Digital image transforms

peachpuceAI and Robotics

Nov 6, 2013 (3 years and 5 months ago)

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Digital image processing




Digital image transforms





4. DIGITAL IMAGE TRANSFORMS





4.1. Introduction




4.2. Unitary orthogonal two
-
dimensional transforms




Separable unitary transforms





4.3. Properties of the unitary transforms




Energy conservation




Energy compaction; the variance of coefficients




De
-
correlation




Basis functions and basis images




4.4. Sinusoidal transforms




The 1
-
D discrete Fourier transform (1
-
D DFT)




Properties of the 1
-
D DFT




The 2
-
D discrete Fourier transform (2
-
D DFT)




Properties of the 2
-
D DFT




The discrete cosine transform (DCT)




The discrete sine transform (DST)




The Hartley transform




4.5. Rectangular transforms




The Hadamard transform = the Walsh transform




The Slant transform




The Haar transform




4.6. Eigenvectors
-
based transforms




The Karhunen
-
Loeve transform (KLT)




The fast KLT






The SVD




4.7. Image filtering in the transform domain




4.8. Conclusions

4.1 INTRODUCTIO
N


Definition:

Image transform

=
operation
to
change the default representation
space of a digital image
(
spatial domain
-
> another domain)

so that
:

(1)

all the information present in the image is preserved in the
transformed domain, but represe
nted differently;

(2)

the transform is reversible, i.e., we can revert to the spatial doma
in


Generally
:
in the transformed domain

-
>
image information is represented
in a
more compact form

=>
main goal of the transforms
:

image
compress
ion.



Other usage:
imag
e analysis
-
a
new type of representation of different types
of information present in the image.


Note:
Most image transforms
=
“generalizations” of frequency transforms
=>
the
representation of the image by a DC component and several AC components.


Def
inition
:
“original representation space” of the image
U
[M×N]
=
a MN
-
dimensional space:

-

each coordinate of the space
=
a spatial location (m,n) in the digital
image
;

-

the value of the coordinate of
U
on
an
axis
=
the grey level in
U
in this
spatial locati
on (m,n).

x
1
=(0,0); x
2
=(0,1);

x
3
=(0,2);

.
..
x
MN
=(M
-
1,N
-
1).


=> A

unitary transform
of
the image
U
=
a
rotation
of the MN
-
dimensional space,
defined by a rotation matrix

A
in MN
-
dimensions.


Digital image processing




Digital image transforms








{u(n),
0
n
N
1}



;
A


unitary matrix
,
T
*
1
A
A
















1
N
0
n
1
N
k
0
,
n)u(n)
a(k,
v(k)
r
o
,
Au
v

(4.1)













1
N
0
k
*
T
*
1
N
n
0
,
n)v(k)
(k,
a
u(n)
or
v
A
u
(4.2)






}
1
N
n
0
,
n)
(k,
{a
a
*
*
k




,


4.2
UNITARY ORTHOGONAL TWO
-
DIMENSIONAL TRANSFORMS







v(k,
l)
a
(m,
n)
u(m,
n)
,
0
k,
l
N
1
k,
l
n
0
N
1
m
0
N
1














(4.3)






u(m,
n)
a
(m,
n)
v(k,
l)
,
0
k,
l
N
1
k,
l
*
l
0
N
1
k
0
N
1













(4.4)




ort
h
onormali
ty
:











1
N
0
m
1
N
0
n
*
'
l
,
'
k
l
,
k
)
'
l
l
,
'
k
k
(
)
n
,
m
(
a
)
n
,
m
(
a


(4.5)




compl
eteness
:
a
m
n
a
m
n
m
m
n
n
k
l
k
l
l
N
k
N
,
,
*
(
,
)
(
'
,
'
)
(
'
,
'
)











0
1
0
1


(4.6)





u
(m,
n)
a
(m,
n)
v(k,
l)
,
P
N,
Q
N
P,
Q
k,
l
*
l
0
Q
1
k
0
P
1












(4.7)










e
2
n
0
N
1
m
0
N
1
P,
Q
2
[u(m,
n)
u
(m,
n)]











(4.8)







N
Q
P
f
0
2
e



i


Digital image processing




Digital image transforms


Unitary separable transforms







a
m
n
a
m
b
n
a
k
m
b
l
n
k
l
k
,
(
,
)
(
)
(
)
(
,
)
(
,
)


1



(4.9)



where
1}
N
0,...,
l
(n),
{b
and
1}
N
0,...,
k
(m),
{a
1
k




are the
orthonormal sets of basis vectors.







