Constant-time filtering using shiftable kernels

It was recently demonstrated in [5] that the non-linear bilateral filter [14] can be efficiently implemented using a constant-time or O(1) algorithm. At the heart of this algorithm was the idea of approximating the Gaussian range kernel of the bilateral filter using trigonometric functions. In this letter, we explain how the idea in [5] can be extended to few other linear and non-linear filters [14, 17, 2]. While some of these filters have received a lot of attention in recent years, they are known to be computationally intensive. To extend the idea in [5], we identify a central property of trigonometric functions, called shiftability, that allows us to exploit the redundancy inherent in the filtering operations. In particular, using shiftable kernels, we show how certain complex filtering can be reduced to simply that of computing the moving sum of a stack of images. Each image in the stack is obtained through an elementary pointwise transform of the input image. This has a two-fold advantage. First, we can use fast recursive algorithms for computing the moving sum [15, 6], and, secondly, we can use parallel computation to further speed up the computation. We also show how shiftable kernels can also be used to approximate the (non-shiftable) Gaussian kernel that is ubiquitously used in image filtering.