are still satisfied by these filters are: linearity shift invariance (integer shifts) non-creation of local extrema (zero-crossings) in one dimension non-enhancement of local extrema in any number of dimensions positivity normalization The following are only approximately satisfied, the approximation being better for larger numbers of pole pairs: existence of an infinitesimal generator A (the infinitesimal generator of the discrete Gaussian, or a filter approximating it, approximately maps a recursive filter response to one of infinitesimally larger t) the semi-group structure with the associated cascade smoothing property (this property is approximated by considering kernels to be equivalent when they have the same t value, even if they are not quite equal) rotational symmetry scale invariance This recursive filter method and variations to compute both the Gaussian smoothing as well as Gaussian derivatives has been described by several authors. Tan et al. have analyzed and compared some of these approaches, and have pointed out that the Young and van Vliet filters are a cascade (multiplication) of forward and backward filters, while the Deriche and the Jin et al. filters are sums of forward and backward filters. At fine scales, the recursive filtering approach as well as other separable approaches are not guaranteed to give the best possible approximation to rotational symmetry, so non-separable implementations for 2D images may be considered as an alternative. When computing several derivatives in the N-jet simultaneously, discrete scale-space smoothing with the discrete analogue of the Gaussian kernel, or with a recursive filter approximation, followed by small support difference operators, may be both faster and more accurate than computing recursive approximations of each derivative operator. Finite-impulse-response (FIR) smoothers For small scales, a low-order FIR filter may be a better smoothing filter than a recursive filter. The symmetric 3-kernel , for t ≤ 0.5 smooths to a scale of t using a pair of real zeros at Z < 0, and approaches the discrete Gaussian in the limit of small t. In fact, with infinitesimal t, either this two-zero filter or the two-pole filter with poles at Z = t/2 and Z = 2/t can be used as the infinitesimal generator for the discrete Gaussian kernels described above. The FIR filter's zeros can be combined with the recursive filter's poles to make a general high-quality smoothing filter. For example, if the smoothing process is to always apply a biquadratic (two-pole, two-zero) filter forward then backwards on each row of