Terminology and Other Pitfalls

Since the need for interpolation arises so often in practice, its droning use makes it look like a simple operation. This is deceiving, and the fact that the interpolation terminology is so perplexing hints at hidden difficulties. Let us attempt to befuddle the reader: The cubic B-spline is a piecewise polynomial function of degree 3. It does not correspond to what is generally understood as cubic convolution, the latter being a Keys' function made of piecewise polynomials of degree 3 (like the cubic B-spline) and of maximal order 3 (contrary to the cubic B-spline, for which the order is 4). No B-spline of sufficient degree should ever be used as an interpolant ^int, but a high-degree B spline makes for a high-quality synthesis function The Appledorn function of degree 4 is no polynomial and has order zero. There is no degree that would be associated to the sinc function, but its order is infinite. Any polynomial of a given degree can be represented by splines of the same degree, but, when the spline degree increases to infinity, the cardinal spline tends to the sinc function, which can at best represent a polynomial of degree zero. In order to respect isotropy in two dimensions, we expect that the synthesis function itself must be endowed with rotational symmetry; yet the best possible function (sinc) is not. Kriging is known to be the optimal unbiased linear interpolator, yet it does not belong to the category of linear systems; an example of linear system is the Dodgson synthesis function made of quadratic polynomials. Bilinear transformation and bilinear interpolation have nothing in common. Your everyday image is low-pass, yet its most important features are its edges. And finally, despite more than 10 years of righteous claims to the contrary [31] some authors (who deserve no citation) persist in believing that every synthesis function ^ built to match Eq. (4) can be used in Eq. (1) as well, in the place of ^int. Claims that the use of a cubic B-spline blurs data, which are wrongly made in a great majority of image processing textbooks, are a typical product of this misunderstanding.


We hope that the definitions of order and degree, given later in this chapter, will help to clarify this mess. We also hope to make clear when to use Eq. (1) and when to use the generalized approach of Eq. (4).

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