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Unser et al. (1993) noticed that the basis function idea has an
especially simple implementation if the basis is convolutional and
satisfies the equation
 
(65) 
In other words, the basis is constructed by integer shifts of a single
function . Substituting expression () into
equation () yields
 
(66) 
Evaluating the function f(x) in equation () at an
integer value n, we obtain the equation
 
(67) 
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 f(x). We can invert equation () to obtain the
coefficients c_{k} from f(n) by inverse recursive filtering
(deconvolution). In the case of a noncausal filter , an
appropriate spectral factorization will be
needed prior to applying the recursive filtering.
According to the convolutional basis idea, forward interpolation
becomes a twostep procedure. The first step is the direct inversion
of equation (): the basis coefficients c_{k} are found by
deconvolving the sampled function f(n) with the factorized filter
. The second step reconstructs the continuous (or arbitrarily
sampled) function f(x) according to formula (). 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 .
scheme
Figure 12 Schematic relationship among
different variables for interpolation with a convolutional basis.

 
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Stanford Exploration Project
12/28/2000