Can Aliasing Happen in Laplacian Pyramids?

Question

I want to use the Laplacian pyramid framework for representing images. The The Laplacian Pyramid as a Compact Image Code's language seems to provide freedom in the choice of blurring/interpolation kernel, subject to certain constraints. However, as we are performing subsampling when constructing pyramids, would aliasing be a problem with an injudicious choice?

I read in the Toronto lecture notes on image pyramids (Slide 8) that a laplacian pyramid can reconstruct an image exactly, agnostic of the kernel. Can i take this to be a guarantee against aliasing, or can aliasing be "hidden" by the reconstruction process?

Answer 1

The reconstruction will remove aliasing, no matter which filter kernel is used.

The lower layer of the 2-layer Laplacian pyramid on page 8 of the Toronto lecture notes, adapted here:

Figure 1. A 2-layer Laplacian pyramid.

can be redrawn as:

Figure 2. An alternative presentation of the lower layer of the Laplacian pyramid of fig. 1.

This layer is an identity operation: what is subtracted is also added. This allows to draw the next (and in this case the only) layer above it the same way and it also can be identified to be an identity operation. And so on until the whole process becomes an identity operation.

Answer 2

Yes, most of the known multiscale or multirate decompositions, as long as they combine at one stage a non-ideal filter and a subsampling operator, induce some kind of aliasing at the analysis stage. And the Laplacian pyramid does so. But proper designs allow perfect reconstruction, so the aliasing can be reversed or cancelled, and kept reasonable in the case of processing between analysis and synthesis.

The simplest form of alias cancellation arises with Haar sum and difference filters. Consider $h_l = [1,1]$ and $h_h = [1,-1]$, the sum and difference filters, followed by a 2-fold downsampling. Both filters have very poor frequency responses, and cause aliasing with downsampling.

Take signal $X=[a,b]$. With the above scheme, you get two outputs $y_l = a+b$ and $y_h = b-a$, from which you can recover $X$ with $(y_l-y_h)/2 = a$ and $(y_l+y_h)/2 = b $. The latter operations are filtering by an averaged difference filter, and an averaging filter, e.g. the same as above, with a different scaling factor.

That is the magic of perfect reconstruction filter-banks: from well-chosen imperfect analysis filter-banks (with some decimation), to be able to design a synthesis filter-bank, itself imperfect, that cancels aliasing and distorsion of the whole analysis-synthesis system. It may happen as long as you have two or more channels, and with a little oversampling in the system, you can even find optimized inverse filter-banks.