Time frequency analysis_journey

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7 January 2016
© 2014We value our relationship
7 January 2016
Chandrashekhar Padole
Title for Presentation
Journey from
Vector Algebra to Wavelet Transform
(Time-Frequency Analysis)

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Presenter: Chandra7 January 2016 2
Objective
Topics to be covered (Various Transforms):
• Vector Algebra
•Discrete Fourier Transform
•KL transform
•PCA
•Wavelet Transform
•Wigner Distribution

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Applications
•Pattern Recognition Problems
• Biometrics: Face, Fingerprint, IRIS etc
• Automotive: Lane , Vehicle, Pedestrian, Sleeping Pattern, Signal
etc
• Manufacturing: object, documents etc
• GIS, Healthcare, Military and so on
•Analysis-Synthesis (Graphic Equalizer, noise removal,
image restoration etc)
•Data-Mining (content based image retrieval, audio mining
based on emotions etc)
•Transmission and Compression of data

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Agenda
• Vector Algebra
•QA
•Fourier Transform
•QA
•K-L Transform
•PCA
•QA
•QA
•QA

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Take-off for Vector Algebra

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Preliminaries in Vector Algebra
•Vector P=3 i + 4 j
•P quantity to be analyzed or represented as P
•Projection or mapping of P onto unit vectors i and j will be
3 and 4 resp.
•Measurement of projection is obtained by using
mathematical tool or operator.
•In this case, operator is dot
product (inner product)

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Contd..
• e.g. P.i=[ 3i+4j] . [ i]=[3i+4j].[1i+0j]=3
•And P.j=[ 3i+4j] . [ j]=[3i+4j].[0i+1j]=4
•Significance of i and j-
•In 2D space problem
• i is the vector which represents horizontal property
(feature/dimension/axis),
• j is for vertical property
• k- depth property in 3D

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Let’s Look at some Real-life Problems
Two Cupboards

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Contd..
Chemist

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Contd..
Carpenter

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Rewind
• P=3i + 4j
•Px = P. i= 3
•Py= P . j= 4
•Operator we used for measurement of projection is dot
product or inner product
•[ 3 4 2][2 1 2]T = 3.2 + 4.1+ 2.2=6 +4+4= 14
•It is also a correlation (zero-shift) between two vectors
• P properties component value/coefficient
•It’s a analysis process

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Contd..
•To synthesize combine component values along with
property vectors to get original P
i.e. combine ( 3 and 4) as 3i+4j= P
•What we had till now , is 2D problem. Same terminology and
process can be extended for 3 D space problem….

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3D Space Problem
• P= ai+bj+ck
• analysis : find { a,b,c}
• synthesis : combine a,b,c
•In 2D space problem – 2 analysis vectors
•In 3D space problem – 3 analysis vectors

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Investigation in A-S Problem
• Can there be more than 3 analysis vectors ..?
if so , give examples..
•Can each analysis vector be multi valued ( as opposed to
double/triple valued vector in 2D /3D analysis) ?
•If both answers are positive, can vector algebra analysis
theory be extended to those problems?

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Stop for Queries

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Replacing VA by DFT
•Use of DFT – frequency analysis
•Why it is needed?
• to measure/analyze the variation of physical quantity
(such as pressure, intensity, voltage , current etc) with
respect to one or more independent variables
•Representation of physical quantity with respect to
independent variables is called as function or signal
•Variation need not to be periodic and mostly it is aperiodic
in real life problems
• For aperiodic DFT will analyze variations in signal

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Take-off for Vector Algebra

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Fourier Idea

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Frequency Analysis
• Definition of DFT
• Computations steps in DFT
• Its physical significance
•Definition of DFT
for N-point DFT
∑
−
=
−
=
1
0
2
)()(
N
n
N
nkj
enxkX
π

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Computations in DFT
• Signal to be analyzed is 64 points in length
x(n)
analog discrete
0 10 20 30 40 50 60 70
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0 10 20 30 40 50 60 70
-0 .5
-0 .4
-0 .3
-0 .2
-0 .1
0
0 .1
0 .2
0 .3
0 .4
0 .5

