data (and on each column in the 2D case), the poles and zeros can each do a part of the smoothing. The zeros limit out at t = 0.5 per pair (zeros at Z = –1), so for large scales the poles do most of the work. At finer scales, the combination makes an excellent approximation to the discrete Gaussian if the poles and zeros each do about half the smoothing. The t values for each portion of the smoothing (poles, zeros, forward and backward multiple applications, etc.) are additive, in accordance with the approximate semi-group property. The FIR filter transfer function is closely related to the discrete Gaussian's DTFT, just as was the recursive filter's. For a single pair of zeros, the transfer function is where the t parameter here is related to the zero positions Z = z via: and we require t ≤ 0.5 to keep the transfer function non-negative. Furthermore, such filters with N pairs of zeros, are an even better approximation to the exponential and extend to higher values of t : where the stable zero positions are adjusted by solving: These FIR and pole-zero filters are valid scale-space kernels, satisfying the same axioms as the all-pole recursive filters. Real-time implementation within pyramids and discrete approximation of scale-normalized derivatives Regarding the topic of automatic scale selection based on normalized derivatives, pyramid approximations are frequently used to obtain real-time performance. The appropriateness of approximating scale-space operations within a pyramid originates from the fact that repeated cascade smoothing with generalized binomial kernels leads to equivalent smoothing kernels that under reasonable conditions approach the Gaussian. Furthermore, the binomial kernels (or more generally the class of generalized binomial kernels) can be shown to constitute the unique class of finite-support kernels that guarantee non-creation of local extrema or zero-crossings with increasing scale (see the article on multi-scale approaches for details). Special care may, however, need to be taken to avoid discretization artifacts. Other multi-scale approaches For one-dimensional kernels, there is a well-developed theory of multi-scale approaches, concerning filters that do not create new local extrema or new zero-crossings with increasing scales. For continuous signals, filters with real poles in the s-plane are within this class, while for discrete signals the above-described recursive and FIR filters satisfy these criteria. Combined with the strict requirement of a continuous semi-group structure, the continuous Gaussian and the discrete Gaussian constitute the unique choice for