In particular, let n be an index for the filter samples. The two filters must have a specific relationship to each other. In the QMF pyramid, we apply two filters (hi- and lo- pass) and subsample BOTH by a factor of 2, thus eliminating the excess coefficients of the Laplacian pyramid. The lowpass band is subsampled by a factor of 2, but the highpass band is NOT subsampled. Recall that the Laplacian pyramid is formed by simple hi/low splitting at each level. These are closely related to Wavelets (essentially, they are approximate wavelet filters). One type that arose in the speech coding community is based on a particular pairs of filters known as a “Quadrature Mirror Filters” or QMFs. Secondly, the “bandpass” images (fineN) do not segregate information according to orientation. Specifically, the 1-dimensional transform is overcomplete by a factor of 4/3, and the 2-dimensional transform is overcomplete by a factor of 2. First, there are more pixels (coefficients) in the representation than in the original image. Two things about Laplacian pyramids are a bit unsatisfactory. ‘binom3’ with ‘binom5’ or ‘binom7’ below), and you’ll see that the aliasing is much worse for the 3 tap filter. Try this for 2 different filters (by replacing If there’s no aliasing, then the blur and shift operations should commute (i.e., shift-filter-downsample-upsample-filter is the same as filter-downsample-upsample-filter-shift).
Then blur (filter-downsample-upsample-filter) the original image and blur the shifted image. We choose an image and shift it by some number of pixels. One way to see the consequences of the aliasing artifacts is by examining variations that occur when the input is shifted. However, it can be a serious problem if we intend to process each of the subbands independently. So it’s not a problem if the only thing we want to do is reconstruct. When reconstructing, the pyramid is designed in such a way that these aliasing artifacts cancel out. This is true even though the Laplacian pyramid can be used to reconstruct the original image perfectly. Unless one is careful, the subsampling operations will introduce aliasing artifacts in these pyramid transforms. ALIASING in the Gaussian and Laplacian pyramids: