lect wavelet filt

7/26/2019 Lect Wavelet Filt

1/44

Wavelet Transform


2/44

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).


3/44

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


4/44

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.


5/44

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,


6/44

= 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:


7/44

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=


8/44

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


9/44

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


10/44

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


11/44

Block diagram of the methodology of 1-D DWT.


12/44

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


13/44

FrequencyResponseof 2-channel

Daubeschies8-taporthogonalwavelet filters.

Low-Pass

High-Pass

Frequency Response


14/44

Frequency Responseof a 3-channelorthogonal wavelet

filters.

Channel - I

Channel - II

Channel - III

Frequency Response


15/44

Frequency Responseof a 4-channelorthogonal wavelet

filters.

Channel - I

Channel - IV

Channel - II

Channel - III

Two-level maximally decimated filter bank


16/44

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


17/44

Illustrations to demonstrate thedifference between:

FT, STFT and WT

1002cos()502cos()252cos()102cos()( ttttt +++=


18/44

1002cos()502cos()252cos()102cos()( ttttt +++=


19/44


20/44

Note that the FT gives what frequency components (spectral


21/44

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


22/44

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


23/44

dtftjtttxftSTFT )2exp()]'()([)(


24/44

= dtftjtttxftSTFTx )2exp()]().([),(

)

2

*exp()(

2ta

tw

=


25/44


26/44

Broader Window, w

Narrow Window, w


27/44

Still larger window, w


28/44


29/44


30/44


31/44


32/44

Time and requency Resolutions


33/44

Frequency

Amplitude (Fourier)Time

Amplitude

Frequency

Time (STFT/Gabor)

Scale

Time (Wavelet)

y

(STFT/Gabor)


34/44

Frequency

T 2T

1

21

Time

Time

(Wavelet)

T 2T 4T0

0/2

0/4

Frequency

Scale

Two-dimensional Wavelet Transform


35/44

2LPF

HPF 2

LPF

LPF

HPF

HPF

2

2

2

2

Image

LL

LH

HL

HH

Level I wavelet decomposition of an image


36/44

Level II wavelet decomposition of an image


37/44

References:


38/44

Multirate Systems and Filter banks, P. P. Vaidyanathan; PrenticeHallInc., 1993.


39/44

Wavelet based analysis

of texture Images


40/44

Problem of

Shape from

3-D Textures


41/44

2-D Textures

3-D Textures


42/44

Real world 3-D Texture image

REFERENCES

1. M. Clerc and S. Mallat, The Texture Gradient Equation for Recovering Shape from Texture, IEEE


43/44

, 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.


44/44

lect wavelet filt

Documents