1. Introduction to Lifting Wavelet Transform (computationally efficient filterbank implementation) and Homework 3 Feb. 2, 2010

2. Lazy Wavelet Transform-I x 1 : even samples of x[n] and x 2 = x odd [n-1] x'[n] = x[n-1] It is lazy because it does not filter the input signal

3. Lazy Wavelet Transform-II x1: even samples; x2: odd samples of x[n] x'[n] = x[n] It is lazy because it does not filter the input signal

4. Lifting Idea Use filters in the down-sampled rate: Perfect reconstruction: x'[n] = x[n-1] You can add more branches and filters You can use Noble identity

5. Example: Halfband filters H(z)+H(-z) = 1 => where

6. Example: Both low- and highpass filters: Analysis filterbank: Synthesis filterbank:

7. Lifting summary: Computationally efficient ! (Don't compute the samples that you are going to drop during down- sampling) Perfect reconstruction property is trivially true It is used in JPEG-2000 image coding standard We will discuss Lifting later in more detail: Theorem (Sweldens and Daubechies): Every perfect reconstruction filterbank can be implemented using lifting stages

8. Homework 3 1.Show that lazy wavelet transforms achieve perfect reconstruction 2.a) Implement a 2-channel Perfect Reconstruction Filter Bank (PRFB) using Matlab b) Apply an input signal to your PRFB and show that your filter reconstructs the input. Plot the subsignals.