Haar Wavelets

1
Haar Wavelets

• A first look
• Ref Walker (ch1)
• Jyun-Ming Chen, Spring 2001

2
Introduction

• Simplest hand calculation suffice
• A prototype for studying more sophisticated
  wavelets
• Related to Haar transform, a mathematical
  operation

3
Haar Transform

• Assume discrete signal (analog function occurring
  at discrete instants)
• Assume equally spaced samples (number of samples
  2n)

• Decompose the signal into two sub-signals of half
  its length
• Running average (trend)
• Running difference (fluctuation)

4
Haar transform, 1-level

• Running difference
• Denoted by
• Meaning of superscript explained later

• Running average
• Multiplication by is needed to ensure
  energy conservation (see later)

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Example
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Inverse Transform
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Small Fluctuation Feature

• Magnitudes of the fluctuation subsignal (d) are
  often significantly smaller than those of the
  original signal
• Logical samples are from continuous analog
  signal with very short time increment
• Has application to signal compression

8
Energy Concerns

• Energy of signals

• The 1-level Haar transform conserves energy

9
Proof of Energy Conservation
10
Haar Transform, multi-level
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Compaction of Energy

• Compare with 1-level
• Can be seen more clearly by cumulative energy
  profile

12
Cumulative Energy Profile

• Definition

13
Algebraic Operations

• Addition subtraction
• Constant multiple
• Scalar product

14
Haar Wavelets

• 1-level Haar wavelets
• wavelet plus/minus wavy nature
• Translated copy of mother wavelet
• support of wavelet 2
• The interval where function is nonzero

Property 1. If a signal f is (approximately)
constant over the support of a Haar wavelet, then
the fluctuation value is (approximately) zero.
15
Haar Scaling Functions

• 1-level scaling functions
• Graph translated copy of father scaling function
• Support 2

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Haar Wavelets (cont)

• 2-level Haar scaling functions
• support 4

• 2-level Haar wavelets
• support 4

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Multiresolution Analysis (MRA)

• Natural basis
• Therefore

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MRA
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MRA
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Example
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Example (cont)
Decomposition coefficients obtained by inner
product with basis function
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Haar MRA
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More on Scaling Functions (Haar)

• They are in fact related
• Pj is called the synthesis filter (more later)

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Ex Haar Scaling Functions
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Ex Haar Scaling Functions
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More on Wavelets (Haar)

• They are in fact related
• Qj is called the synthesis filter (more later)

27
Ex Haar Wavelets
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Ex Haar Wavelets
Synthesis Filter Q1
Synthesis Filter Q2
29
Analysis Filters

• There is another set of matrices that are related
  to the computation of analysis/decomposition
  coefficient
• In the Haar case, they are the transpose of each
  other
• Later well show that this is a property unique
  to orthogonal wavelets

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Analysis/Decomposition (Haar)
A2
A3
B2
Analysis Filter Aj
Analysis Filter Bj
B3
A1
B1
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Synthesis Filters

• On the other hand, synthesis filters have to do
  with reconstructing the signal from MRA results

32
Synthesis/Reconstruction (Haar)
Synthesis Filter Pj
Synthesis Filter Qj
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Conclusion/Exercise