 Microsphere
projection is based on the physical model of an infinitesimally
small sphere located at the point of
interpolation. This tiny sphere is then ‘illuminated’ by
the surrounding sample points. Based on the degree of illumination
on various parts of the sphere by various sample points, a series
of weights for all the sample points are assigned. These weights,
when applied, yield the interpolated value for the location.
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 The surface of the Microsphere is divided into
a large number of equally-spaced regions. Each region records for
itself which sample point has illuminated it the most, and what
illumination that sample point has provided. Each surface region
is represented by a single unit vector pointing out from the center
of the sphere to the center of that region. “S[i].Vector” will
be used to represent the unit vector for surface region i. The
more regions used, the greater the precision of the interpolation.
For each region, two values are recorded: one recording the index
of which sample point has illuminated this section the greatest,
and the second recording the degree of illumination from this point.
These will be referred to as “S[i].Brightest_Sample” and “S[i].
Max_Illumination”, respectively.
Since determining an arbitrarily large number of equally-spaced
regions on the surface of a sphere is no small task, we accept
that a large number of randomly placed unit vectors will provide
a fairly uniform distribution. The vectors are generated using
the following algorithm:
do
// x,y,z are uniformly-distributed
random numbers in
the range (-1,1)
x := rand(-1,1)
y := rand(-1,1)
z := rand(-1,1)
vectorSize := sqrt (x*x + y*y + z*z)
// if the vector these points form
is outside the unit
sphere,
// disregard and find a new vector.
while ( vectorSize > 1 )
// normalize the vector, so that it forms a unit vector for the
surface of our sphere.
x := x / vectorSize
y := y / vectorSize
z := z / vectorSize
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 Net illumination is applied to the microsphere
by iterating through each of the sample points, and applying illumination
to the sphere one-by-one. It should be noted that illumination
on various parts of the sphere decreases proportionally
to the acuteness
of the angle between the surface of the sphere and the direction
of the sample point. Illumination also decreases
as the distance between the microsphere and the sample point increase. Much like Shepard’s
Method, this inverse
relationship between distance and ‘brightness’ is governed
by a power value ‘p’ specified by the user where p>0,
p=1 and p=2 are typical values. p=1 yields an interpolation that
is C0 (non-differentiable),
p > 1 is C1 (first-derivative is continuous). Similar
to Shepard’s
Method, as p→∞, the closest points dominate the interpolation
and the algorithm becomes the equivalent of Nearest
Neighbor.
Intesnsity Projection Function:
for i := 0 to Number of Samples
// vector connecting the current sample to
the interpolation location
vector1 := sample[i].XYZLocation - interpolation.XYZLocation
// the distance-modified weight of this point
// p > 0, typically p=1 or p=2.
weight := pow(vector1.Size, -p)
// the value of ‘Precision’ represents
how many subdivisions
// of the surface of the microsphere we are working
with.
for j := 0 to Precision
// each sample only
'shines' on one hemisphere.
// as the angle becomes
more acute, the intensity
// of that shine
decreases as the cosine function
cosValue := CosValueBetweenVectors(vector1,
S[j].Vector)
// if the brightness
of the shine on this section of the sphere
// is more than any
other point thus far checked, update our
// 'Brightest_Sample'
and 'Illumination' data.
if (cosValue * weight > S[j].Max_Illumination)
S[j].Max_Illumination
:= cosValue * weight
S[j].Brightest_Sample
:= i
endif
endfor
endfor
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Once all the calculations are complete regarding
the maximum illuminations on the various sections of the sphere,
we must make use of this data to produce a single interpolated
value. To do this, we assign a weight to each sample point equal
to the total illumination that point provided to the sections of
the sphere. Note that each section of the sphere only records data
regarding the point which provided the most illumination; sample
points which did not out-shine any other points on any section
of the sphere are assigned a weight of 0. If you are having problems
understanding this, please visit the walkthrough.
Accumulation Pseudo-Code:
// accumulate the data from our sphere, and determine final interpolation
value := 0
totalWeight := 0
for i := 0 to Precision
value := value + S[j].Max_Illumination * sample[S[j].Brightest_Sample].SampledValue
totalWeight := totalWeight + S[j].Max_Illumination
endfor
// the final interpolated value generated by the algorithm
interpolation := value / totalWeight
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Example 1, Top-Left: Illumination of a 2D
Microsphere with a single sample point known.
Example 2, Top-Middle-Left: Illumination of a
2D Microsphere with a single sample point known. Sample point is
closer than in Example
1, thus illumination is brighter.
Example 3, Top-Middle-Right: Illumination of a
2D Microsphere with 3 sample points known. Note the exclusivity
of each section of
the sphere (ie, all top sections are either red or green. No greenish-red
or redish-green).
Example 4, Top-Right: Illumination of a 2D Microsphere with 3 sample
points known. Similar to the previous example, note that the red
points are not additive.
Example 5, Bottom: Illumination of a 2D Microsphere with 4 sample
points known. Note that since the distance of the green point is
greater than red and blue points, it dominates fewer sections of
the sphere.
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Click HERE to
go to the algorithm walkthrough.
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