Lifting scheme versus filter banks

Traditional (analysis) filter banks are banks of linear filters followed by downsampling. This is not so efficient, because you convolve and then subsample: possibly a waste of operations. People have found ways to invert the operations, downsample and convolve, with other filters involve the polyphase components (even and odd ones in the two-channel case) of the signal (polyphase matrix). Such matrix can be factored into a product simpler matrices, like diagonal, upper and lower diagonal ones, acting more separately on the polyphase components.

A reinterpretation consists in using the upper matrix on some polyphase component (say, odds) to predict and combine outcome to other components (say, even) to get what is called an update, possibly with scaling factors.

Properties of polyphase matrices tell that this is always possible, yet not unique. And one can save some operations, compute inverses more easily (upper and lower matrices have simpler inverses). So all invertible linear filter banks can be converted in lifting.

But more importantly, one can insert non-linear blocks and still be invertible and efficient. So the lifting is more generic that "linear" filter banks (as long as they have finite support). An example of the above is known as the S-transform: given a sequence of integers $c[n]$, you can decompose it into ($lfloor cdotrfloor$ denotes rounding down):

• $l[n] =Largelfloor frac{c[2n]+c[2n+1]}{2}Largerfloor$
• $h[n] = c[2n]-c[2n+1]$

How to design a lifting scheme from scratch requires to:

• understand how to split the data into several subsets
• design proper predict and update operators
• choose how to iterate them on different levels.

I do like a couple of papers from people at KU Leuven:

• Integer wavelet transfroms using the lifting scheme, Geert Uytterhoeven, Dirk Roose, Adhemar Bultheel
• Wavelet Transforms Using the Lifting Scheme, Uytterhoeven, Geert and Roose, Dirk and Bultheel, Adhemar
• WAILI: Wavelets with integer lifting, G. Uytterhoeven, F. Van Wulpen, M. Jansen, D. Roose and A. Bultheel