The 2-D DWT 2.2 2003/05/01 2003/06/23 Nick Kingsbury ngk10@cam.ac.uk Liqun Wang liqun@rice.edu Nick Kingsbury ngk10@cam.ac.uk 1-D Haar transform 2-D DWT This module introduces 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 the rows 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 4 bands (Lo-Lo = y 0 0 , Lo-Hi = y 0 1 , Hi-Lo = y 1 0 , and Hi-Hi = 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 is then used as the input image to the next level. Typically 4 levels are used, as for the Haar transform.

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

Filtering of the rows of an image by Ha z1 and of the columns by Hb z2 , where a, b = 0 or 1, is equivalent to filtering by the 2-D filter: H a b z1 z2 Ha z1 Hb z2 In the spatial domain, this is equivalent to convolving the image matrix with the 2-D impulse response matrix h a b h a h b where h a and 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 h a b from h 0 and h 1 using with a b = 00, 01, 10 and 11.

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.

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: The LeGall 3,5-tap filters: H0 and H1 from these equations, and these equations in our discussion of Good Filters / Wavelets. The near-balanced 5,7-tap filters: substituting Z 1 2 z z into this previous equation. The near-balanced 13,19-tap filters: substituting this equation into this equation. Note the sharp points in (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 (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 (in mesh form) and (in contour form). These are obtained by substituting z1 ω1 and z2 ω2 into . demonstrates that the 2-D frequency response is just the product of the responses of the relevant 1-D filters.

Mesh frequency response plots of the 2-D level 1 filters, derived from the LeGall 3,5-tap filters.

Contour plots of the frequency responses of .

and 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.

Mesh frequency response plots of the 2-D level 1 filters, derived from the near-balanced 13,19-tap filters.

Contour plots of the frequency responses of .