Mathematica 9 is now available
Wolfram Mathematica Documentation Center
DOCUMENTATION CENTER SEARCH
Function Index /
Previous section-----Next section

DiscreteWaveletTransform

DiscreteWaveletTransform uses Mallat's algorithm.
DiscreteWaveletTransform computes the periodic wavelet transform.
In the forward transform (i.e., decomposition) operation, the coefficients ck and dk at different scales are calculated based on the following convolution-like expressions
where j denotes the resolution and k is the index for the samples. This operation is a linear digital filtering operation followed by downsampling. The coefficients ckj and dkj are known, respectively, as the level j scaling and wavelet coefficients. The top-level coefficients cJ represent the original signal. For a signal of length N, where N is a power of two, we get J=Log[2,N]. In such a case, the iteration may be repeated J times with the last stage being of length one, one scaling coefficient, and one wavelet coefficient. Note that the iteration is performed over the scaling coefficients only.
Similarly, the 2D wavelet transform is computed using 2D filtering and downsampling operations. At each scale, the coefficients are given by the following formulas:
where the filters h0, h1, h2, and h3 are formed by vector outer products of the quadrature mirror filters h and g.
h0 = h hT,     h1 = h gT,    h2 = g hT,    h3= g gT.
At each scale j, the array cj contains the low-frequency information from the previous stage, while dj, ej, and fj contain the horizontal, vertical, and bidirectional edge information (high-frequency), respectively.
The argument img may be any list or an ImageData expression.
See also User's Guide 8.6.

Example

This loads the package.
In[1]:=
This creates a 2D Gaussian-shaped sequence.
In[2]:=
This computes the DWT.
In[3]:=
This shows the result of the 2-level decomposition of the Gaussian image.
In[4]:=
Out[4]=Out[4]=


Any questions about topics on this page? Click here to get an individual response.Buy NowMore Information
THIS IS DOCUMENTATION FOR AN OBSOLETE PRODUCT.