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   -> Volume 10, Issue 4

Thesis: Complex Wavelet Transforms with Controllable Redundancy
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"Fernandes, Felix" (

PostPosted: Wed Oct 03, 2001 10:08 pm    
Subject: Thesis: Complex Wavelet Transforms with Controllable Redundancy
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#10 Thesis: Complex Wavelet Transforms with Controllable Redundancy

My Ph.D. dissertation entitled

Directional, Shift-Insensitive, Complex Wavelet Transforms with
Controllable Redundancy

is available at

Felix Fernandes.

Abstract: Although the Discrete Wavelet Transform (DWT) is a powerful
tool for signal and image processing, it has three serious
disadvantages. First, the DWT is shift sensitive because input-signal
shifts generate unpredictable changes in DWT coefficients. Second,
the DWT suffers from poor directionality because DWT coefficients
reveal only three spatial orientations. Third, DWT analysis lacks the
phase information that accurately describes non-stationary signal
behavior. To overcome these disadvantages, we introduce the notion of
projection-based complex wavelet transforms.

These two-stage, projection-based complex wavelet transforms consist
of a projection onto a complex function space followed by a DWT of the
complex projection. Unlike other popular transforms that also mitigate
DWT shortcomings, the decoupled implementation of our transforms has
two important advantages. First, the controllable redundancy of the
projection stage offers a balance between degree of shift sensitivity
and transform redundancy. This allows us to create a directional,
non-redundant, complex wavelet transform with potential benefits for
image coding systems. To the best of our knowledge, no other complex
wavelet transform is simultaneously directional and non-redundant.
The second advantage of our approach is the flexibility to use
emph{any} DWT in the transform implementation. We exploit this
flexibility to create the Complex Double-density DWT (CDDWT): a
shift-insensitive, directional, complex wavelet transform with a low
redundancy of $frac{3^m - 1}{2^m - 1}$ in $m$ dimensions. To the best
of our knowledge, no other transform achieves all these properties at
a lower redundancy. Besides the mitigation of DWT shortcomings, our
transforms have unique properties that will potentially benefit a
variety of signal processing applications. As an example, we
demonstrate that our projection-based complex wavelet transforms
achieve state-of-the-art results in a seismic signal-processing
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