Transcript An Overview of Wavelets Part I

De-Noising Signals: A
Wavelet Approach
S. Ezekiel, W. Oblitey, R Trimble
Computer Science Department
Indiana University of Pennsylvania
Indiana, PA 15705
Introduction
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Past 3 decades– DSP fast growing fields
Signal is a function of independent variable ( time,
distance, position..)
Analog signal- operation done on time domain
Digital signal- both domain( time and frequency)
Multi-band spectral subtraction– Wu, Hidden Markov
model– Sameti et al. --- Noise Masking algorithm –
Virag---- Weighted Linear
These Models--Variable degree of successdepending on the goals and assumptions
Our goal:-develop a model to reduce various
noises and improve the quality of a signal
What is Wavelet? ( Wavelet Analysis)
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Wavelets are functions that satisfy certain mathematical
requirements and are used to represent data or other functions
Idea is not new--- Joseph Fourier--- 1800's
Wavelet-- the scale we use to see data plays an important role
FT non local -- very poor job on sharp spikes
Wavelet db10
Sine wave
History of wavelets
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1807 Joseph Fourier- theory of frequency analysis-- any 2pi
functions f(x) is the sum of its Fourier Series
1909 Alfred Haar-- PhD thesis-- defined Haar basis function---- it is
compact support( vanish outside finite interval)
1930 Paul Levy-Physicist investigated Brownian motion ( random
signal) and concluded Haar basis is better than FT
1930's Littlewood Paley, Stein ==> calculated the energy of the
function 1960 Guido Weiss, Ronald Coifman-- studied simplest
element of functions space called atom
1980 Alex Grossmann (physicist) Jean Morlet( Engineer)-- broadly
defined wavelet in terms of quantum mechanics
1985 Stephane Mallat--defined wavelet for his DSP ( Ph.D. thesis)
Y. Meyer constructed first non trivial wavelet.
1988 Ingrid Daubechies-- used Mallat work constructed set of
wavelets
The name emerged from the literature of geophysics, by a route
through France. The word onde led to ondelette. Translation wave
led to wavelet-- by Jean Morlet
Fourier Series and Energy
f ( x )  a0 

 (a
k 1
k
cos kx  bk sin kx )
where the coefficients are calculated by
a0
2
1

2
ak 
1

f ( x ) dx
0
2
 
f ( x ) cos( kx ) dx
0
bk 
1
2
 
f ( x ) sin( kx ) dx
0
Energy of a function
1
energy 
2
2

0
2
f ( x) dx
f ( x)
Fourier, Haar
time  amplitude , frequency
 1965 Cooley and Tukey – Fast Fourier
Transform
1 
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 Haar
1
0

x<

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 Amplitude,
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
 ( x)  1

0
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
2

1

<x  1 
2
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otherwise 


 ( x),
 (2 x), (2 x  1),
 (4 x), (4 x  1), (4 x  2), (4 x  3),
Wavelets

F ( )   f (t )e jwt dt
Fourier Transform

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CWT =
C( scale, position)=

f (t ) ( scale, position, t) dt

3
Scaling wave means
(or Shrinking) it
2.5
2
1
1.5
0.5
1
2
3
4
5
2.5
6
-0.5
-1
5
7.5
10
12.5
15
simply Stretching
17.5
Shifting
f (t) f(t-k)
Wavelet Decomposition and Reconstruction
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Calculating coefficients for C at every scale– lots of
work
Choose a subset– powers of 2’s dyadic scales
==> DWT
Efficient way of calculating coefficients– by using
filters– defined by Mallat-- => FWT( 2 channel subband coder)=>decompose signal into 2 parts:
approximate (low= actual content) and detail
(high=flavor)
The above is called decomposition of Level 1
This process can be iterated => signal broken down
into many low-resolutions.
The procedure to reconstruct the original signal from
its decomposed parts is called Reconstruction
Transforms
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Transform of a signal is a new representation of that
signal
 Example:- signal x0,x1,x2,x3 define y0,y1,y2,y3
 Questions
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1. What is the purpose of y's
2. Can we get back x's
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Answer for 2: The Transform is invertible-- perfect
reconstruction
 Divide Transform in to 3 groups
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1. Lossless( Orthogonal)-- Transformed Signal has the same
length
2. Invertible (bi-orthogonal)-- length and angle may change-no information lost
3. Lossy ( Not invertible)--
Answer to Q1: Purpose
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IT SEES LARGE vs SMALL
X0=1.2, X1= 1.0, x2=-1.0, x3=-1.2
Y=[2.2 0 -2.2 0]
Key idea for wavelets is the concept of " SCALE"
We can take sum and difference again==> recursion
=> Multiresolution
Main idea of Wavelet analysis– analyze a function at
different scales– mother wavelet use to construct
wavelet in different scale and translate each relative to
the function being analyzed
Z=[ 0 0 4.4 0 ]
Reconstruct =====>compression 4:1
Scaling function and Wavelets
corresponding to low pass--> there is scaling function  (t )
 corresponding to high pass--> there is wavelet function  (t )
 dilation equation--> scaling function
N
 (t )  2  c(k ) (2t  k )
 In terms of original low pass filters
k 0
N
 we have
 (t )  2 h(k ) (2t  k )
k  01/2
 for h(0) and h(1) =
we have
 (t )   (2t )and
  (2
t 1)
 the graph
compressed by 2 gives
shifted by
 (2t )
1/2 gives  (t )
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 (2t  1)
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By similar way the wavelet equation
 (t )   (2t )   (2t 1)
Multiresolution Analysis
Signals and Signal Applications
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are various types of signals:
Heart Rate Variability (HRV)
Seismic Signals
Speech Signals
 Noise
is mixed during the data collection
phase
 x(t) = a(t) + d(t)
ECG Signal with Noise
T
Thresholding
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Transformation of an input signal I to an output
signal BI where T is the Threshold value
Hard Threshold
 I (i ) if I (i)  T
BI (i )  
0 otherwise
Soft Threshold
 sign( I )(| I (i )  T |)
BI (i )  
0 otherwise
if
I (i)  T
Signal Enhancement Model
 Several
traditional methods, such as
median filtering, averaging with various
masks, and histogram have been studied
and implemented
 Each of these methods are designed to
filter a specific type of noise, for example
white, Gaussian, impulsive, additive,
quantization, or multiplicative noise.
 Step
1: Data Collection. A total of 57
signals were collected for study. We
collected 11 speech, 14 HRV, 30 Seismic,
and 2 Historic Stock Market signals.
 Step 2: Resample. We added noises to
test our model against.
 River
 car engine
 air conditioner
 highway traffic
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Step 3: Wavelet Decomposition. Choose a
wavelet. Decide the number of levels N , then,
compute wavelet decomposition x(n) for N levels.
This process separates the signal into two parts,
low frequency (approximate coefficients ) and
high frequency (detailed coefficients).
 Step 4: Remove Noises by Thresholding
 Choose the threshold value and type.
 Apply the thresholding technique to only the
high frequency part of level
 This clears out most background noise
 Step 5: Reconstruction. Perform wavelet
reconstruction from the approximate coefficients
in step 4.
Error Measures
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We used the two commonly used measure to test
the quality of our signal
 Root Mean Square (RMS) error
RMS 
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1 N
2
 xi  yi 

