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Twin Dragon Wavelet

scaling function
wavelet
iteration
8
edges
tile
zoom %
100
This Demonstration shows the iteration process that approximates a continuous-time, twin dragon wavelet basis from the discrete-time one. The iteration is performed on the so-called quincunx lattice. At every iteration, the resulting function tiles the space, even though it has a "fractal" boundary.
In discrete time, two filters, low pass and high pass, are needed to create a wavelet basis. When iterated on the sampling lattice, they will lead to an equivalent discrete-time scaling function (obtained by repeated upsampling and filtering by the low pass) and an equivalent discrete-time wavelet (obtained by repeated upsampling and filtering by the low pass and a final step by the high pass). In the Demonstration, the low pass and high pass filters are of Haar form (two-point average and difference). The equivalent approximated continuous-time scaling function and wavelet are shown in each iteration; the darker color denotes where the function equals
1
, and the lighter where it is
-1
(for the wavelet). The equivalent discrete-time scaling function and wavelet are just single points in each tile (which can be seen by turning on edges). The plots are shown in the new coordinate system at each iteration (this is why they seem to rotate about the origin). The meaning of the zeroth iteration is explained in the Details.
You can observe two effects:
1. Both the scaling function and the wavelet in the current iteration are obtained from two copies of the scaling function from the previous iteration. This is called the "two-scale equation", and can most easily be seen by observing the supports of the wavelet at
1
and
-1
; these supports correspond to the support of the scaling function and its shift from the previous iteration.
2. At every iteration, the support of the scaling function/wavelet and its copies on the lattice (white points) tile the space; this is called a fractal tiling of the plane (check "tile" and only the eight closest tiles are shown).
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