![226 Computer Science & Information
Many transformation techniques such as DFT[1], DCT[2], DWT[3] and others are already
implemented in digital world and widely used in today’s technique such
data compression and many more.
This paper proposed a frequency domain based technique where the digital content such as image
is converted form spatial domain to Z
major to generate 2x2 real and 2x2 imaginary matrices through less amount of computation and
without the trigonometric complexity. This paper also emphasis on transmitting the image over
network where half of the Z domain coefficients are sufficient to regenerate the or
Various parametric tests are performed based on the original and reconstructed image at
destination which shows that both the images are identical in terms of MSE as zero, PSNR as
infinity and that of image fidelity as one.
Section 2 of this paper deals with the proposed technique through five sub sections. Results and
discussions are outlined in section 3, conclusions are drawn in section 4 and references are given
at end.
2. THE TECHNIQUE
The IZEB technique is divided into three major
2.1, inverse Z-Transform given in section 2.2, and that of section 2.5 elaborate the procedure of
bandwidth minimization. Traditional and fast Z
example in section 2.3 and 2.4 respectively.
2.1 Forward Z-Transformation
Z-transform in signal processing converts a discrete time domain signal which is a sequence of
real or complex numbers into a complex frequency domain representation. Z
defined in two ways, unilaterally or bilaterally.
In bilateral or two sided Z-transform of discrete time signal x[n] is the formal power series X(
defined by eq(1).
Where n is an integer and z is, in general, a complex number.
Alternatively, in cases where x[n] is defined only for
transform is defined by eq(2).
Where r is the magnitude of ᵶ, j is the imaginary unit, and
by substituting the value of ᵶ in eq(2).
Computer Science & Information Technology (CS & IT)
Many transformation techniques such as DFT[1], DCT[2], DWT[3] and others are already
implemented in digital world and widely used in today’s technique such as steganography[4],
data compression and many more.
This paper proposed a frequency domain based technique where the digital content such as image
is converted form spatial domain to Z-domain using a 2x2 sliding window based mask in a row
ate 2x2 real and 2x2 imaginary matrices through less amount of computation and
without the trigonometric complexity. This paper also emphasis on transmitting the image over
network where half of the Z domain coefficients are sufficient to regenerate the original image.
Various parametric tests are performed based on the original and reconstructed image at
destination which shows that both the images are identical in terms of MSE as zero, PSNR as
infinity and that of image fidelity as one.
his paper deals with the proposed technique through five sub sections. Results and
discussions are outlined in section 3, conclusions are drawn in section 4 and references are given
The IZEB technique is divided into three major tasks. Forward Z-transform as describe in section
Transform given in section 2.2, and that of section 2.5 elaborate the procedure of
bandwidth minimization. Traditional and fast Z-transform calculations are also given with
n 2.3 and 2.4 respectively.
Transformation
transform in signal processing converts a discrete time domain signal which is a sequence of
real or complex numbers into a complex frequency domain representation. Z-transform can be
two ways, unilaterally or bilaterally.
transform of discrete time signal x[n] is the formal power series X(
(1)
Where n is an integer and z is, in general, a complex number.
x[n] is defined only for n ≥0, the single sided or
(2)
, j is the imaginary unit, and ω is the angle in radians. We get eq(3)
in eq(2).
Many transformation techniques such as DFT[1], DCT[2], DWT[3] and others are already
as steganography[4],
This paper proposed a frequency domain based technique where the digital content such as image
domain using a 2x2 sliding window based mask in a row
ate 2x2 real and 2x2 imaginary matrices through less amount of computation and
without the trigonometric complexity. This paper also emphasis on transmitting the image over
iginal image.
Various parametric tests are performed based on the original and reconstructed image at
destination which shows that both the images are identical in terms of MSE as zero, PSNR as
his paper deals with the proposed technique through five sub sections. Results and
discussions are outlined in section 3, conclusions are drawn in section 4 and references are given
transform as describe in section
Transform given in section 2.2, and that of section 2.5 elaborate the procedure of
transform calculations are also given with
transform in signal processing converts a discrete time domain signal which is a sequence of
transform can be
transform of discrete time signal x[n] is the formal power series X(ᵶ)
or unilateral Z-
is the angle in radians. We get eq(3)](https://image.slidesharecdn.com/csit3625-180201085041/85/IMAGE-CODING-THROUGH-ZTRANSFORM-WITH-LOW-ENERGY-AND-BANDWIDTH-IZEB-2-320.jpg)
![Computer Science & Information Technology (CS & IT)
On applying eq (3) for forward transformation on 2x2 mask of cover image in a row major order,
four frequency component generates such as lower, horizontal, vertical and complex conjugate
pair of horizontal frequency as shown in figure 1.a which is similar
frequency coefficient in lower to higher frequency bands are complex number in the format of ‘a
+ j b’. Separation of real and imaginary parts is shown in figure 1.b and 1.c respectively.
Lower
Frequency
(LF)
Horizontal
frequency
(HF)
Vertical
frequency
(VF)
Complex
conjugate
pair of
(HF)
(a) Z-coefficient quadrants
of complex value ‘a + j b’
Fig. 1. Structural representation of forward Z
2.2 Inverse Z-Transformation
Every transform technique exists with pair of equations, forward and inverse. The inverse Z
transform can be obtained by eq(4).
where C is a counter clockwise closed path encircling the origin and entirely in the
convergence (ROC). A special case of this contour integral occurs when C is the unit circle. The
inverse Z-transform simplifies to eq (5).
