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Microsphere Projection Walkthrough ↑ ↑ ↑ 

Microsphere Projection Walkthrough





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.

Looking
at the figure to the right, for simplicity's sake, the sphere segments
have been 'unrolled' starting
with θ = 0 at the top, and working our way around towards θ =
2π (360 degrees). Compare the color values in the picture to
the right to the colors of the circle from Step
1 if you still don't understand.
Ok, so the KEY thing to remember about assigning these values is
that ONLY the sample point with the LARGEST value will be represented
in that section of the sphere. Note that there are no redgreens
or bluereds, or bluegreens. Either a section is red, green, or
blue. No mixing. Only the largest value dominates the section.
The values shown in this figure are calculated by:
cos(θ) / dist^{2}. For example, the sections of the sphere nearly
exactly facing the red sample point have angles nearly 0, thus their values
[cos(~0)/1^{2}] are nearly 1 (0.98). Similarly, the sections nearly facing
the green sample point are nearly 0.25.
The "2" in dist^{2} is arbitrary. A value ≥ 2 Provides differentiability,
but in
reality
any
positive number will suffice, and any value ≥ 1 will provide a good interpolation.
I personally use p=2 and sometimes p=1 if differentiability is not required.

Now that the tough work is out of the way, we
simply have to determine the weights to assign to the original
sample points.
To do this, we first accumulate and then normalize. The weights
are accumulated from all the sections of the sphere (as seen in
the figure to the left). The accumulation is performed by a simple
summation. Then to make the weights add up to '1', we divide each
weight by the total. The result is shown in the figure on the
right.
Simple Summation (before normalizing) 
Normalized Weights 

If we look back up to Step 1 we can see our
original values for "red", "green", and "blue". Our final step
involves applying those known values to the weight distribution
we have created for them.
As you can see, the final resultant value from all of this is 8.8dB which completely agrees with our original intuitive guess of "somewhere
near 7.5, but definitely higher because of the 40 nearby."
Congratulations, You have just completed the walkthrough! If you
still have questions, please feel free to browse the other
materials or email me.






