Inverse B-spline interpolation |

The two simplest forms of the forward interpolation operators are the
1-point nearest neighbor interpolation with the weight

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 (2) and (3) and their spectra are plotted in Figures 1 and 2. The spectra plots show that both interpolants act as low-pass filters, preventing the high-frequency energy from being correctly interpolated.

nnint
Nearest neighbor interpolant (left) and its spectrum
(right).
Figure 1. |
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linint
Linear interpolant (left) and its spectrum
(right).
Figure 2. |
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On the other side of the accuracy scale, there is the infinitely long
sinc interpolant:

sincint
Sinc interpolant (left) and its spectrum
(right).
Figure 3. |
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Several approaches exist for extending the nearest neighbor and linear
interpolation to more accurate (albeit more expensive) methods. One
example is the 4-point cubic convolution suggested by Keys (1981).
The cubic convolution interpolant is a local piece-wise cubic
function, which approximates the ideal sinc equation (4).
Another popular approach is to taper the ideal sinc function in a
local window. For example, one can use the Kaiser window (Kaiser and Shafer, 1980)

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. Observing Figures 5, 6, and 7, we can see the interpolation error steadily decreasing as we go subsequently from 1-point nearest neighbor to 2-point linear, 4-point cubic convolution, and 8-point windowed sinc 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. | |
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binlin
Interpolation error of the nearest
neighbor interpolant (dashed line) compared to that of the linear
interpolant (solid line).
Figure 5. | |
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lincub
Interpolation error of the linear
interpolant (dashed line) compared to that of the cubic convolution
interpolant (solid line).
Figure 6. | |
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cubkai
Interpolation error of the cubic
convolution interpolant (dashed line) compared to that of the
8-point windowed sinc interpolant (solid line).
Figure 7. | |
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The differences among different methods are also clearly visible from the discrete spectra of the corresponding interpolants. The left plots in figures 8 and 9 show discrete interpolation responses: the function for a fixed value of . The right plots compare the corresponding discrete spectra. We can see that the spectrum gets flatter and wider as the accuracy of the method increases.

speclincub
Discrete interpolation responses of
linear and cubic convolution interpolants (left) and their
discrete spectra (right) for .
Figure 8. | |
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speccubkai
Discrete interpolation responses
of cubic convolution and 8-point windowed sinc interpolants (left)
and their discrete spectra (right) for .
Figure 9. | |
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Inverse B-spline interpolation |

2014-02-15