I
B
B
BB
,
I
A
A
AA
*
T
*T
*
T
*T





(4.10)
















1
N
0
m
1
N
0
n
r
o
l),
a(n,
n)
u(m,
m)
a(k,
l)
v(k,
T
AUA
V

(4.11)















1
N
0
n
1
N
0
l
*
*
r
o
n),
(l,
a
l)
v(k,
m)
(k,
a
n)
u(m,
*
T
*
VA
A
U
(4.12)








V
A
UA
M
N






(4.13)









U
A
VA
M
*T
N
*T






(4.14)








V
AUA
,
V
A[AU]
T
T
T



(4.15)


Digital image processing




Digital image transforms

4.3
PROPERTIES OF UNITARY TRANSFORMS



Energy

conservation




v
v
v
u
A
A
u
u
u
u
2
*T
*T
*T
*T
2
















v(k)
2
0
1
0
1
k
N
n
N
u
n
(
)
2
(4.16)






u
m
n
v
k
l
k
l
N
m
n
N
(
,
)
(
,
)
,
,
2
2
0
1
0
1











(4.17)



Energy compaction
and
the variance of coefficients









v
u
E
E
E




[
]
[
]
[
]
v
Au
A
u
A



(4.18)




R
v
v
A
u
u
A
A
R
A
v
v
v
T
u
u
T
T
u
T
E
E









[(
)(
)
]
(
[(
)(
)
])
*
*
*
*




(4.19)









v
v
k
k
u
T
k
k
k
2
(
)
[
]
[
]
,
*
,


R
AR
A


(4.20)






v
k
W
v
T
v
u
T
T
u
u
n
N
k
n
(
)
(
)
*
*
*
2
0
1
2
0
1
















A
A

(4.21)







v
k
N
u
T
u
u
n
N
k
n
2
0
1
2
0
1
(
)
[
]
[
]
(
)
*









Tr
AR
A
Tr
R



(4.22)













E
v
k
E
u
n
n
N
k
N
(
)
(
)
2
2
0
1
0
1










(4.23)





Digital image processing




Digital image transforms

Energy compaction and the variance of coefficients




v
u
n
m
k
l
a
k
m
a
l
n
m
n
(
,
)
(
,
)
(
,
)
(
,
)





(4.24)















m
n
m
n
v
v
n
l
a
m
k
a
n
m
n
m
r
n
l
a
m
k
a
l
k
l
k
v
E
l
k
'
'
*
*
2
2
)
'
,
(
)
'
,
(
)
'
,
'
;
,
(
)
,
(
)
,
(
)
,
(
)
,
(
)
,
(



(4.25)







r
m
n
m
n
r
m
m
r
n
n
(
,
;
'
,
'
)
(
,
'
)
(
,
'
)


1
2



(4.26)









v
T
k
k
T
l
l
k
l
k
l
2
1
2
2
2
1
2
(
,
)
(
)
(
)
*
,
*
,



AR
A
AR
A
where
R
1


r
1
(m,m')

and
R
2


r
2
(n,n')

.



Decorrelation


1
0
,
1
1
where
3
1
1
3
2
1
u
























R
u
v
,
,



















2
3
1
2
2
2
3
1
v
/
/
/
/
R







v
v
v
E
v
v
(
)
[
(
)
(
)]
(
)
(
)
(
)
0,1
0
1
0
1
2
1
3
4
2
1
/
2



,
A








1
2
1
1
1
1


Digital image processing




Digital image transforms



Basis functions and basis images

Basis functions (basis vectors)

Basis images (e.g.): DCT, Haar, ….

