4.3 Wavelets

The process of creating the outputs $y_1$ to $y_{0000}$ from $x$ 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 bandpassoutput. The frequency response of the wavelet at level $k$ is obtained by substituting $z=e^{i\omega T_s}$ in the z-transfer function $H_{k,1}$ from this equation and this equation from our discussion of the binary filter tree. $T_s$ 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 longerby the same factor at each level, BUT their shapes remain very similar. The basic impulse response waveshape is almost independent of scale and known as the mother wavelet .

The impulse response to a lowpass output $H_{k,0}$ is called the scaling function at level $k$ .

shows these effects using the impulse responses and frequency responses for the fiveoutputs 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 Haarwavelets. These result in noticeable blocking artefacts in decompressed images (as in part (b) of this previous figure ).