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Constrained Visualization Using the Shepard Interpolation Family

This paper discusses the problem of visualizing data where there are underlying constraints that must be preserved. For example, we may know that the data are inherently positive. We show how the Modified Quadratic Shepard method, which interpolates scattered data of any dimensionality, can be constrained to preserve positivity. We do this by forcing the quadratic basis functions to be positive. The method can be extended to handle other types of constraints, including lower bound of 0 and upper bound of 1—as occurs with fractional data. A further extension allows general range restrictions, creating an interpolant that lies between any two specified functions as the lower and upper bounds.