Signal Processing Stack Exchange is a question and answer site for practitioners of the art and science of signal, image and video processing. It only takes a minute to sign up.
Sign up to join this community
I am having some trouble understanding the intuitive connection between the two. I worked through all the theory and learned that Sweldens showed how to derive lifting steps when having admissible filter taps (Daubechies, etc.).
The opposite is simple mathematically, i.e. deriving the filter taps when having discrete lifting steps. However, I dont get how one would come up with 'admissible' lifting steps in the first place, if not by getting them from valid filter taps like Sweldens showed.
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 ($\lfloor \cdot\rfloor$ denotes rounding down):
How to design a lifting scheme from scratch requires to:
I do like a couple of papers from people at KU Leuven:
