Smoothing on the Surface
Smoothing on the Surface
Spatial smoothing on the surface can be tricky because the spatial
grid of locations where the data are sampled is neither uniformly
sampled nor rectangular. In addition, it is difficult convolve the
data with a spatial kernel because it is diffucult to compute the
distance between any two points on the grid. A simple cartesian
distance cannot be computed because the surface is folded in 3D
space. One can compute the distance based on the shortest path between
the two points, but this is computationally prohibative because the
large number of points (on the order of 100,000) leads to a huge
number of paths to be searched.
We have two methods for smoothing on the surface. In the first method
(nearest-neighbor averaging or NNA), the value at a vertex is
recomputed as the average of itself and it's nearest neighbors. More
smoothing is obtained through iteration. While this is fast and
convenient, it is not clear what the size of the actual smoothing
kernel is. In the second method (spherical smoothing), the surface is
inflated so that all points lie on a sphere. This allows the distance
between two points to be easily computed as the angle between them
times the radius and so allows for convolution with a filter
kernel. The problem with this method is that there is inevitably
metric distortion in inflated to the sphere.
The data presented here compare these two methods in order to
determine how many iterations in NNA are equivalent to a given
gaussian FWHM. White guassian noise was generated on a 7th order
icosahedron (163842 vertices, radius = 100 mm). Various levels of NNA
and spherical smoothing were then applied. The reduction in the
standard deviation (RStdDev) with respect to the unsmoothed data set
was then computed across all vertices for each result. For the
spherical smoothing, the RStdDev as a function of FWHM was fit using a
slope and intercept. For NNA, the RStdDev as a function of number of
iterations was fit using an intercept, slope, and square root. The raw
data and best fit are shown in Figure 1. These best fits of the
RStdDev were then used to compute the FWHM as a function of number of
nearest-neighbor iterations (ie, the FWHM that would result in the
same RStdDev as a given number of iterations). The equivalence plot is
show in Figure 2 (to see a table, click here.
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Figure 1:
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Figure 2:
Table of equivalent values.