4.5 The 2-d dwt

We have already seen in our discussion of The Haar Transform how the 1-D Haar transform (or wavelet) could be extended to 2-D by filtering therows and columns of an image separably.

All 1-D 2-band wavelet filter banks can be extended in a similar way. shows two levels of a 2-D filter tree. The input image at each level is split into 4bands (Lo-Lo = $y_{00}$ , Lo-Hi = $y_{01}$ , Hi-Lo = $y_{10}$ , and Hi-Hi = $y_{11}$ ) using the lowpass and highpass wavelet filters on the rows and columns in turn. The Lo-Lo band subimage $y_{00}$ is then used as the input image to the next level. Typically 4 levels are used, as for the Haar transform.

Filtering of the rows of an image by $H_a(z_1)$ and of the columns by $H_b(z_2)$ , where $a$ , $b$ = 0 or 1, is equivalent to filtering by the 2-D filter:

To obtain the impulse responses of the four 2-D filters at each level of the 2-D DWT we form $h_{a, b}$ from $h_0$ and $h_1$ using with $ab$ = 00, 01, 10 and 11.

shows the impulse responses at level 4 as images for three 2-D wavelet filtersets, formed from the following 1-D wavelet filter sets:

• The LeGall 3,5-tap filters: $H_0$ and $H_1$ from these equations , and these equations in our discussion of Good Filters / Wavelets.
• The near-balanced 5,7-tap filters: substituting $Z=\frac{1}{2}(z+z^{(-1)})$ into this previous equation .
• The near-balanced 13,19-tap filters: substituting this equation into this equation .

The 2-D frequency responses of the level 1 filters, derived from the LeGall 3,5-tap filters, are shown in figs (in mesh form) and (in contour form). These are obtained by substituting $z_1=e^{i{\omega }_1}$ and $z_2=e^{i{\omega }_2}$ into . demonstrates that the 2-D frequency response is just the product of the responses of the relevant1-D filters.

and are the equivalent plots for the 2-D filters derived from the near-balanced 13,19-tap filters. Wesee the much sharper cut-offs and better defined pass and stop bands of these filters. The high-band filters no longer exhibitgain peaks, which are rather undesirable features of the LeGall 5-tap filters.