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Wavelet Transform
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Wavelet Transform
The wavelet transform corresponds to the
decomposition of a quadratic integrable function s(x) L2(R)
in a family of scaled and translated functions k,l(t),
The function (x) is called wavelet function and shows
band-pass behavior. The wavelet coefficients da,b are derived
as follows:
where k R+, l Rand * denotes the complex conjugate function
)()( 2/1,k
ltktlk
=
)dxk
l-x(s(x)
k
1=d
*
-
lk,
The discrete wavelet transform (DWT) represents a 1-D
signal s(t) in terms of shifted versions of a lowpass scaling function(t) and shifted and dilated versions of a prototype bandpass wavelet
function (t).
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For special choices of (t) and (t), the functions:
for j and k Z, form an orthonormal basis, and we have therepresentation:
)2(2)(),2(2)(
,
2/
,
kttktt
jj
kj
jj
kj
=
=
=+=
k jj k
kjkjkjkj ttutz0
)()()( ,,,0,0
where,
= dttts kjkj )()( *
,, = dtttsu kjkj )()( *
,, and
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Relook at F.T. expressions:
dueuFxf xuj
=
2
)()(dxexfuF xuj
= 2)()(
Thus f(x) is represented here as a linear combination of the basis
functions: exp(jx)
Wavelet transform on the other hand, represents f(x) (or f(t)) as a
linear combination of:
)2(2)( 2/ ltt kkkl =
where (t) is called the mother wavelet.
Parameters k and l are integers which generates the basis
functions as the dilated and shifted variations of the mother wavelet.
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The parameter k plays the role of frequency and l plays the
role of time. Hence by varying k and l, we have different frequency
and different time or space hence the term m u l t i - ch a n n e l m u l t i - r e s o l u t i o n a p p r o a ch .
)1(,....,1,0;)(1)(1
0
2=
=
= NxN
u
NuxjeuFN
xf
Compare in discrete case:
The F.T.:
2/ kk )]2(2[),()( ltlkXtfk l
DWT = The DWT:
= dtk
lttxk
lkXCWT )()(1),( where,
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= dtlTtahtxalkX kkDWT )()(),( 2/
)]([),()( 2/ lTtafalkXtx kk
k l
DWT =
DWT Discrete Wavelet Transform:
Forward:
and Inverse:
Take, T = 1 and time is continuous.
Synthesis filters:Analysis filters:
)()( 2/ tafatf kkk =)()( 2/ tahath kk
k=
Functions h(t) and f(t) are derived by dilation of a single filter. Thus
the basis functions are dilated (t -> a-kt) and shifted (t -> t - la-kt)versions of:
)()()( 2/ lTtaattf kkkl ==
)()( * thtf kk =Synthesis filters for perfect reconstruction:
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Visualize pseudo-frequency correspondingto a scale. Assume a center frequency Fc of thewavelet and use the following relationship:
where a is the scale. is the sampling period and
Fc is the center frequency of a wavelet in Hz. Fa
is the pseudo-frequency corresponding to thescale a, in Hz.
=
.a
FF
c
a
The highpass and lowpass filters are notindependent of each other, and they are related bythe following expression:
)(.)1(]1[ nhNLg n=
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QMF bank and typical magnitude responses
H0(z) G0(z)
G1(z)H1(z)
x(t)
2
2
2
2
)(^ tx
Analysis Bank Synthesis Bank
Decimators Expanders
1H0(z) H1(z)
/2
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M-channel (M-band) QMF bank
H0(z) G0(z)
G1(z)H1(z)
x(t)
M
)(^
tx
Analysis Bank Synthesis Bank
Decimators Expanders
GM-1(z)HM-1(z) M
M
M
M
M
Th DWT l th i l t diff t f b d
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The DWT analyzes the signal at different frequency bandswith different resolutions by decomposing the signal intoc o a r se a p p r o x im a t i o n a n d d e t a i l i n f o r m a t i o n .
DWT employs two sets of functions, called scaling functionsand wavelet functions, which are associated with low passand highpass filters, respectively.The decomposition of the signal into different frequency
bands is simply obtained by successive highpass and lowpassfiltering of the time domain signal.The original signal x[n] is first passed through a half-bandhighpass filter g[n] and a lowpass filter h[n].
After the filtering, half of the samples can be eliminated(according to the Nyquists rule) since the signal now has ahighest frequency of fmax/2 radians instead of fmax.The signal can therefore be sub-sampled by 2, simply by
discarding every other sample. This constitutes one level ofdecomposition and can mathematically be expressed asfollows:
=
=
]2[].[][
]2[].[][
nkhnxky
nkgnxky
lo
hi
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Block diagram of the methodology of 1-D DWT.
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Frequency responses (bandwidths) of the different
output channels of the wavelet filter bank, for a = 2
and three or more levels of decomposition
24 = 2
F
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FrequencyResponseof 2-channel
Daubeschies8-taporthogonalwavelet filters.
Low-Pass
High-Pass
Frequency Response
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Frequency Responseof a 3-channelorthogonal wavelet
filters.
Channel - I
Channel - II
Channel - III
Frequency Response
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Frequency Responseof a 4-channelorthogonal wavelet
filters.
Channel - I
Channel - IV
Channel - II
Channel - III
Two-level maximally decimated filter bank
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Two-level maximally decimated filter bank
G0(z)
G1(z)
)(
^
tx
H0(z)
H1(z)
x(t)
H0(z)
H1(z)
H0(z)
H1(z)
G0(z)
G1(z)
2
G0(z)
G1(z)
2
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Illustrations to demonstrate thedifference between:
FT, STFT and WT
1002cos()502cos()252cos()102cos()( ttttt +++=
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1002cos()502cos()252cos()102cos()( ttttt +++=
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Note that the FT gives what frequency components (spectral
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components) exist in the signal. Nothing more, nothing less.
When the time localization of the spectral components are
needed, a transform giving the TIME-FREQUENCY REPRESENTATION of thesignal is needed.
What is Wavelet Transform and how does it solve the problem?
View WT as a plot on a 3-D graph, where time is one axis, frequency the
second and amplitude is the third axis.
This will show us what frequencies, f, exist at which time, T.
There is an issue, called "u n ce r t a i n t y p r i n c ip l e ", which states that, we
cannot exactly know what frequency exists at what time instance , but we can
only know what frequency bands exist at what time intervals.
The uncertainty principle, originally found and formulated by
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Heisenberg, states that, the momentum and the position of a moving particle
cannot be known simultaneously. This applies to our subject as follows:
The frequency and time information of a signal at some certain point inthe time-frequency plane cannot be known.
In other words: We cannot know what spectral component exists at any
given time instant. The best we can do is to investigate what spectral componentsexist at any given interval of time.
This is a problem of resolution, and it is the main reason whyresearchers have switched from STFT to WT.
STFT gives a fixed resolution at all times, whereas WT gives a variable (or
suitable) resolution as follows:
Higher frequencies are better resolved in time, and lower frequencies are betterresolved in frequency.
This means that, a certain high frequency component can be located
better in time (with less relative error) than a low frequency component. On thecontrary, a low frequency component can be located better in frequency
compared to high frequency component
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dtftjtttxftSTFT )2exp()]'()([)(
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= dtftjtttxftSTFTx )2exp()]().([),(
)
2
*exp()(
2ta
tw
=
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Broader Window, w
Narrow Window, w
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Still larger window, w
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Time and requency Resolutions
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Frequency
Amplitude (Fourier)Time
Amplitude
Frequency
Time (STFT/Gabor)
Scale
Time (Wavelet)
y
(STFT/Gabor)
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Frequency
T 2T
1
21
Time
Time
(Wavelet)
T 2T 4T0
0/2
0/4
Frequency
Scale
Two-dimensional Wavelet Transform
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2LPF
HPF 2
LPF
LPF
HPF
HPF
2
2
2
2
Image
LL
LH
HL
HH
Level I wavelet decomposition of an image
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Level II wavelet decomposition of an image
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References:
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Multirate Systems and Filter banks, P. P. Vaidyanathan; PrenticeHallInc., 1993.
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Wavelet based analysis
of texture Images
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Problem of
Shape from
3-D Textures
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2-D Textures
3-D Textures
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Real world 3-D Texture image
REFERENCES
1. M. Clerc and S. Mallat, The Texture Gradient Equation for Recovering Shape from Texture, IEEE
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, q g p ,
Transactions on Pattern Analysis and Machine Intelligence, Vol. 24, No. 4, pp. 536-549, April 2002.
2. J. Garding, Surface Orientation and Curvature from Differential Texture Distortion, Proceedings of the IEEEConference on Computer Vision (ICCV 95), 1995, pp. 733-739.
3. J. S. Kwon, H. K. Hong and J. S Choi, Obtaining a 3-D orientation of Projective textures using a Morphological
Method, Pattern Recognition, Vol. 29, No. 5, pp. 725-732, 1996.
4. T. Leung and J. Malik, On Perpendicular textures, or: Why do we see more flowers in the distance?,Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR 97), 1997, San Juan,
Puerto Rico, pp. 807-813.
5. J. Malik and R. Rosenholtz, Computing Local Surface Orientation and Shape from texture for Curved
Surfaces, International Journal of Computer Vision, Vol. 23(2), pp. 149-168, 1997.
6. E. Ribeiro and E. R. Hancock, Shape from periodic Texture using the eigenvectors of local affine distortion,
IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 23, No. 12, pp. 1459 1465, Dec. 2001.
7. B. J. Super and A. C. Bovik, Planar surface orientation from texture spatial frequencies, Pattern Recognition,
Vol. 28, No. 5, pp. 729-743, 1995.
8. Sukhendu Das and Thomas Greiner; Wavelet based separable analysis of texture images for extracting
orientation of planar surfaces; Proceedings of the second IASTED International Conference on Visualization,
Imaging and Image Processing (IASTED-VIIP); September 9-12, 2002, Malaga, Spain, pp. 607612.
9. Thomas Greiner and Sukhendu Das; Recovering Orientation of a textured planar surface using wavelettransform; Indian Conference on Computer Vision, Graphics and Image Processing, 2002 (ICVGIP '02),
December 16 - 18, 2002, Space Applications Centre (SAC-ISRO), Ahmedabad, INDIA, pp. 254-259.
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