Recursive algorithms for two-dimensional smoothing using bicubic Hermite polynomial

In the past, smoothing splines originated from approximation theory have been successfully applied to data filtering and image smoothing problems. Even though the nonrecursive technique of smoothing splines provides an optimal solution, the amount of computation increases rapidly with the size of the two-dimensional data. Here, we present a derivation of quarter-plane filtering algorithms which obtain smoothed estimates of function values and their derivatives by fitting two-dimensional smoothing splines in a recursive manner. The derivation procedure will highlight specific problems encountered in two-dimensional filtering problem. Also, the amount of computation for this recursive processor increases only linearly with the size of the two-dimensional data. Due to some approximations introduced in its derivation, this recursive processor becomes suboptimal.