In this example we illustrate fitting an RBF to scattered noisy data in 3D.
The locations of the data are illustrated in Figs (a) & (b). The magnitude (value) of the
attribute is depicted by the colour at each location. Values range from 0 to 1.
Fig (b) is identical to (a) except that uniform random noise of magnitude 10% (+/- 0.1) has been added to the data in (a).
Figs (c) & (d) correspond to RBF grid evaluations of the respective data sets.
(a) Noiseless data
(b) Noisy data
(c) Exact fit to noiseless data
(d) Exact fit to noisy data
Aliasing - sub-sampling an RBF
In this figure we consider a low resolution grid evaluation of the RBFs depicted in Figs (c) and (d).
The sampling frequency meets the Nyquist criterion for the noiseless data shown in (e).
However, this condition is no longer met when evaluating the RBF fitted to the noisy data. The added noise
represents the addition of high frequency components to the signal. Evaluating on the low resolution grid corresponds to sub-sampling the RBF depicted in (d). Fig (f) is a normal FastRBFTM
grid evaluation and Fig (g) shows the effect of low pass filtering (anti-aliasing) with a cut-off frequency appropriate to the grid spacing (the colour scales are not quite comparable in this figure and were determined by the highest and lowest value in the respective data volumes).
(e) Noiseless grid evaluation
(f) Grid evaluation of noisy data without anti-aliasing
(g) Grid evaluation of noisy data with anti-aliasing
Low pass filtering and smoothest restricted range (error-bar) fitting
In these figures we consider evaluating an isosurface using the FastRBFTM
isosurfacer. Fig (h) is an isosurface computed from the RBF fitted to the noiseless data. Fig (i) is the corresponding isosurface for the noisy fit. Fig (j) illutrates how low pass filtering in the
FastRBFTM grid evaluation can reduce noise in the data.
Fig (k) illustrates the alternative strategy of fitting an approximating RBF to the noisy data.
In this case the error-bar fitter is used to fit the smoothest RBF within +/- 0.1 of the noisy data.
The advantage of the low pass filter is that the degree of smoothing can be varied interactively
during RBF evaluation while with the error bar fitter noise rejection occurs during the fitting process.
The detail in the raw data can not be recovered without fitting a new RBF. However, the error bar fitter results in a more compact representation of the data.
These issues are discussed further in the smoothing and approximating FAQ.
(h) Isosurface of noise-free RBF
(i) Isosurface of exact fit RBF
(j) Low pass filtered RBF
(k) Error-bar fit
Further information can be found on the following ARANZ web pages:
