# Image Transformations

Τεχνίτη Νοημοσύνη και Ρομποτική

5 Νοε 2013 (πριν από 4 χρόνια και 8 μήνες)

108 εμφανίσεις

Image Transformations

DFT,DCT,Hotelling,Wavelet ...

Image processing in spectral domain

Enhancement

Restoration

(from a known defect)

Spatial

Domain

Spectral

Domain

Point Processing

E.g.

>

>
Histogram equalization

Spatial filtering

E.g.

Filtering

E.g.

>Freq. domain ops.

Inverse filtering

Wiener filtering

Spectral

Domain

Fourier Transform (1D)

)
f
(
j
ft
2
j
e

)
f
(
W
)
f
(
W
)
f
(
Y

j
)
f
(
X
)
f
(
W
dt
e

)
t
(
w
)
t
(
w
)
f
(
W

F
Continuous Fourier Transform (CFT)

Frequency, [Hz]

Amplitude

Spectrum

Phase

Spectrum

df
e

)
f
(
W
)
f
(
W
)
t
(
w
ft
2
j
1
-

F
Inverse Fourier Transform (IFT)

Discrete Fourier Transform (1D)

Discrete Domains

Discrete Time:

k = 0, 1, 2, 3, …………, N
-
1

Discrete Frequency:

n = 0, 1, 2, 3, …………, N
-
1

Discrete Fourier Transform

Inverse DFT

Equal time intervals

Equal frequency intervals

1
N
0
k
nk
N
2
j
;
e

]
k
[
x
]
n
[
X

1
N
0
n
nk
N
2
j
;
e

]
n
[
X
N
1
]
k
[
x
n = 0, 1, 2,….., N
-
1

k = 0, 1, 2,….., N
-
1

2D

D
iscrete

Fourier Transform

1
0
,
/
2
1
1
0
,
/
2
1
]
,
[
]
,
[
]
,
[
]
,
[
N
m
n
N
ml
nk
j
N
N
l
k
N
ml
nk
j
N
e
m
n
x
l
k
X
e
l
k
X
m
n
x

Base
-
functions are
sinusaoidal
waves

2D
-

DFT

Base
-
functions are waves

l

k

N
ml
nk
j
e
/
2

Amplitude

and
Phase

original

amplitude

phase

)
(
x
F
)
(
x
F

Separability property of DFT

1
0
/
2
1
1
0
/
2
1
1
0
,
/
2
1
]
,
[
]
,
[
]
,
[
]
,
[
]
,
[
]
,
[
N
l
N
ml
j
N
N
k
N
nk
j
N
N
l
k
N
ml
nk
j
N
e
l
n
y
m
n
x
e
l
k
X
l
n
y
e
l
k
X
m
n
x

2D FFT=1D FFT for all lines and 1D FFT for all columns

(DFT in row direction)

(DFT in column direction)

Fast Fourier Transform

recursive algorithm

decimation in time = odd even in freq. domain

decimation in freq. domain = odd even in time

N
N
N
log
2
4

Linear Digital Filters

in Freq. Domain

digital filter

input

output

X
H
Y
x
h
y
F

impulse
-
response

transfer
-
function

amplification

phase
-
shift

]
,
[
]
,
[
]
,
[
]
,
[
m
n
H
m
n
H
m
n
H
j
i
h

Visualizing the

Discrete Fourier
t
ransform

Consider the
power
spectrum of the
1D
square
wave

0
1
2
3
4
5
6
7
0
50
100
150
200
250
300
350
Visualizing the

Discrete Fourier
t
ransform

(circular shifting)

The FT is centered about the origin

But the DFT is centered about N/2

We need to correct with a circular shift operation

Or, multiply by (
-
1) prior to taking the transform

0
1
2
3
4
5
6
7
0
50
100
150
200
250
300
350
k

Filtering in Frequency Domain: Basic Steps

Multiply pixel f(x,y) of the input image by (
-
1).

Compute
the DFT:
F(u,v)

G(u,v)=F(u,v)H(u,v)

g
1
(x,y)=F

{G(u,v)}

g(x,y) = g
1
(x,y)*(
-
1)

x+y

x+y

-
1

Ideal Low Pass Filters

D :
The cut
-
of
f
frequency

Image power as a function of distance from
the origin of DFT (5, 15, 30, 80, 230)

o

Ringing e
ffect of Ideal Low Pass Filters

Example 2: Original and images
LPFed with ideal filters

(with cut
-
off freqs getting larger)

Butterworth

Low Pass F
ilter

k
r
m
n
m
n
H
2
2
2
2
/
1
1
]
,
[

low
-
pass filter

cut
-
r

order; sharpness
k

k=1

k=5

r=0.1

Gaussian Low Pass Filter

D
(
u,v
): distance from

the origin of

Fourier transform

2
2
2
)
,
(
exp
)
,
(

v
u
D
v
u
H
High Pass Filters

Ideal high pass filter

Butterworth high pass filter

Gaussian high pass filter

.
1
)
,
(
0
)
,
(
otherwise
D
v
u
D
if
v
u
H
o

n
v
u
D
D
v
u
H
2
0
)
,
(
/
1
1
)
,
(

2
0
2
2
)
,
(
exp
1
)
,
(
D
v
u
D
v
u
H
Alternative to Fourier Transformation ?