N i 1
The Peak Signal-to-Noise Ratio
b
PSNR  20* Log10
RMS
where b is the maximum value of the signal.
Speech Signal (male subject)
Heart Rate Variability
(male, age 48, 8% heart disease, 24 hour sample)
Seismic Signal (24 hour sample)
Dow Jones
(5 day sample from 1888-1990)
Speech Signal (female)
 We
calculated the PSNR for the de-noised
signals and we obtained the values
56.1370, 38.6432, 62.5, 32.568, and
27.1034 respectively. This clearly shows
that our model efficiently removed the
various noises.
Further our model
retained 98.8015%, 99.9623%, 99%,
99.47%, and 88.4015% of the original
content of the signal even after removing
26.8812%, 79.5370%, 82%, 82.1181%,
and 80.95% of the detail coefficients,
respectively.
Conclusion
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A framework using wavelet and multiresolution
analysis based noise removal was presented.
 This approach offers high quality and flexibility
since:
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Parameters can be easily modified to remove various
noises
Threshold values can be automatically calculated and
adjusted based on the waveform.
Although these techniques have been shown to
de-noise signals, further experimental analysis is
needs fine-tune the various parameters.
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JPEG (Joint Photographic Experts Group)
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1. Color images ( RGB) change into luminance, chrominance, color
space
2. color images are down sampled by creating low resolution pixels –
not luminance part– horizontally and vertically, ( 2:1 or 2:1, 1:1)– 1/3
+(2/3)*(1/4)= ½ size of original size
3. group 8x8 pixels called data sets– if not multiple of 8– bottom row
and right col are duplicated
4. apply DCT for each data set– 64 coefficients
5. each of 64 frequency components in a data unit is divided by a
separate number called quantization coefficients (QC) and then
rounded into integer
6. QC encode using RLE, Huffman encoding, Arithmetic Encoding (
QM coder)
7. Add Headers, parameters, and output the result
• interchangeable format= compressed data + all tables need for
decoder
• abbreviated format= compressed data+ not tables ( few tables)
• abbreviated format =just tables + no compressed data
DECODER DO THE REVERSE OF THE ABOVE STEPS
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JPEG 2000
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divide into 3 colors
each color is partitioned into rectangular, non-overlapping
regions called tiles– that are compressed individually
A tile is compressed into 4 main steps
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or JPEG Y2k
1. compute wavelet transform – sub band of wavelets– integer, fp,---L+1
levels, L is the parameter determined by the encoder
2. wavelet coeff are quantized, -- depends on bit rate
3. use arithmetic encoder for wavelet coefficients
4. construct bit stream– do certain region, no order
Bit streams are organized into layers, each layer contains
higher resolution image information
thus decoding layer by layer is a natural way to achieve
progressive image transformation and decompression
Wavelet Packet
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Walsh-Hadamard transform-- complete binary tree -->
wavelet packet
 "Hadamard matrix"==> all entries are 1 and -1 and all
rows are orthogonal-- divide two time by sqrt(2)==>
orthogonal & symmetric
sums z0=0
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Compare with wavelet-- computations
sums y0 and y2
difference z2=4.4
x
sums z1=0.4
difference y1 and y3
difference z3=0
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Filters and Filter Banks
Filter is a linear time-invariant operator
 It acts on input vector x --- Out put vector y is the convolution of
x with a fixed vector h
 h--> contains filter coefficients-- our filters are digital not analog- h(n) are discrete time t= nT,
 T is sampling period assume it is 1 here
 x(n) and y(n) comes all the time t= 0, +_ 1....
 y(n) = Σh(k) x(n-k) = convolution h* x in the time domain
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Filter Bank= Set of all filters
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Convolution by hand--- arrange it as ordinary multiplication -- but don't carry
digits from one column to another
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x= 3 2 4 h= 1 5 2
x * h = 3 17 20 24 8
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