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 the rows and columns of an image separably.

All 1-D 2-band wavelet filter banks can be extended in a similar way. Figure 1 shows two levels of a 2-D filter tree. The input image at each level is split into 4 bands (Lo-Lo = y 0 0 y 0 0 , Lo-Hi = y 0 1 y 0 1 , Hi-Lo = y 1 0 y 1 0 , and Hi-Hi = y 1 1 y 1 1 ) using the lowpass and highpass wavelet filters on the rows and columns in turn. The Lo-Lo band subimage y 0 0 y 0 0 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 Ha ⁢ z1 Ha z1 and of the columns by Hb ⁢ z2 Hb z2 , where aa, bb = 0 or 1, is equivalent to filtering by the 2-D filter:

In the spatial domain, this is equivalent to convolving the image matrix with the 2-D impulse response matrix where ha h a and hb h b are column vectors of the 1-D filter impulse responses. However note that performing the filtering separably (i.e. as separate 1-D filterings of the rows and columns) is much more computationally efficient.

To obtain the impulse responses of the four 2-D filters at each level of the 2-D DWT we form ha,b h a b from h0 h 0 and h1 h 1 using Equation 2 with a b a b = 00, 01, 10 and 11.

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

Note the sharp points in Figure 2(b), produced by the sharp peaks in the 1-D wavelets of this previous figure (Impulse and frequency responses of the 4-level tree of near-balanced 5,7-tap filters). These result in noticeable artefacts in reconstructed images when these wavelets are used. The smoother wavelets of Figure 2(c) are much better in this respect.

The 2-D frequency responses of the level 1 filters, derived from the LeGall 3,5-tap filters, are shown in figs Figure 3 (in mesh form) and Figure 4 (in contour form). These are obtained by substituting z1 =ei⁢ ω1 z1 ω1 and z2 =ei⁢ ω2 z2 ω2 into Equation 1. Equation 1 demonstrates that the 2-D frequency response is just the product of the responses of the relevant 1-D filters.

Figure 5 and Figure 6 are the equivalent plots for the 2-D filters derived from the near-balanced 13,19-tap filters. We see the much sharper cut-offs and better defined pass and stop bands of these filters. The high-band filters no longer exhibit gain peaks, which are rather undesirable features of the LeGall 5-tap 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 10:04 am -0500.