extrema or spurious responses to higher-order derivative operators), then the options are to decrease the value of ε such that a larger finite kernel is used, with cutoff where the support is very small, or to use a tapered window. Recursive filters Since computational efficiency is often important, low-order recursive filters are often used for scale-space smoothing. For example, Young and van Vliet use a third-order recursive filter with one real pole and a pair of complex poles, applied forward and backward to make a sixth-order symmetric approximation to the Gaussian with low computational complexity for any smoothing scale. By relaxing a few of the axioms, Lindeberg concluded that good smoothing filters would be "normalized Pólya frequency sequences", a family of discrete kernels that includes all filters with real poles at 0 < Z < 1 and/or Z > 1, as well as with real zeros at Z < 0. For symmetry, which leads to approximate directional homogeneity, these filters must be further restricted to pairs of poles and zeros that lead to zero-phase filters. To match the transfer function curvature at zero frequency of the discrete Gaussian, which ensures an approximate semi-group property of additive t, two poles at can be applied forward and backwards, for symmetry and stability. This filter is the simplest implementation of a normalized Pólya frequency sequence kernel that works for any smoothing scale, but it is not as excellent an approximation to the Gaussian as Young and van Vliet's filter, which is not normalized Pólya frequency sequence, due to its complex poles. The transfer function, H1, of a symmetric pole-pair recursive filter is closely related to the discrete-time Fourier transform of the discrete Gaussian kernel via first-order approximation of the exponential: where the t parameter here is related to the stable pole position Z = p via: Furthermore, such filters with N pairs of poles, such as the two pole pairs illustrated in this section, are an even better approximation to the exponential: where the stable pole positions are adjusted by solving: The impulse responses of these filters are not very close to gaussian unless more than two pole pairs are used. However, even with only one or two pole pairs per scale, a signal successively smoothed at increasing scales will be very close to a gaussian-smoothed signal. The semi-group property is poorly approximated when too few pole pairs are used. Scale-space axioms that