Properties of Microsphere Projection
1. MS Projection works in any Cartesian coordinate system. The algorithm works as well in 1 dimension as it does in 10 or 20 dimensions.
2. MS Projection exhibits the Maximum
In other words, the interpolated value is guaranteed to lie in
the range between the minimum sampled value and the maximum sampled
value. Other interpolation techniques which demonstrate this quality
Neighbor interpolation, and Shepard’s
Method (naïve inverse-distance weighting).
This feature was chosen because it is guaranteed to provide intuitive
results for bounded
3. MS Projection is guaranteed to preserve monotonic and
monotonic behavior over any set or subset of sample
points. For example, if the set or subset of sample points is increasing
or strictly increasing over a range, then the interpolation is
guaranteed to be increasing or strictly
increasing over the same range.
4. MS Projection demonstrates no oscillatory behavior between
sample points, unlike functional approximations which
are designed to
preserve high differentiability.
5. MS Projection provides a stable extrapolation ability.
Functional approximations tend to produce extremely volatile extrapolation
results beyond the range of the data points. This
can cause serious issues in higher dimensions where the differentiation
between interpolation and extrapolation within the volume is difficult
to determine. Because MS Projection provides
a stable extrapolation, it has considerable benefits over functional
approximations when visualizing higher dimensional data.
1. MS Projection is only proven to be class
long as p>1, C0 otherwise.
This means that the interpolation will have a continuous first
derivative, however no guarantees are made of
the second derivative of the interpolation. MS Projection may actually be better than C1, but furhter research must be done.
2. Depending on the nature of the problem, the fact that
MS Projection exhibits the Maximum
Principle can be an issue. That the interpolation
method is unable to interpolate a value beyond the minimum and maximum sampled values can cause problems depending on the context.
3. Depending on the size of the data set
and other considerations, MS Projection can require more computation
time than some of the
other interpolation algorithms.
Though the overall runtime is O(P*N), this set of calculations must
be run every time a point is to be interpolated. Radial
Basis Function (RBF) interpolations such as Thin-Plate
Spline, Multiquadric, and
Volume Spline are all O(N2) (using Gaussian elimination)
for the first interpolation and O(N) (with a very small overhead)
4. As interpolation location approaches a sample point, the
first derivative in
all dimensions approaches 0 when p>1. In most contexts
this is undesirable behavior; however it is necessary if we wish
Principle in conjunction with differentiability.
p is the inverse distance propagation constant (p=1 is linear)
N is the number of samples of "real" data
P is a precision constant representing the number of faces of the microsphere