Inverse B-spline interpolation |

As I discussed in an earlier paper (Fomel, 1997b), a
general approach for constructing the interpolant function in
equation (1) is to select an appropriate function basis
for representing the function . The functional basis
representation has the general form

Unser et al. (1993a) noticed that the function basis idea has an
especially simple implementation if the basis is
convolutional and satisfies the equation

Evaluating the function in equation (8) at an integer value , we obtain the equation

which has the exact form of a discrete convolution. The basis function , evaluated at integer values, is digitally convolved with the vector of basis coefficients to produce the sampled values of the function . We can invert equation (9) to obtain the coefficients from by inverse recursive filtering (deconvolution). In the case of a non-causal filter , an appropriate spectral factorization will be needed prior to applying the recursive filtering.

According to the convolutional basis idea, forward interpolation becomes a two-step procedure. The first step is the direct inversion of equation (9): the basis coefficients are found by deconvolving the sampled function with the factorized filter . The second step reconstructs the continuous (or arbitrarily sampled) function according to formula (8). The two steps could be combined into one, but usually it is more convenient to apply them separately. I show a schematic relationship among different variables in Figure 10.

scheme
Schematic relationship among
different variables for interpolation with a convolutional basis.
Figure 10. | |
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Inverse B-spline interpolation |

2014-02-15