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

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feb 2 2010
Feb. 2, 2010Introduction to Lifting Wavelet Transform (computationally efficient filterbank implementation)and Homework 3
lazy wavelet transform i
Lazy Wavelet Transform-I
  • x1: even samples of x[n] and x2 = xodd[n-1]
  • x'[n] = x[n-1]
  • It is lazy because it does not filter the input signal
lazy wavelet transform ii
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
lifting idea
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
example halfband filters
Example: Halfband filters
  • H(z)+H(-z) = 1 =>
  • where
example both low and highpass filters
Example: Both low- and highpass filters:
  • Analysis filterbank:
  • Synthesis filterbank:
lifting summary
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
homework 3
Homework 3
  • Show that lazy wavelet transforms achieve perfect reconstruction
  • 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.