Our problem will be defined as follows:
Noise in a room has been sampled in three locations at levels of
40, 10, and 5dB as noted in the figure. Each of the sample points
has been assigned a unique color to aid in understanding the example.
We wish to interpolate the
dB intensity at the point located at the center of the "X" in
the figure.
Note that this algorithm applies to much
more than just noise: we could just as easily be talking about heat
disperison within
a container, wind velocity over sections of a wing, or density at
various depths and positions in the earth  it has many applications.
Since all of our points are within the same plane,
we can use a 2D microsphere to interpolate (if the points were
in 3D we
would
use a 3D sphere,
4D etc.). For simplicity, the sphere (circle) has been divided
into 16 segments. In a situation where accuracy was more important,
the
sphere would be given far more segments.
To make sure that our final result will be in agreement with
our intuition, we should form some kind of idea of what the expected
value is. Ignoring the 40dB sample for a moment, we see that
the interpolating location is halfway between 10dB and 5dB,
suggesting that it would be approximately 7.5dB. Now, reintroducing
the 40dB sample, we assume that the estimate of 7.5dB is probably
a little
smaller than the actual value because the influence
of a strong 40dB will probably bring up our estimate.
So, in all we would expect the interpolation to provide a result
somewhere near 7.5, but definitely higher because of the 40
nearby.
