The process of creating the outputs y1 y1 to y0000 y0000 from xx is known as the discrete wavelet transform (DWT); and the reconstruction process is the inverse DWT.

The word wavelet refers to the impulse response of the cascade of filters which leads to a given bandpass output. The frequency response of the wavelet at level kk is obtained by substituting z=ei⁢ω⁢ Ts z ω Ts in the z-transfer function H k , 1 H k , 1 from this equation and this equation from our discussion of the binary filter tree. Ts Ts is the sampling period at the input to the filter tree.

Since the frequency responses of the bandpass bands are scaled down by 2:1 at each level, their impulse responses become longer by the same factor at each level, BUT their shapes remain very similar. The basic impulse response wave shape is almost independent of scale and known as the mother wavelet.

The impulse response to a lowpass output H k , 0 H k , 0 is called the scaling function at level kk.

Figure 1 shows these effects using the impulse responses and frequency responses for the five outputs of the 4-level tree of Haar filters, based on the z-transforms given in this group of equations. Notice the abrupt transitions in the middle and at the ends of the Haar wavelets. These result in noticeable blocking artefacts in decompressed images (as in part (b) of this previous figure).

Figure 1: Impulse responses and frequency responses of the 4-level tree of Haar filters.

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This work is licensed by Nick Kingsbury under a Creative Commons Attribution License (CC-BY 1.0), and is an Open Educational Resource.

Last edited by Elizabeth Gregory on Jun 8, 2005 10:30 am -0500.