The Story of Wavelets Theory and Engineering Applications

The Story of WaveletsTheory and Engineering Applications

• Stationary discrete wavelet transform

• Two-dimensional wavelet transform

• 2D-DWT using MATLAB

• Implementation issues

• Image compressing using 2D-DWT

Stationary Wavelet Transfporm(SWT)

DWT is not time invariant… Not Good ! What makes DWT time varying? Decimation (down

sampling) DWT can be made time invariant, however, the transform must be

redundant!!! Stationary wavelet transform -decimated DWT

-decimated DWT???At any given level you can have two different

DWT, due to choice in discarding even or odd indexed elements during subsampling

At J levels, you can have N=2J different DWTs. The particular DWT chosen can be denoted by =[1 2 … N], j=1, if odd indexed elements are chosen, j=0, if even indexed elements are chosen

SWT

SWT is defined as the average of all -decimated DWTs

For 6 level DWT64 DWTsFor 10 level DWT 1024 DWTs….

An efficient algorithm:Hj

Gj

cAj=a(j,n)

cAj+1=a(j+1,n)

cDj+1=d(j+1,n)where

2 HjHj-1

2 GjGj-1

Note: No subsampling is involved!!!.

SWT

Does it work…? MATLAB DEMO

Applications of SWT

Denoising… denoising…denoising MATLAB demo: noisy doppler & noisy quadchirp

Interval dependent thresholds

1D-DWT2D-DWT

Recall fundamental concepts of 1D-DWTProvides time-scale (frequency) representation of non-

stationary signalsBased on multiresolution approximation (MRA)

Approximate a function at various resolutions using a scaling function, (t)

Keep track of details lost using wavelet functions, (t) Reconstruct the original signal by adding approximation and

detail coeff. Implemented by using a series of lowpass and highpass filters

Lowpass filters are associated with the scaling function and provide approximation

Highpass filters are associated with the wavelet function and provide detail lost in approximating the signal

2-D DWT

How do we generalize these concepts to 2D? 2D functions images f(x,y) I[m,n] intensity function What does it mean to take 2D-DWT of an image? How do we

interpret?How can we represent an image as a function?How do we define low frequency / high frequency

in an image?How to we compute it?

Why would we want to take 2D-DWT of an image anyway?CompressionDenoisingFeature extraction

2D Scaling/Wavelet Functions

We start by defining a two-dimensional scaling and wavelet functions:

If (t) is orthogonal to its own translates,

is also orthogonal to its own translates. Then, if fo(x,y) is the projection of f(x,y) on the space Vo generated by s(x,y):

)()(),( yxyxs

Zlklykxs ,:),(

),(),,(),(

),(),(),(

jyixsyxfjia

jyixsjiayxf

o

i joo

)()(),( yxyxs

2D-DWT

Just like in 1D we generated an approximation of the 2D function f(x,y). Now, how do we compute the detail lost in approximating this function?

Unlike 1D case there will be three functions representing the details lost:Details lost along the horizontal directionDetails lost along the vertical directionDetails lost along the diagonal direction

1D Two sets of coeff.; a(k,n) & d (k,n) 2D Four sets of coefficients: a(k,n), b(k, n), c(k, n) & d(k,n)

Four Faces of 2D-DWT

One level of 2D DWT reconstruction:

)()(),(

)()(),(

)()(),(

)()(),(),(1

nxnxpnd

nxnxpnc

nxnxpnb

nxnxpnayxf

o

o

o

n po

Approximation coefficients

Detail coefficients along the horizontal direction

Detail coefficients along the vertical directionDetail coefficients along the diagonal direction

Implementation of 2D-DWT

INPUTIMAGE…

…

……

RO

WS

COLUMNSH~ 2 1

G~ 2 1

H~ 1 2

G~ 1 2

H~ 1 2

G~ 1 2

ROWS

ROWS

COLUMNS

COLUMNS

COLUMNS

LL

LH

HL

HH

)(1

hkD

)(1

vkD

)(1

dkD

1kA

INPUTIMAGE

LL LH

HL HH

LLLH

HL HH

LHH

LLH

LHL

LLLLH

HL HH

LHH

LLH

LHL

Up and Down … Up and Down

2 1Downsample columns along the rows: For each row, keep the even indexed columns, discard the odd indexed columns

1 2Downsample columns along the rows: For each column, keep the even indexed rows, discard the odd indexed rows

2 1

1 2

Upsample columns along the rows: For each row, insert zeros at between every other sample (column)

Upsample rows along the columns: For each column, insert zeros at between every other sample (row)

Implementing 2D-DWT

Decomposition

ROW i

CO

LU

MN

j

Reconstruction

)(1

hkD

)(1

vkD

)(1

dkD

1kA 1 2

1 2

1 2

1 2

H

G

H

G

2 1

2 1

H

G

ORIGINALIMAGE

LL

LH

HL

HH

2-D DWT ON MATLAB

Load Image

(must be.mat file)

Choosewavelet type

HitAnalyze

Choosedisplayoptions