Scale Space Implementation - Finite-impulse-response (FIR) Smoothers

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 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 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 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.