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Mixed Fourier Transforms (MxFTs)
and Image Encryption
Nan Du, Sree Devineni,
Artyom M Grigoryan
EE dept. UTSA, San Antonio TX, USA
Presented by Sree Devineni and Nan Du
IEEE SMC’09 San Antonio
Overview
Introduction
Mixed Transformations
Discrete-Mixed Transformations
General MxDFT
Series of Fourier matrices
Conclusion
References
Q&A
2Nan, Sree, and Art, ECE UTSA 2009
Introduction
Representations of signals help us to solve many problems in
signal processing, image encryption, linear systems theory, etc.
Understanding them is easy in the light of their interpretation as
rotations which makes possible to define new transformations.
In this paper, we focus on a general concept of the mixed Fourier
transformations (MxFT), when signals are transformed to the
time-frequency domain, where the difference between of time and
frequency is disappeared.
Mixed transformations allow for effective calculation of different
roots of the Fourier and identity transformations, which can be
used in signal processing, image filtration and encryption.
3Nan, Sree, and Art, ECE UTSA 2009
Interpretation of the CTFT as a rotation
x(t)
F[x(t)]=X(ω)
F2[x(t)]=x(-t)
F3[x(t)]=X(-ω)
F4[x(t)]=x(t)
By transformation, a point in
time domain space gets
transformed to a point in
frequency domain space
whose spaces are “uncoupled”,
represented as “perpendicular”
axes.
time
frequency
4Nan, Sree, and Art, ECE UTSA 2009
Fractional FT
Arbitrarily rotate time-frequency axes through a predefined
angle „alpha‟ in L2 space
Time-frequency are coupled generally.
CTFT is a special case of FRFT where α=90o .
FRFT has the following closed form expression:
)2/sin(1 and )cot( where
2),(
)(2
))cot(1())(()(
22
),(
CB
tBtCBtQ
dtetfj
fFFtQj
α
5Nan, Sree, and Art, ECE UTSA 2009
Mixed FTs
Transform f(t) mf(t) followed by Mf(t)=F[mf(t)],
e.g. simply f(t)f(t)+F(t)
gives Mf(t)=m*f(t/2π).
in Space M1,1
Rotation by 180o → time and frequency axis are the same.
Generally, transform f(t) a·f(t)+b·F(t)
in Space Ma,b
6Nan, Sree, and Art, ECE UTSA 2009
Properties
FT of the mixed transformed signal is time-scaled by 1/(2π)
conjugate of the mixed transformed signal itself in M1,1
linear convolution between signals in M1,1
.)2()2(
)()2(
)()()(
)()()(
**
*
thtx
tHtx
tHtXtY
thtxty
7Nan, Sree, and Art, ECE UTSA 2009
Discrete mixed Transformation
MxDFT is defined as a linear combination of the real signal
f(n), its DFT, and its DFT2.
Transform matrix:
Inverse Transform matrix:
.2 cFbFaIS
).( , , ,
,
22
21
bac
tb
sa
p
tFsFpIS
8Nan, Sree, and Art, ECE UTSA 2009
512-point mxDFT
Nan, Sree, and Art, ECE UTSA 2009 9
Fig.1: (a) Original signal of length 512, and (b) the real part
and (c) imaginary part.
Applications
Image encryption
Nan, Sree, and Art, ECE UTSA 2009 10
Fig. 2: Amplitude spectrums of the MxDFT, when (a) a = 0.2 and
b = 0.8, (b) a = 0.75 and b = 0.25, (c)a = 0.25 and b = 0.25,
and (d) a = 0.75 and b = −0.5.
Applications (contd…)
Nan, Sree, and Art, ECE UTSA 2009 11
Fig.3.(a) The tree image f and 2-D mixed Fourier transforms (b) S1(f),
(c) S2(S1(f)), and (d) S3(S2(S1(f))).
Fig.4.(a) Lena image f and 2-D mixed Fourier transforms (b)S1(f),
(c) S2(S1(f)), and (d) S3(S2(S1(f))).
General Concept of the MxDFTThe mixed transformation with respect to the discrete Fourier
transformation is defined by
where a, b, c and d are coefficients, I is the identity matrix, F is the
matrix of the discrete Fourier transformation.
