April 2019
Intermediate to advanced
624 pages
16h 54m
English
Content preview from Discrete Wavelet Transformations, 2nd Edition Become an O’Reilly member and get unlimited access to this title plus top books and audiobooks from O’Reilly and nearly 200 top publishers, thousands of courses curated by job role, 150+ live events each month,
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CHAPTER 11LIFTING
If you worked Problem 4.36 in Section 4.4, then you have had a brief introduction to lifting and how to use it in lossless compression.
The lifting method for computing wavelet transforms was introduced by Wim Sweldens1 in [92]. For a quick introduction to the topic, see Sweldens' tutorial paper [91]. Another good reference on lifting is the book by Jensen and la Cour‐Harbo [61].
This chapter begins with a section that introduces lifting via the LeGall biorthogonal filter pair. This filter pair is a modification of the (5, 3) biorthogonal spline filter pair introduced in Section 7.1. Section 11.2 introduces Z‐transforms and Laurent polynomials. We introduce and consider several properties of Z‐transforms and Laurent polynomials. In particular, we discuss the identification of a greatest common divisor of two Laurent polynomials obtained via the Euclidean algorithm. The matrix formulation of this greatest common divisor is a critical component in the derivation of a lifting method for a lowpass/highpass filter pair. In Section 11.3, we introduce the concept of a polyphase matrix. The factorization of this matrix, due to Daubechies and Sweldens [31], leads to a lifting scheme for a wavelet transform. We state this result and provide a constructive proof in this section. The final section contains several examples of lifting methods for lowpass/highpass filter pairs.
11.1 The LeGall Wavelet Transform
One of the goals of the JPEG2000 standard was to include lossless compression ...
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