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The 2-D DWT

Module by: Nick Kingsbury. E-mail the author

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Summary: This module introduces 2-D DWT.

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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.

Figure 1: Two levels of a 2-D filter tree, formed from 1-D lowpass ( H0 H0 ) and highpass ( H1 H1 ) filters.
Figure 1 (figure13.png)

Filtering of the rows of an image by Haz1 Ha z1 and of the columns by Hbz2 Hb z2 , where aa, bb = 0 or 1, is equivalent to filtering by the 2-D filter:

H a b z1z2=Haz1Hbz2 H a b z1 z2 Ha z1 Hb z2 (1)
In the spatial domain, this is equivalent to convolving the image matrix with the 2-D impulse response matrix
hab=hahbT h a b h a h b (2)
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 hab 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: 2-D impulse responses of the level-4 wavelets and scaling functions derived from the LeGall 3,5-tap filters (a), and the near-balanced 5,7-tap (b) and 13,19-tap (c) filters.
Figure 2 (figure14.png)

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:

  1. The LeGall 3,5-tap filters: H0 H0 and H1 H1 from these equations, and these equations in our discussion of Good Filters / Wavelets.
  2. The near-balanced 5,7-tap filters: substituting Z=12z+z-1 Z 1 2 z z into this previous equation.
  3. The near-balanced 13,19-tap filters: substituting this equation into this equation.
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=ⅇⅈω1 z1 ω1 and z2=ⅇⅈω2 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 3: Mesh frequency response plots of the 2-D level 1 filters, derived from the LeGall 3,5-tap filters.
Figure 3 (figure15.png)
Figure 4: Contour plots of the frequency responses of Figure 3.
Figure 4 (figure16.png)

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.

Figure 5: Mesh frequency response plots of the 2-D level 1 filters, derived from the near-balanced 13,19-tap filters.
Figure 5 (figure17.png)
Figure 6: Contour plots of the frequency responses of Figure 5.
Figure 6 (figure18.png)

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