A radial version of the Central Limit Theorem

The idea was extended in two different directions in [1]. In one direction, it was argued that, by adjusting the widths of the box function along each radial direction, one could approximate anisotropic Gaussians. In particular, for the special case of the so-called four-directional box splines, a simple algorithm was developed that allowed one to control the covariance by simply adjusting the widths of the box distributions. In a different direction, an algorithm for spacevariant filtering was developed which, unlike convolution filtering, allowed one to change the shape and size of the box spline at each point in the image. This was also done at the expense of just O(1) operations (independent of the shape and size of the Gaussian) using a single global pre-integration followed by local finite-differences; cf. Algorithm 1 in [1].

A probabilistic interpretation of this result is as follows. Let X be random vector on the plane that is distributed on a line passing through the origin (e.g., one of the coordinate axes). Thus, X is completely specified by a probability measure µ(t) on the real line. Suppose that t dµ(t) = 0, and t 2 dµ(t) = 1.

For 0 ≤ θ < π, let us denote the rotation matrix on the plane by

Guided by the idea behind Theorem 2.2 in [1], we can then conclude the following. * kchaudhu@math.princeton.edu

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