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Feb. 2, 2010

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

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

  2. 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

  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 • 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.

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