O
ther orthogonal transformations
:

Cosine transform

Haar transform

Wavelet transform

Karhunen
-
Loeve

Slant transform

cosine
-
functions

block
-
waves

hierarchical block
-
functions

Discrete
Cosine
T
ransform

1
(2 1)
2
0
( ) ( ) ( ) cos
N
x u
N
x
C u u f x

 

 

1
1
2
( )
N
N
u

for
u = 0,1,....,N
-
1

1D DCT:

1
(2 1)
2
0
( ) ( ) ( ) cos
N
x u
N
u
f x u C u

 

 

for
x = 0,1,....,N
-
1

1D IDCT:

for u=0

for u=1,2,...,N
-
1

1
(2 1) (2 1)
2 2
0
(,) ( ) ( ) (,) cos cos
N
x u x v
N N
x
C u v u v f x y
 
 

 

   

   

for
u,v = 0,1,....,N
-
1

2D DCT:

1
(2 1) (2 1)
2 2
0
(,) ( ) ( ) (,) cos cos
N
x u x v
N N
u
f x y u v C u v
 
 

 

   

   

for
x,y = 0,1,....,N
-
1

2D IDCT:

Hotelling Transform (aka Karhunen Lowe
Transform or Principle Component
Analysis)

1
2
.
.
n
x
x
x
x
 
 
 
 

 
 
 
 
1
1
{ }
M
x k
k
m E x x
M

 

x
,........,

M data points

1

M

1
1
{( )( ) }
M
T T
T
x x x k k k k
k
C E x m x m x x m m
M

    

Mean:

Covariance:

Hotelling Transform:

( )
x
y A x m
 
The rows of matrix A are the eigen vectors of the covarience matrix

arranged in descending order (The first row corresponds to the eigen vector

corresponding to the largest eigen value of C, ...)

Hotelling T. Example: x vectors are
coordinates of points

Hotelling T. Example: x vectors are point
values in several spectral bands (channels)

From channels to principle components

Wavelet Transform

WT
:

Unification

of

Several

Techniques

Filter

Bank

Analysis

Pyramid

Coding

Subband

Coding

Three

Types

of

WT

CWT

(Continuous

WT)

Wavelet

series

expansion

DWT

(Discrete

WT)

Discrete WT

DWT

Most closely resembles unitary transforms

Most useful in image compression

Given a set of orthonormal basis fct.’s, DWT just like unitary
transform

Orthonormal wavelets with compact support

(by
Daubechies):

j,k
: integers

compact

support
:

[
0
,
2
r
-
1
]

shift
:

k

dilation

(scaling)
:

N
-
point

signal

N

coefficients

image

coefficients

)}
2
(
2
/
2
{
)}
(
{
k
x
j
r
j
x
r

j
2
0
20
40
60
80
100
120
140
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
a=1

a=6

N
N

2
N

Scaling

Image Processing applications of
DWT

Block

Diagram

2
-
D Wavelet
transform for
image

decomposition

Quantization

Coding of
quantized
coefficients

Input
image

Image Compression

Processing in Wavelet Domain

(e.g. compute cooccurence matrices)

for texture segmentation

.

.

.

DWT for Image Compression

Image Decomposition

Scale 1

4 subbands:

Each coeff. a 2*2 area in the original image

Low frequencies:

High frequencies:

LL
1

HL
1

LH
1

HH
1

1
1
1
1
,
,
,
HH
LH
HL
LL

2
/
0

2
/
DWT for Image Processing

Image
Decomposition

Scale 2

4 subbands:

Each coeff. a 2*2
area in scale 1 image

Low Frequency:

High frequencies:

HL
1

LH
1

HH
1

HH
2

LH
2

HL
2

LL
2

2
,
2
,
2
,
2
HH
LH
HL
LL

4
/
0

2
/
4
/

At a coarser scale, coefficients represent a larger spatial area of the
image but a narrow band of frequencies.

DWT for Image Processing

Image Decomposition

Parent

Children

Descendants: corresponding
coeff. at finer scales

Ancestors: corresponding
coeff. at coarser scales

HL
1

LH
1

HH
1

HH
2

LH
2

HL
2

HL
3

LL
3

LH
3

HH
3

Parent
-
children dependencies of subbands: arrow points
from the subband of parents to the subband of children.