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Properties of Microsphere Projection


Properties of Microsphere Projection


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Useful Characteristics 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 Principle
. 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 include Nearest Neighbor interpolation, and Shepard’s Method (naïve inverse-distance weighting). This feature was chosen because it is guaranteed to provide intuitive results for bounded data.

3. MS Projection is guaranteed to preserve monotonic and strict 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.


Not-So Useful Characteristics of Microsphere Projection

1. MS Projection is only proven to be class C1 so 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) for subsequent interpolations.

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 to preserve the Maximum Principle in conjunction with differentiability.

Definitions:
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


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