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K=0
)/2sin()/2cos(
2
NnkjNnke N
nk
j
ππ
π
+=
−
0 10 20 30 40 50 60 70
-1
-0.5
0
0.5
1
SINE SEQUENCE
Amplitude
n
0 10 20 30 40 50 60 70
0
0.5
1
COSINE SEQUENCE
Amplitude
n

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K=1
)/2sin()/2cos(
2
NnkjNnke N
nk
j
ππ
π
+=
−
0 10 20 30 40 50 60 70
-1
-0.5
0
0.5
1
SINE SEQUENCEAmplitude
n
0 10 20 30 40 50 60 70
-1
-0.5
0
0.5
1
COSINE SEQUENCE
Amplitude
n

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K=2
)/2sin()/2cos(
2
NnkjNnke N
nk
j
ππ
π
+=
−
0 10 20 30 40 50 60 70
-1
-0.5
0
0.5
1
SINE SEQUENCE
Amplitude
n
0 10 20 30 40 50 60 70
-1
-0.5
0
0.5
1
COSINE SEQUENCE
Amplitude
n

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K=3
)/2sin()/2cos(
2
NnkjNnke N
nk
j
ππ
π
+=
−
0 10 20 30 40 50 60 70
-1
-0.5
0
0.5
1
SINE SEQUENCEAmplitude
n
0 10 20 30 40 50 60 70
-1
-0.5
0
0.5
1
COSINE SEQUENCE
Amplitude
n

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K=31
)/2sin()/2cos(
2
NnkjNnke N
nk
j
ππ
π
+=
−
0 10 20 30 40 50 60 70
-1
-0.5
0
0.5
1
SINE SEQUENCE
Amplitude
n
0 10 20 30 40 50 60 70
-1
-0.5
0
0.5
1
COSINE SEQUENCE
Amplitude
n
0 10 20 30 40 50 60 70
-1
-0.5
0
0.5
1
SINE SEQUENCE
Amplitude
n
0 10 20 30 40 50 60 70
-1
-0.5
0
0.5
1
COSINE SEQUENCE
Amplitude
n

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DFT- Overall Physical Significance
0 10 20 30 40 50 60 70
-0. 5
-0. 4
-0. 3
-0. 2
-0. 1
0
0. 1
0. 2
0. 3
0. 4
0. 5
0 10 20 30 40 50 60 70
-1
-0.5
0
0.5
1
SINE SEQUENCE
Amplitude
n
0 10 20 30 40 50 60 70
0
0.5
1
COSINE SEQUENCE
Amplitude
n
0 10 20 30 40 50 60 70
-1
-0.5
0
0.5
1
SINE SEQUENCE
Amplitude
n
0 10 20 30 40 50 60 70
-1
-0.5
0
0.5
1
COSINE SEQUENCE
Amplitude
n
0 10 20 30 40 50 60 70
-1
-0.5
0
0.5
1
SINE SEQUENCE
Amplitude
n
0 10 20 30 40 50 60 70
-1
-0.5
0
0.5
1
COSINE SEQUENCE
Amplitude
n
0 10 20 30 40 50 60 70
-1
-0.5
0
0.5
1
SINE SEQUENCE
Amplitude
n
0 10 20 30 40 50 60 70
-1
-0.5
0
0.5
1
COSINE SEQUENCE
Amplitude
n
0 10 20 30 40 50 60 70
-1
-0.5
0
0.5
1
SINE SEQUENCE
Amplitude
n
0 10 20 30 40 50 60 70
-1
-0.5
0
0.5
1
COSINE SEQUENCE
Amplitude
n
a1+jb1
a5+jb5
Dot
Product
a2+jb2
a3+jb3
a4+jb4
X(1)
X(31)
X(2)
X(3)
X(4)
∑
−
=
−
=
1
0
2
)()(
N
n
N
nkj
enxkX
π

