The Wavelet Transform - Sparse Image and Signal Processing - page 29

w 1 w 1

w 2 w 2

w 1

w 3 w 3

w 2

c 3

w 3

Figure 2.7. The image “Einstein” (top right) and its DWT decomposition (schematic, top
left; coefficients, bottom left). The bottom right image is the result of the compression-
decompression JPEG 2000 algorithm, employing the 7/9 DWT, using a compression ratio
of 40.

Using this method, we have only one wavelet image at each scale, and not three,
as in the previous separable DWT algorithm. Figure 2.8 shows the organization
of the wavelet subimages and the smoothed image. For more effectively visualiz-
ing the entire transformation, coefficients are reorganized in a compact way. Fig-
ure 2.10 summarizes this reorganization. The low-pass filter h is centered at the
samples denoted by crosses, which provides the image at the lower resolution, while
the high-pass filter g is centered at the samples indicated by circles, which allows
the wavelet coefficients to be obtained. The shift due to the filter g is made during
down-sampling.
Figure 2.9 shows the nonzero coefficients of the 2-D low-pass filter h and its dual
h used in the Feauveau wavelet transform. Figure 2.10 displays the overall flowchart