Forward interpolation

Interpolation with convolutional bases

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

$$\psi_k (x) = \beta (x-k)\;.$$ (40)

In other words, the basis is constructed by integer shifts of a single function $\beta(x)$. Substituting expression (40) into equation (8) yields

$$f (x) = \sum_{k \in K} c_k \beta (x - k)\;.$$ (41)

Evaluating the function $f (x)$ in equation (41) at an integer value $n$, we obtain the equation

$$f (n) = \sum_{k \in K} c_k \beta (n-k)\;,$$ (42)

which has the exact form of a discrete convolution. The basis function $\beta(x)$, evaluated at integer values, is digitally convolved with the vector of basis coefficients to produce the sampled values of the function $f (x)$. We can invert equation (42) to obtain the coefficients $c_k$ from $f (n)$ by inverse recursive filtering (deconvolution). In the case of a non-causal filter $\beta(n)$, 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 (42): the basis coefficients $c_k$ are found by deconvolving the sampled function $f (n)$ with the factorized filter $\beta(n)$. The second step reconstructs the continuous (or arbitrarily sampled) function $f (x)$ according to formula (41). 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 12.

scheme Figure 12. Schematic relationship among different variables for interpolation with a convolutional basis.

Forward interpolation