IntervalWavelets is a Julia package for computing ordinary/interior Daubechies scaling functions and the moment preserving boundary scaling functions of Cohen, Daubechies and Vial. See the enclosed document for further description of these functions.

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add IntervalWavelets

Except for the explicitly defined Haar wavelet, Daubechies scaling functions can only be calculated at the dyadic rationals, i.e., rational numbers of the form k/2^R (where R >= 0). In IntervalWavelets we refer to R as the resolution of the dyadic rationals.

Let us first load IntervalWavelets as well as the Plots package.

using IntervalWavelets
using Plots

The ordinary Daubechies scaling functions are called interior scaling functions in IntervalWavelets. An interior scaling function is defined using a filter where we specify the number of vanishing moments (of the associated wavelet) and the phase. A second argument for interior_filter specifies the phase to be either minimum or symmlet (the default).

Besides the filter we specify the resolution at which we wish to compute the scaling function:

h = interior_filter(2)
phi = interior_scaling_function(h, 8)
plot(phi)

The scaling function phi can be evaluated in all dyadic rationals (where it is computed). As an example we evaluate phi in -0.5:

phi(DyadicRational(-1, 1))

0.9330127018921028

The boundary scaling functions are either modified near the left or right side of their support. The side is specified with an enum with values LEFT and RIGHT. There is no choice of filter for the boundary scaling functions so we just specify the number of vanishing moments and the resolution of the scaling functions:

L = boundary_scaling_functions(LEFT, 2, 8)
plot(L)

Similarly, we get the right boundary scaling functions:

R = boundary_scaling_functions(RIGHT, 2, 8)
plot(R)

The boundary scaling functions are indexed from 0 through p - 1, where p is the number of vanishing moments.

L[0]

Left scaling function number 0 with 2 vanishing moments at resolution 8

Just as the interior scaling functions the individual boundary scaling functions can be evaluated in all dyadic rationals.

A basis of scaling functions is computed by specifying the number of vanishing moments (here 2), the scale of the functions (here 3) and their resolution (here 8):

B = interval_scaling_function_basis(2, 3, 8)
plot(B)

By default, the basis is on the interval [0, 1], but this can be specified with optional arguments. The basis functions are indexed from 1 through 2^J where J is the scale. We do not return the functions as above, but the function values in the support. This is to make reconstruction faster.

From a set of coefficients representing a function in a basis we can compute its reconstruction.

x, y = reconstruct(B, ones(8));

If the coefficients are a matrix the reconstruction is also a matrix (representing an image).

This README is generated with the Weave package using the command

weave("README.jmd", doctype = "github", fig_path = "figures")