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--- visual3d:documentation:pipeline:signal_commands:gcvspl [2024/10/24 15:11] – [Cutoff Frequency] wikisysop
+++ visual3d:documentation:pipeline:signal_commands:gcvspl [2024/10/24 15:15] (current) – wikisysop
@@ Line 3: / Line 3: @@
 GCVSPL was implemented based on the following article:
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 **H.J. Woltring** (1986), A FORTRAN package for generalized, cross-validatory spline smoothing and differentiation. Advances in Engineering Software 8(2):104-113 (U.K.).
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 The command is used like all other filter commands in Visual3D.
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 [[http://biomch-l.isbweb.org/threads/17045-Re-Woltring-Filter-cutt-off-frequency-VAL-P-value?highlight=spline+filter|From Biomch-L]]
+<code>
-First account for a pass butterworth filter (e.g. a fourth order butterworth filter)\\
+First account for a pass butterworth filter (e.g. a fourth order butterworth filter)
-Modified_cut_off_freq= cut_off_freq / 0.802\\
+Modified_cut_off_freq= cut_off_freq / 0.802
 Variance = sampling_freq/(2*PI*Modified_cut_off_freq)^(2*order)
+</code>
 === Cutoff Frequency Test ===
@@ Line 132: / Line 130: @@
 The ISB website contains the following information on the algorithm
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 For large datasets (N >> 0) and negligible boundary artefacts, the behaviour of a natural spline approximates that of a periodic spline. For the latter case, the frequency characteristic in the equidistantly sampled, uniformly weighted case is that of a double, phase-symmetric Butterworth filter, with transfer function H(w) = [1 + (w/wo)^2M]^-1, where w is the frequency, wo = (p*T)^(-0.5/M) the filter's cut-off frequency, p the smoothing parameter, T the sampling interval, and 2M the order of the spline. If T is expressed in seconds, the frequen- cies are expressed in radians/second.
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 It has been found empirically, that the effective number of estimated spline parameters Np is related to the Butterworth cut-off frequency wo as Np ~ M/2 + KM * wo * N * T, where Np ranges between M and N, and where KM is the integral over x from 0 to infinity of (1 + x^2M)^-1 divided by PI. For large M, KM approaches 1/PI from above; values for small M are: K1 = 1/2, K2 = 1/V8, K3 = 1/3. This relation has also been found to apply for uniformly weighted data which are sampled slightly anequidistantly, with T taken as the average sampling inter- val. For large Np, the relation with wo * N * T becomes nonlinear.