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DFT- Phase Information
clear all;
close all;
N=64;
t=0:N-1;
f=1;
ws=sin(2*pi*f*t/N);
figure,
subplot(2,1,1)
stem(ws);
wc=cos(2*pi*f*t/N);
subplot(2,1,2)
stem(wc);
%ps=sin(2*pi*f*t/N);
%ps=sin(2*pi*f*t/N + (pi/8));
ps=sin(2*pi*f*t/N + (3*pi/8));
figure,
stem(ps);
a=sum(ps.*ws);
b=sum(ps.*wc);
ph_ang=atan(b/a);

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Contd..
0 10 20 30 40 50 60 70
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0 10 20 30 40 50 60 70
-1
-0.5
0
0.5
1
SINE SEQUENCE
Amplitude
n
0 10 20 30 40 50 60 70
-1
-0.5
0
0.5
1
COSINE SEQUENCE
Amplitude
n
Dot
Product
a=32
b= 0
ph_ang=0
a=0
b= 32
ph_ang=pi/2
a= 22.67
b= 22.67
ph_ang=pi/4
0 10 20 30 40 50 60 70
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0 10 20 30 40 50 60 70
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
ph=0
ph=3*pi/8
ph=pi/2
0 10 20 30 40 50 60 70
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
ph=pi/8
a= 29.56
b= 12.24
ph_ang=pi/8
0 10 20 30 40 50 60 70
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
ph=pi/4
a= 12.24
b= 29.56
ph_ang=pi/4

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Relation bet VA and DFT
Vector algebra -2/3D space DFT
No of analysis vectors 2/3 N
No of elements in each
vector
2/3 N
Projection measurement
method
Inner Product Inner Product

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Fourier Idea
Jean Baptiste Joseph
Fourier (1768 - 1830).
Fourier was a French
mathematician, who was
taught by Lagrange and
Laplace

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Application –Example
Music Equalizer
Audio file recorded
with studio setting
N-point FFT
N-point
User setting
Modified FFT
IFFT

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Stop for Queries and Discussion

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New terminology:
•quantity to be analyzed: signal /function
•Analysis vectors : basis vectors
• Thus, collection of basis vectors forms basis set
•Properties of basis set
• Completeness
• Orthogonal
• orthonormal

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Completeness
• e.g. we have a function P=3i+4j+2k
• while analyzing we used only two basis vectors , i and k;
• using projection over these two vectors will not give
proper reconstruction
• But having so , we can achieve dimensionality reduction
which useful in data compression
•Dimensionality extension is possible?
XOR Problem

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Orthogonal
•Information carried by one basis function should not be included in any
other basis function from basis set
that’s a orthogonality
•Mathematically, if i.j=0, then I and j are orthogonal .
• In Fourier transform , and are orthogonal if
•In Fourier transform within basis function , two sub basis
function ,sin and cosine, are also orthogonal and used for
determining the phase of that frequency
N
nk
j
e
π2
N
mk
j
e
π2
nm ≠
∑
∞
−∞=
=
n
wmwn 0sinsin ∑
∞
−∞=
=
n
wnwn 0cossin

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2D Fourier Transform
2D FT basis images for 8x8

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Thank you
Queries?

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Objective
Topics to be covered (Various Transforms):
• Vector Algebra( part I)
•Discrete Fourier Transform( part I)
•KL transform (part II)
•PCA ( part II)
•Wavelet Transform( part II)

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Take-off for KL-Transform

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Agenda
• Vector Algebra
•QA
•Fourier Transform
•QA
•K-L Transform
•PCA
•QA
•QA
•QA

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KL Transform
• Basis vectors adopt to the data
•Powerful tool when reducing the dimensionality
•de-correlates the data => less redundancy
•Key idea: Represent the data in a more compact manner
•Used for PCA (to be discussed later)
•Thus , it provides the good and compact representation in
transformed domain
•Basis functions are derived from data itself

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Contd..
Procedure to obtain KL transform basis functions
•Collect data
• subtract mean from the data (optional)
•Organize the data in matrix
• If sources are M and elements/dimensions in each
source data are N
• Then place a data in NxM matrix
•Calculate covariance of the matrix to get NxN matrix
•Calculate eigen vectors and eigen values
• N eigen vector and each vector will have N elements
• N eigen values corresponding to each eigen vector

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PCA –Example

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Contd..

