Wavelet Transform. What Are Wavelets? In general, a family of representations using: hierarchical...
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Transcript of Wavelet Transform. What Are Wavelets? In general, a family of representations using: hierarchical...
What Are Wavelets?
In general, a family of representations using:
• hierarchical (nested) basis functions
• finite (“compact”) support
• basis functions often orthogonal
• fast transforms, often linear-time
MULTIRESOLUTION ANALYSIS (MRA)
• Wavelet Transform– An alternative approach to the short time Fourier transform to
overcome the resolution problem – Similar to STFT: signal is multiplied with a function
• Multiresolution Analysis – Analyze the signal at different frequencies with different
resolutions– Good time resolution and poor frequency resolution at high
frequencies– Good frequency resolution and poor time resolution at low
frequencies– More suitable for short duration of higher frequency; and longer
duration of lower frequency components
PRINCIPLES OF WAVELET TRANSFORM
• Split Up the Signal into a Bunch of Signals
• Representing the Same Signal, but all Corresponding to Different Frequency Bands
• Only Providing What Frequency Bands Exists at What Time Intervals
Wavelet Transform (WT)
• Wavelet transform decomposes a signal into a set of basis functions.• These basis functions are called wavelets• Wavelets are obtained from a single prototype wavelet (t) called
mother wavelet by dilations and shifting:
where a is the scaling parameter and b is the shifting parameter
)(1
)(, a
bt
atba
• The continuous wavelet transform (CWT) of a function f is defined as
• If is such that
f can be reconstructed by an inverse wavelet transform:
dta
bttf
afbaTf ba )()(
1,),( *
,
dC2
)(
20
,1 )(),()(
a
dadbtbaTfCtf ba
SCALE
• Scale– a>1: dilate the signal
– a<1: compress the signal
• Low Frequency -> High Scale -> Non-detailed Global View of Signal -> Span Entire Signal
• High Frequency -> Low Scale -> Detailed View Last in Short Time
• Only Limited Interval of Scales is Necessary
Wavelet transform vs. Fourier Transform
• The standard Fourier Transform (FT) decomposes the signal into individual frequency components.
• The Fourier basis functions are infinite in extent.
• FT can never tell when or where a frequency occurs.
• Any abrupt changes in time in the input signal f(t) are spread out over the whole frequency axis in the transform output F() and vice versa.
• WT uses short window at high frequencies and long window at low frequencies (recall a and b in (1)). It can localize abrupt changes in both time and frequency domains.
RESOLUTION OF TIME & FREQUENCY
Time
Frequency
Better time resolution;Poor frequency resolution
Better frequency resolution;Poor time resolution
• Each box represents a equal portion • Resolution in STFT is selected once for entire analysis
Discrete Wavelet Transform
• Discrete wavelets
• In reality, we often choose
• In the discrete case, the wavelets can be generated from dilation equations, for example,
(t)h(0)(2t) + h(1)(2t-1) + h(2)(2t-2) + h(3)(2t-3)]• Solving equation (2), one may get the so called scaling function (t).
• Use different sets of parameters h(i)one may get different scaling functions.
),( 02
0, ktaa jj
kj ., Zkj
.20 a
2
Discrete WT Continued
• The corresponding wavelet can be generated by the following equation
(t)[h(3)(2t) - h(2)(2t-1) + h(1)(2t-2) - h(0)(2t-3)]. (3)
• When and
equation (3) generates the D4 (Daubechies) wavelets.
2
,24/)31()0( h ,24/)33()1( h ,24/)33()2( h
24/)31()3( h
Discrete WT continued
• In general, consider h(n) as a low pass filter and g(n) as a high-pass filter where
• g is called the mirror filter of h. g and h are called quadrature mirror filters (QMF).
• Redefine– Scaling function
).1()1()( nNhng n
).2()(2)( nxnhxn
Discrete Formula
– Wavelet function
• Decomposition and reconstruction of a signal by the QMF.
where and is down-sampling and is up-sampling
).2()(2)( nxngxn
2
2
2
2
f(n) +
f(n)
)(ng
)(nh )(nh
)(ng
)()( nhnh ).()( ngng
Generalized Definition
• Let be matrices, where are positive integers
is the low-pass filter and is the high-pass filter.
• If there are matrices and which satisfy:
where is an identity matrix. Gi and Hi are called a discrete wavelet pair.
• If and
The wavelet pair is said to be orthonormal.
,...)2,1(, iHG ii...2,1,0, iN i1 ii NN
ii NN 1iG
iH
iiiii IGGHH
iI11 ii NN
Tii
Tii GGHH , i
Tii
Tii IHHGG 0 T
iiTii GHHG
ii GH ,
iH iG
• For signal let and• One may have
• The above is called the generalized Discrete Wavelet Transform (DWT) up to the scale
is called the smooth part of the DWT andis called the DWT at scale
• In terms of equation
),,...,( 21 Nffff NN 0 .,...2,1 Ji
.J
fHHH JJ 11....
fHHG JJ 11... .J
),,,......,....( 1121231111 fGfHGfHHGfHHGfHHH JJJJ
).2()()(1
0
12
0, ktrttf j
p
j kkj
j
Multilevel Decomposition
• A block diagram
2
2
2
f(n) 2
)(nh
)(nh
)(ng
)(ng
2
2
)(nh
)(ng
Haar Wavelets
1 0 0 1
Scaling Function Wavelet
]2
1,
2
1[)( nh ]
2
1,
2
1[)( ng
Example: Haar Wavelet
[18]
Summary on Haar Transform• Two major sub-operations
– Scaling captures info. at different frequencies– Translation captures info. at different locations
• Can be represented by filtering and downsampling• Relatively poor energy compaction
1
x
2D Wavelet Transform
• We perform the 2-D wavelet transform by applying 1-D wavelet transform first on rows and then on columns.
Rows Columns LL
f(m, n) LH
HL
HH
H 2
G
2 G
2 H
2
2
G
H
2
Applications
• Signal processing– Target identification.
– Seismic and geophysical signal processing.
– Medical and biomedical signal and image processing.
• Image compression (very good result for high compression ratio).
• Video compression (very good result for high compression ratio).
• Audio compression (a challenge for high-quality audio).
• Signal de-noising.
Original Video Sequence Reconstructed Video Sequence
3-D Wavelet Transform for Video Compression