and more straightforward. Moreover, it makes us possible to explicit the scale space behaviour of spline smoothing.