KLT

Haar

Walsh

Slant

DCT

Digital image processing




Digital image transforms

Imagine originala
Imagine originala
V(9,9)
V(1,1)
=

+

=

V(1,3)
+

+

+

+

+

+

+

+

+

+

+

+

+



+

V(1,5)
V(1,7)
V(1,9)
V(1,13)
V(1,15)
V(2,1)
V(2,9)
V(3,1)
V(3,5)
V(5,1)
V(5,2)
V(5,6)
V(5,8)
V(16,15)
Imagine aproximata
Keeping only 50% of coefficients

4.4
SINUSOIDAL TRANSFORMS




The

discrete Fourier transform
(DFT)



1
-
D
DFT

of a sequence


u(n), n

0,..., N
-
1


is defined as
:







v(k)
u(n)
W
,
k
=
0,1,
.
.
.
N
-
1
N
kn
n
0
N
1






(4.28)


where
:








W
j
N
N








exp
2



(4.29)




The inverse
DFT
(
IDFT
)
:






u(n)
1
N
v(k)
W
,
n
0,1,
.
.
.
N
1
N
kn
k
0
N
1









(4.30)









v(k)
1
u(n)
W
,
k
=
0,1,
.
.
.
N
-
1
N
kn
n
0
N
1





N

(4.31)








u(n)
1
N
v(k)
W
,
n
0,1,
.
.
.
N
1
N
kn
k
0
N
1









(4.32)








F
1
N
W
,
0
k,
n
N
1
N
kn












(4.33)

Digital image processing




Digital image transforms


DFT properties







v(k)
W
u(n)
W
(n)
u
k)
(N
v
1
N
0
n
kn
N
1
N
0
n
k)n
(N
N
*
*














(4.34)






v
N
2
k
v
N
2
k
,
k
0,
.
.
.
,
N
2
1
*



















(4.35)







v
N
2
k
v
N
2
k


















(4.36)


v(0)
,

R
e


v(k)

, k

1,...,N/2
-
1

,

i
m

v(k), k

1,...,N/2
-
1

, v(N
/2)
(4.37)

(Conjugate symmetry

the
DFT of a real sequence is conjugate
symmetric
about N/2).



The 2
-
D DFT:







v(k,
l)
u(m,
n)
W
W
,
0
k,
l
N
1
N
km
n
0
N
1
N
ln
m
0
N
1













(4.38)





u(m,
n)
1
N
W
W
v(k,
l)
,
0
m,
n
N
1
2
N
km
l
0
N
1
N
ln
k
0
N
1















(4.39)







1
N
l
k,
0
,
W
W
N
1
l)
v(k,
1
N
0
m
ln
N
1
N
0
n
km
N












)
n
,
m
(
u

(4.40)





u(m,
n)
1
N
v(k,
l)
W
W
v(k,
l)
,
0
m,
n
N
1
N
km
l
0
N
1
N
ln
k
0
N
1















(4.41)


Digital image processing




Digital image transforms

Properties of 2
-
D DFT




Symmetry
:





F
F
,
F
F
T
1



*


(4.42)




Periodicity
:






l
,
k
)
l
,
k
(
v
)
N
l
,
N
k
(
v







(4.43)







u
m
N
n
N
u
m
n
m
n
(
,
)
(
,
)
,







(4.44
)




The sampled Fourier spectrum
:


If

0
n)
(m,
u
si
1
N
n
m,
0
n),
u(m,
n)
(m,
u





~
~
otherwise, =>:







~
,
(
,
)
(
,
)
U
k
N
l
N
DFT
u
m
n
v
k
l
2
2












(4.45)


where
~
(
,
)
U
w
w
1
2
is the Fourier transform of
u(m,n).




Fast Fourier transform (FFT):
s
ince 2
-
D
DFT
i
s separable =>

equations (4.40)
and
(4.41)
a
re equivalent to

2N

1
-
D
DFT
s
;

e
ach of them can be
computed in
Nlog
2
N

o
perations through
FFT.

=> The total number of operations for
2
-
D
DFT:

N
2
log
2
N
.

Digital image processing




Digital image transforms

Properties of

2
-
D
DFT
(cont
inued
)




C
onjugate symmetry
:

t
he 2
-
D
DFT and unitary 2
-
D
DFT
o
f a real image

exhibit
conjugate symmetry
:





v
N
2
k,
N
2
l
v
N
2
k,
N
2
l
,
0
k,
l
N
2
1
*





















(4.46)



or



v(k,
l)
v
(N
k,
N
l)
,
0
k,
l
N
1
*









(4.47)


1 (N/2 -1) N/2 N-1
l

0

k
(N/2)-1
N/2
N-1
N/2


Fig. 4.2

T
he conjugate symmetry of the
2
-
D
DFT
coefficients

Digital image processing




Digital image transforms

Digital image processing




Digital image transforms



The discrete Cosine transform (DCT)



FDCT:































2N
1)l
(2n
cos
2N
1)k
(2m
cos
n)
u(m,
(l)
(k)
l)
v(k,
1
N
0
m
1
N
0
n

(4.47)



where
k, l

0, 1, ... N
-
1
.


