Interaction Splines with Regular Data: Automatically Smoothing Digital Images

The idea of interaction spline was first proposed by Barry [Ann. Statist., 14 (1986), pp. 934–953] in the context of nonparametric Bayesian multivariate regression. Wahba [Computer Science and Statistics, American Statistical Association, 1986, pp. 75–80] shows that Barry’s model is a special case of a wider class of smooth functions which can be constructed from the tensor product of reproducing kernel Hilbert spaces. Algorithms for computing interaction splines with arbitrarily scattered data have been developed by Gu, Bates, Chen, and Wahba [SIAM J. Matrix Anal. Appl.,10 (1989), pp. 457–480] and Gu and Wahba [SIAM J. Sci. Statist. Comput.,12 (1991), pp. 383–398]. These algorithms require O(n³ ) flops for sample size n. The purpose of this work is to develop fast algorithms for computing interaction splines with data sampled from a regular product mesh. The algorithm is of order $O(n)$ for practical problem size. Smoothing parameters can be tuned using the generalized cross-validation. A possible application of the technique is image smoothing. In image smoothing, a periodic interaction spline is observed to be a low-pass filter of a certain form. Simulated examples are presented and practical aspects of the algorithms are discussed.