*dFcFbEaIS
FEFFFFEFandIFFFFEEFFE *,
Square roots of the Fourier transform: .
The coefficients a, b, c and d are calculated by
2S F
4
s w jp jta
4
s w jp jtb
4
s w p tc
4
s w p td
The solution is not unique because of ambiguity of the square roots.
12Nan, Sree, and Art, ECE UTSA 2009
.p , t,1 ,1 2222 jjws
Case 1: [+,+,+,+]
Consider the following values of the square roots:
1s w j1
2
jp j
1
2
jt j
Coefficients are obtained:
Transformation defined by this set of coefficients is called
1st square root discrete Fourier transformation (1-SQ DFT)
1(1 2 )
4a j
1(1 2 )
4b j
*c a *d b
Case 2: [-,+,+,+]
1( 1 2 )
4a j
1( 1 2 )
4b j
*c a *d b
For the following values of the square roots:
1s w j 1
2
jp j
1
2
jt j
the coefficients of the transform S are calculated by
The corresponding transformation is called the 2nd square root
discrete Fourier transformation (2-SQ DFT)13Nan, Sree, and Art, ECE UTSA 2009
Replacing the coefficients c and d in the square root [1/ 2]F* 1/ 2 *( )F aI bE dF cF
[1/ 2] [1/8]F I
0 50 100 150 200 250 300 350 400 450 500
0
20
40
(a)
0 50 100 150 200 250 300 350 400 450 5000
1000
2000
(b)
512-point DFT
0 50 100 150 200 250 300 350 400 450 5000
100
200
(c)
512-point DFT[1/2]
Fig. 6.
Original signal of length 512
14Nan, Sree, and Art, ECE UTSA 2009
The Fourier Transform
The square root Fourier
transform.
4F I
Square root of the inverse Fourier matrix
0 50 100 150 200 250 300 350 400 450 500-2
0
2
(a)
0 50 100 150 200 250 300 350 400 450 500-100
0
100
(b)
0 50 100 150 200 250 300 350 400 450 500-2
0
2
(c)
0 50 100 150 200 250 300 350 400 450 500-2
0
2
(d)
0 50 100 150 200 250 300 350 400 450 500-5
0
5
(a)
0 50 100 150 200 250 300 350 400 450 500-5
0
5
(b)
0 50 100 150 200 250 300 350 400 450 500-5
0
5
(c)
Fig. 6:
Original signal of length 512
Phase of the (a) DFT
1-SQ DFT
2-SQ DFT.
.50,0,64/sin2 216
ttetxt
15Nan, Sree, and Art, ECE UTSA 2009
real part of the DFT
1-SQ DFT
2-SQ DFT
Series of Fourier matrixes
A matrix S is represented by
2 3
0 1 2 3S a I a F a F a F
with real or complex coefficients , k=0,1,2,3.
The square of this matrix can be written as
ka
Where the coefficients are calculated by the cyclic convolution nb
3
mod 4
0
, 0,1, 2,3n k n k
k
b a a n
3
0
23
3
2
210
2 )(n
n
nFbFaFaFaIaS
16Nan, Sree, and Art, ECE UTSA 2009
The above cyclic convolution is written as
3
mod 4 ;1
0
, 0,1, 2,3k n k n
k
a a n
where and if n=0,2,3. In the frequency domain,
this convolution has a form 2
24 , 0,1,2,3
jp
pA e p
where is the four-point discrete Fourier transform of the vector-
coefficient pA
0 1 2 3, , ,a a a a
0 1 2 3
1 11, ( 1 ), , (1 )
2 2A A j A j A j
111,1 b 01, nn b
Case 1: FS 2
17Nan, Sree, and Art, ECE UTSA 2009
Images and their 2-D SQ-DFT
(a) (b)
Fig.7. (a) The tree image (b) 2-D SQ-DFT of the image
(a) (b)
Fig. 8: (a) The Lena image and (b) 2-D SQ-DFT of the image
18Nan, Sree, and Art, ECE UTSA 2009
2S I
3 32 2
0 0
( )k k
k k
n n
S a F b F I
and , can be found from cyclic convolution:1 2 3 0b b b 0 1b ka
3
mod 4
0
, 0,1, 2,3k n k n
n
a a n
where if n=0, and if n=1,2,3. In the frequency domain,
this convolution has a form , where p=0,1,2, 3 . The
coefficients can be defined by
Square root of the identity matrix:
1n 0n1
2
pA
ka 3,2,1,0;1 pAp
5.0,5.01,1,1,1 3210
1
aaaaA F
Case 2:
)(2
1 *]2/1[ FFEIIS
19Nan, Sree, and Art, ECE UTSA 2009
S= I, S= E,
?