IDCT:




u(m,
n)
(k)
(l)
v(k,
l)
cos
(2m
1)k
2N
cos
(2n
1)l
2N
l
=
0
N
1
k
=
0
N
1


























(4.48)


where
m, n


0, ... N
-
1
, and the coefficients are:





N
k
1
or
f
N
2
(k)
nd
N
1
(0)






a
(4.49)








T
C
u
C
V




(4.50)
















2N
1)k
(2m
cos
(k)
c
m
k,


(4.51)

Digital image processing




Digital image transforms

Digital image processing




Digital image transforms



T
he discrete Sine transform (DST)
:




v(k,
l)
2
N
1
u(m,
n)
sin
(m
1)(k
1)
N
1
sin
(n
1)(l
1)
N
1
n
0
N
1
m
0
N
1





























(4.52)






u(m,
n)
2
N
1
v(k,
l)
sin
(m
1)(k
1)
N
1
sin
(n
1)(l
1)
N
1
l
0
N
1
k
0
N
1





























(4.53)








s
2
N
1
sin
(m
1)(k
1)
N
1
m,
k














(4.54)




T
h
e
Hartley
transform:






v
(k,
l)
1
N
u(m,
n)
cas
2
N
(mk
nl)
n
0
N
1
m
0
N
1

















(4.55)








u
m
n
(
,
)














1
N
v(k,
l)
cas
2
N
(mk
nl)
l
0
N
1
k
0
N
1



(4.56)






cas(
)
cos(
)
sin(
)
2
cos(
/
4)











(4.57)







h
1
N
m,
k













cas
mk
N
2




(4.58)

Digital image processing




Digital image transforms

4.5 RECTANGULAR TRANSFORMS



The Hadamard transform (= the Walsh transform; the
Walsh
-
Hadamard transform)















1
1
1
1
2
1
H
2

(4.59)















2
/
2
/
2
/
2
/
N
N
N
N
N
H
H
H
H
2
1
H
(4.60)


H
1
2
2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
7
3
4
1
6
2
5
8
























































(
4.
61)

62)

(4.

1
7
6
5
4
3
2
1
0
1
1
1
-
1
1
-
1
1
1
1
1
-
1
1
1
1
1
-
1
1
-
1
1
1
1
1
-
1
1
1
1
1
1
1
1
-
1
-
1
1
-
1
1
1
1
-
1
1
1
1
1
1
-
1
-
1
1
1
1
1
1
1
-
1
1
1
1
1
1
1
1
1
1
1
1
2
2
1
H
ord
8,









































Digital image processing




Digital image transforms

Basis vectors for the

Walsh
-
Hadamard transform

Digital image processing




Digital image transforms

Original image

Digital image processing




Digital image transforms

Ordered Hadamard

Non
-
ordered Hadamard


The Slant transform







S
1
2
1
1
1
1
2










(4.63)













































1
n
1
n
1
-
n
n
n
n
n
1
-
n
1
-
n
n
n
n
n
n
a
b
1
0
a
b
1
0
b
a
0
1
b
a
0
1
2
1
S
S
I
I
I
I
S
1
-
n
0
0
0
0
0
0
0
0
0
0


(4.64)








1
4N
1
N
b
and
1
4N
3N
a
2
2
1
n
2
2
1
n










(4.65)


The Haar transform






k
2
q
1
p







(4.66)

N
1
(x)
h
0

and


















therwise
o
,
0
2
q
x
2
1/2
q
f
i
,
2
2
1/2
q
x
2
1
q
f
i
,
2
N
1
(x)
h
p
p
p/2
p
p
p/2
k

(4.67)

,
N
/
i
x


i=0,1,...,N
-
1



H
r







































1
8
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
2
2
2
0
0
0
0
0
0
0
0
2
2
2
2
2
2
0
0
0
0
0
0
0
0
2
2
0
0
0
0
0
0
0
0
2
2
0
0
0
0
0
0
0
0
2
2
0
1
2
2
2
2
2
2

