mixed fourier transforms (mxfts) and image encryption

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.


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


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


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


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


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


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


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…)


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.


.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


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


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


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


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


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


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.


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


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


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.


(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.


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.


Questions?


mixed fourier transforms (mxfts) and image encryption

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