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Contd..
calculating covariance
Calculating eigen vectors and values

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Contd..

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Contd..
Reconstruction/restoration

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Contd..
Matlab function used in PCA
• cov
• eig

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Stop for Queries and Application

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Presenter: Chandra
1st Person
7 January 2016 50
PCA- Face Recognition
1 2 7
MxNAveraging
2D-> 1D
Matrix
MNx40
2nd Person
1 2 7
40th Person
1 2 3
MNxMN
Cov Eig
MNxMN
MN
X
1
KL Transform
(Eigen Vectors)
Eigen Values
PCA1D-2D

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WAVELET TRANSFORM

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Need of WT
• Gives good resolution in time as well as frequency domain.
•It also gives locations of different frequency spectral
components during that particular instant of time.
•This is the main advantage of wavelet Transform over FT &
STFT.
•WT is used to mainly analyze non stationary signals, i.e.,
whose frequency response varies in time.

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Take-off for Wavelet Transform

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What is wavelet ?
• The term wavelet means a small wave , i.e. a window
function of finite length.
• It a oscillatory in which high frequency components exist
only for a short duration of time and low frequency exist
throughout the signal.
• The main feature of wavelet is that it can be of finite or
infinite duration but most of the energy of wavelet is
confined to a particular interval of time thus making it time
limited window function.

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Properties of Wavelet
For any function to be a wavelet function must satisfy
following properties:
1.The function integrates to zero:
∞
-∞∫Ψ(t) dt = 0.
2.It is square integrable or, equivalently has finite energy:
∞
-∞∫Ψ(t)2 dt < ∞.
3.The admissibility condition:
∞
C ≡ -∞∫ (Ψ(ω)2) / (ω) dω
Such that 0<C<∞.

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Contd..
• Property 1 is suggestive of a function that is oscillatory or
that has a wavy appearance. Thus, in contrast to a sinusoidal
function, it is a “small wave” or a wavelet.
•Property 2 implies that most of the energy inΨ(t) is
confined to a finite duration.
•Property 3 is known as admissibility condition and is
sufficient condition that leads to the Inverse CWT but it is
not a necessary condition to obtain a mapping from the set
of CWTs back to L2R.
These properties are easily satisfied and there are an infinite
number of functions that qualify as mother wavelets.

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Some Mother Wavelets
Mother wavelet: It’s a basic wavelet function at t=0 without
performing any scaling and shifting operation i.e.Ψ(t).
Morlet wavelet Haar
DB4

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Definition of Wavelet Transform
∫
∞
∞
−
= dt
a
bt
a
tfbaW )(
1
)(),( ψ
Where,
x(t) = Input signal which is to be
transformed
Ψ(t) = Mother wavelet
b = Time shift parameter
a = Scaling parameter
1/sqrt(a)=normalizing factor which ensures
that energy of remains constant for all values of
a & b.

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MRA
Signal to be analyzed

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Contd..

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Chirp function

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Time-Frequency Resolution

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DWT Calculation
Mallat Algorithm:

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Stop for Calculation of WT

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2D DWT
Music Equalizer

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Matlab Code
[im1,n]=imread( ‘lena.jpg‘);
im=double(im1);
wname='haar';
[A_1,H_1,V_1,D_1] = DWT2(im,wname);
[A_2,H_2,V_2,D_2] = DWT2(A_1,wname);
[A_3,H_3,V_3,D_3] = DWT2(A_2,wname);
[A_4,H_4,V_4,D_4] = DWT2(A_3,wname);

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Thank you
Queries?

Time frequency analysis_journey

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