0 1 2 3 4-1
-0.5
0
0.5
1
0 1 2 3 4-1
-0.5
0
0.5
1
0 1 2 3 4-1
-0.5
0
0.5
1
0 1 2 3 4-1
-0.5
0
0.5
1
1111
1
1
1
111
111
111
2
12/1I
Fig.9.: The basis functions of the square root of the 4-point identity
transformation S.
20Nan, Sree, and Art, ECE UTSA 2009
Basic functions: N=4 case
0 2 4 6 8-1
0
1
0 2 4 6 8-1
0
1
0 2 4 6 8-1
0
1
0 2 4 6 8-1
0
1
0 2 4 6 8-1
0
1
0 2 4 6 8-1
0
1
0 2 4 6 8-1
0
1
0 2 4 6 8-1
0
1
0.5000 0 0.50000.7071 0.5000 0 1.5000- 0.7071-
0 1.7071 00.7071- 0 0.2929- 0 0.7071-
0.5000 0 0.50000.7071 1.5000- 0 0.5000 0.7071-
0.7071 0.7071- 0.70710.7071- 0.7071 0.7071- 0.7071 0.7071-
0.5000 0 1.5000-0.7071 0.5000 0 0.5000 0.7071-
0 0.2929- 0 0.7071- 0 1.7071 0 0.7071-
1.5000- 0 0.50000.7071 0.5000 0 0.5000 0.7071-
0.7071- 0.7071- 0.7071-0.7071- 0.7071- 0.7071- 0.7071- 0.7071-
2
1]2/1[I
21Nan, Sree, and Art, ECE UTSA 2009
Basic functions: N=8 case
Fig.10.: The basis
functions of the
square root of the 8-
point identity
transformation S.
)(2
1,, *
210 FFEISESIS
The discrete transform defined by is called the square root of
discrete identity transformation (SR-DIT). The transform
is a square root of I , because the Fourier transform is the 4th degree
root of the identity transform. contains the cosine transform
and the difference of transform equals
2S
ES 1
2S
2/)( *FFC
CEIFF
EIFFEIS
)(2
1
2)(
2
1)(
2
1 **
2
There are six square roots of the identity matrix.
CEISS )3(2
112
22Nan, Sree, and Art, ECE UTSA 2009
Square roots of the identity matrix
0 50 100 150 200 250 300 350 400 450 500-2
0
2
(a)
0 50 100 150 200 250 300 350 400 450 500-5
0
5
(b)
0 50 100 150 200 250 300 350 400 450 500-100
0
100
(c)
]50,0[),64/sin(2)( 216
ttetxt
Fig. 11: (a) The original signal of length 512, (b) the square root
of discrete identity transform, and (c) the real part of the square
of the Fourier transform of this signal.
23Nan, Sree, and Art, ECE UTSA 2009
(a) (b) (c)
2-D SR-DIP
Fig. 12: (a) tree image in part, (b) 2-D SR-DIP of the image
(c) second application of the square root over the image in b.
24Nan, Sree, and Art, ECE UTSA 2009
The second square root is the inverse to itself !
Conclusion
In this paper, the concept of the mixed Fourier transform in the
continuous and discrete time cases have been considered. The
mixed transform represents the signals and images in the time-
frequency domain, where the concepts of time and frequency
are united.
Mixed Fourier transformations can be used for calculating
different roots of the Fourier and identity transformations, as
well as other transformations, such as the Hadamard and cosine
transformations.
Our preliminary experimental examples show that the described
mixed and root transformations can be used for signal and
image processing, especially for image encryption.
25Nan, Sree, and Art, ECE UTSA 2009
Questions?
26Nan, Sree, and Art, ECE UTSA 2009