Mathematical interpolation theory considers a function *f*, defined on
a regular grid *N*. The problem is to find *f* in a continuum that
includes *N*. I am not defining the dimensionality of *N* and *f* here
because it is not essential for the derivations. Furthermore, I am
not specifying the exact meaning of ``regular grid,'' since it will
become clear from the analysis that follows. The function *f* is
assumed to belong to a Hilbert space with a defined dot product.

If we restrict our consideration to a linear case, the desired solution will take the following general form

(26) |

**Property 9334**

(27) |

Equality () is necessary to assure that the interpolation
of a single spike at some point *n* does not change the value *f* (*n*)
at the spike.

**Property 9338**

(28) |

This property is the normalization condition. Formula ()
assures that interpolation of a constant function *f*(*n*) remains
constant.

One classic example of the interpolation weight *W* (*x*, *n*) is the
Lagrange polynomial, which has the form

(29) |

(30) |

(31) |

Because of their simplicity, the nearest-neighbor and linear interpolation methods are very practical and easy to apply. Their accuracy is, however, limited and may be inadequate for interpolating high-frequency signals. The shapes of interpolants () and () and their spectra are plotted in Figures and . The spectral plots show that both interpolants act as low-pass filters, preventing the high-frequency energy from being correctly interpolated.

Figure 1

Figure 2

The Lagrange interpolants of higher order correspond to more complicated polynomials. Another popular practical approach is cubic convolution Keys (1981). The cubic convolution interpolant is a local piece-wise cubic function:

(32) |

Figure 3

I compare the accuracy of different forward interpolation methods on a one-dimensional signal shown in Figure . The ideal signal has an exponential amplitude decay and a quadratic frequency increase from the center towards the edges. It is sampled at a regular 50-point grid and interpolated to 500 regularly sampled locations. The interpolation result is compared with the ideal one. Figures and show the interpolation error steadily decreasing as we proceed from 1-point nearest-neighbor to 2-point linear and 4-point cubic-convolution interpolation. At the same time, the cost of interpolation grows proportionally to the interpolant length.

chirp
One-dimensional test signal. Top:
ideal. Bottom: sampled at 50 regularly spaced points. The bottom
plot is the input in a forward interpolation test.
Figure 4 |

binlin
Interpolation error of the
nearest-neighbor interpolant (dashed line) compared to that of the
linear interpolant (solid line).
Figure 5 |

lincub
Interpolation error of the linear
interpolant (dashed line) compared to that of the cubic convolution
interpolant (solid line).
Figure 6 |

12/28/2000