(4.68)

Digital image processing




Digital image transforms

Digital image processing




Digital image transforms

Digital image processing




Digital image transforms

Applying the Haar transform at block level (e.g. 2
×
2 pixels blocks => Hr[2
×
2]):

Rearrange

coefficients
:

Block

transform
:

Applying the Haar transform at block level for a 4
×
4 pixels blocks => Hr[4
×
4]:

Block

transform
:

Rearrange

coefficients
:

4.6
EIGENVECTOR
BASED TRANSFORMS









0
k





A



(4.69)








Av
v
k
k
k






(4.70)











L
l
l
L
1
1
x
m
x




(4.71)














L
1
l
t
t
l
l
T
L
1
)
(
)
(
E
x
x
x
x
x
m
m
x
x
m
x
m
x
C


(4.72)








y
A
x
m
x


(
)




(4.73)








T
A
C
A
C
x
y





(4.74)







C
y
N















1
0
0



(4.75)







x
T
x
1
m
y
A
m
y
A
x






(4
.76)








~
y
B
x





(4.77)









~
~
x
B
y
T





(4.78)









e
k
N
2




k
M
1




(4.79)

Digital image processing




Digital image transforms


T
h
e
Karhunen

Loeve

transform
(
KLT = PCA
)




T
he fast KLT



T
h
e
SVD
transform
(
s
ingular value
s
decomposition
)








A
U
V
T






(4.80)










U
AV
T




(4.81)


A

















0
1
2
1
0
1
3
4
3
1
2
4
5
4
2
1
3
4
3
1
0
1
2
1
0


AA
T


































6
14
18
14
6
14
36
48
36
14
18
48
65
48
18
14
36
48
36
14
6
14
18
14
6
147,07
1,872
0,058
0
0
;



U
























0,186
0,638
0,241
0,695
0,695
0,476
0,058
0,52
0,133
0,128
0,691
0,422
0,587
0
0
0,476
0,058
0,52
0,133
0,128
0,186
0,638
0,241
0,695
0,695























U
AU
T
12,58
0
0
0
0
0
1,142
0
0
0
0
0
0,557
0
0
0
0
0
0
0
0
0
0
0
0

A
U
U
T



















0
1
2
1
0
1
3
4
3
1
2
4
6
4
2
1
3
4
3
1
0
1
2
1
0


Digital image processing




Digital image transforms





KLT (PCA)

Eigenimages


examples:

3 eigenimages and the individual variations on those components

Facial
image
set

Corresponding
“eigenfaces”

Face aproximation,
from rough to
detailed, as more
coefficients are
added

4.7

FIL
T
ERING IN THE TRANSFORM DOMAIN



transform
Haar

Its
Image
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2,83
0
0
0
0
0
0
0
2
0
0
0
0
0
0
0
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1


















































Digital image processing




Digital image transforms

DFT = sinc 2
-
D for the square + cst. (for noise)

Original image = (white square, grey background)

+ aditive noise

DFT

LPF 2
-
D

IDFT

Noisy image; periodic noise as vertical lines

The 2
-
D spectrum of the image and the filters applied:

In the regions corresponding to the vertical lines frequencies

Image restoration through filtering

4.8 CONCLU
SIONS




DFT

-

Fast transform; very useful in digital signal processing, convolution, filtering, image
analysis

-

Good energy compaction; however

requires complex computations




DCT

-

Fast transform and requires only real number operations

-

The opti
mal alternative to the KLT for highly correlated images

-

Used in compression and image restoration by Wiener filtering

-

Excellent energy compaction




Hadamard

-

Faster than sinusoidal transforms since it only implies sums and subtractions

-

Used for hardware i
mplementation of some digital image processing algorithms

-

Applied in image compression, filtering, coding

-

Good energy compaction



Haar

-

Very fast
t
ransform

-

Useful for feature extraction (like
horizontal
or vertical lines), image coding, image
analysis

-

Average energy compaction performance




K
LT

-

Optimal transform as: energy compaction; decorrelation

-

D
oes not have a fast algorithm

-

Generally used for small sized vectors and to evaluate the performances of other
transforms, but also for
image analysis and recognition
(PCA)


Digital image processing




Digital image transforms