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
(1) |
Property 1650
(2) |
Equality (2) 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 1654
(3) |
This property is the normalization condition. Formula (3) 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
(4) |
(5) |
(6) |
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 (5) and (6) and their spectra are plotted in Figures 1 and 2. The spectral plots show that both interpolants act as low-pass filters, preventing the high-frequency energy from being correctly interpolated.
The Lagrange interpolants of higher order correspond to more complicated polynomials. Another popular practical approach is cubic convolution (). The cubic convolution interpolant is a local piece-wise cubic function:
(7) |
I compare the accuracy of different forward interpolation methods on a one-dimensional signal shown in Figure 4. 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 5 and 6 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
Figure 4 One-dimensional test signal. Top: ideal. Bottom: sampled at 50 regularly spaced points. The bottom plot is the input in a forward interpolation test. |
binlin
Figure 5 Interpolation error of the nearest-neighbor interpolant (dashed line) compared to that of the linear interpolant (solid line). |
lincub
Figure 6 Interpolation error of the linear interpolant (dashed line) compared to that of the cubic convolution interpolant (solid line). |