Multiresolution Analysis

Multiresolution Analysis

Background

The wavelet transform can be thought of as a band-pass filter, where the location and width in Fourier space depends on the wavelet scale. Larger scales imply a lower frequency and small bandwidth.

In computing the wavelet transform, you change from small scales to larger scales. At each stage you can stop and compute the inverse wavelet transform using the remaining coefficients, while setting the small-scale coefficients to zero. You can then build up a series of smooth (or low-passed), detailed (or band-passed), or rough (high-passed) versions of your original data.

Method

Details on computing the multiresolution analysis can be found in Lindsay et al. (1996).

Example

Use the "Mantle convection" dataset that is included in the Wavelet sample file. This dataset contains an image of convection within the Earth's mantle.

Try the following steps:

Select the Convection dataset and start up the Multiresolution Analysis viewer using either the Visualize Menu or the Toolbar button.

As you progress from top to bottom the wavelet scale increases in powers of two. At the smallest scale most of the image is still in the Smooth image. Notice that the Rough image contains only the edges or discontinuities which the small scales can pick out.

Change to the Haar wavelet and observe the different structure of the images.