The original gray scale image as shown in figure 2.a ‘Map.pgm’ on forward Z transf
generates four real and four imaginary subband as shown in figure 2.b and 2.d respectively. The
information on the bands are emphasis
and imaginary parts respectively. Threshold increases
present in the band. Inverse Z transform applied on real and imaginary parts generate lossless
image back with a MSE[7] as zero and that of PSNR[7] is infinity.
Computer Science & Information Technology (CS & IT)
(3)
On applying eq (3) for forward transformation on 2x2 mask of cover image in a row major order,
four frequency component generates such as lower, horizontal, vertical and complex conjugate
pair of horizontal frequency as shown in figure 1.a which is similar to subband coding[5]. Every
frequency coefficient in lower to higher frequency bands are complex number in the format of ‘a
b’. Separation of real and imaginary parts is shown in figure 1.b and 1.c respectively.
Real part
of LF
Real part
of HF
Imaginary
part of LF
Imaginary
part of HF
Real part
of VF
Real part
of HF
Imaginary
part of VF
Negation of
Imaginary
part of HF
quadrants (b) Real part of all
frequency component ‘a’
(c) Imaginary part of all
frequency component ‘j b’
Structural representation of forward Z-Transform (FZT) coefficients
Transformation
Every transform technique exists with pair of equations, forward and inverse. The inverse Z
transform can be obtained by eq(4).
(4)
where C is a counter clockwise closed path encircling the origin and entirely in the
convergence (ROC). A special case of this contour integral occurs when C is the unit circle. The
transform simplifies to eq (5).
(5)
The original gray scale image as shown in figure 2.a ‘Map.pgm’ on forward Z transf
generates four real and four imaginary subband as shown in figure 2.b and 2.d respectively. The
information on the bands are emphasis through threshold as shown in figure 2.c and 2.e for real
and imaginary parts respectively. Threshold increases the brightness of the small information
present in the band. Inverse Z transform applied on real and imaginary parts generate lossless
image back with a MSE[7] as zero and that of PSNR[7] is infinity.
227
On applying eq (3) for forward transformation on 2x2 mask of cover image in a row major order,
four frequency component generates such as lower, horizontal, vertical and complex conjugate
to subband coding[5]. Every
frequency coefficient in lower to higher frequency bands are complex number in the format of ‘a
b’. Separation of real and imaginary parts is shown in figure 1.b and 1.c respectively.
Imaginary
part of HF
Negation of
Imaginary
part of HF
b’
Every transform technique exists with pair of equations, forward and inverse. The inverse Z-
where C is a counter clockwise closed path encircling the origin and entirely in the region of
convergence (ROC). A special case of this contour integral occurs when C is the unit circle. The
The original gray scale image as shown in figure 2.a ‘Map.pgm’ on forward Z transform (FZT)
generates four real and four imaginary subband as shown in figure 2.b and 2.d respectively. The
through threshold as shown in figure 2.c and 2.e for real
the brightness of the small information
present in the band. Inverse Z transform applied on real and imaginary parts generate lossless](https://image.slidesharecdn.com/csit3625-180201085041/85/IMAGE-CODING-THROUGH-ZTRANSFORM-WITH-LOW-ENERGY-AND-BANDWIDTH-IZEB-3-320.jpg)
![228 Computer Science & Information
Fig. 2. Forward Z transform followed by inverse
2.3 Traditional Z Transform
In a specific case of Z-transform where we consider value of magnitude of
angular frequency ω € {[0, π/2, π
mask of 2 x 2) is represented in figure 3(a), and the equation is given in eq (6). The elaborate
form of eq (6) is shown in eq (7) where C
X00 X01
X10 X11
(a) Single mask
Fig. 2. Single mask representation of Image and its components after forward Z
Different angular frequency ω
coefficients values Cv = Rlm + jIlm
Here ω depends on the value of lm.
2.4 Fast Z-Transformation
Z-Transform can also be done with minimizing the computation and without the use of
trigonometry, only by applying addition and subtraction
Computer Science & Information Technology (CS & IT)
Fig. 2. Forward Z transform followed by inverse transform over image
Traditional Z Transform
transform where we consider value of magnitude of ᵶ (r = 1), and the
π/2, π, 3π/2]}. The forward Z-transformation for the vector X (single
is represented in figure 3(a), and the equation is given in eq (6). The elaborate
form of eq (6) is shown in eq (7) where Cv is the coefficient value.
R00 R01 I00 I01
R10 R11 I10 I11
(a) Single mask (b) Real value after FZT (c) Imaginary value after FZT
. Single mask representation of Image and its components after forward Z-Transformation (FZT).
(6)
(7)
ω € {[0, π/2, π, 3π/2]} is taken and with r = 1 the complex
lm are calculated by equation set eq(8).
(8)
depends on the value of lm.
m 0 1
l 0 ω = 0 ω = π/2
1 ω = π ω = 3π/2
Transform can also be done with minimizing the computation and without the use of
trigonometry, only by applying addition and subtraction.
(r = 1), and the
transformation for the vector X (single
is represented in figure 3(a), and the equation is given in eq (6). The elaborate
(c) Imaginary value after FZT
Transformation (FZT).
/2]} is taken and with r = 1 the complex
Transform can also be done with minimizing the computation and without the use of](https://image.slidesharecdn.com/csit3625-180201085041/85/IMAGE-CODING-THROUGH-ZTRANSFORM-WITH-LOW-ENERGY-AND-BANDWIDTH-IZEB-4-320.jpg)
![230 Computer Science & Information
most six subbands are required at destination to regenerate the lossless image, due to the entire
zero value imaginary part for lower
to ω). Two more subbands can reduce without any loss due to the complex conjugate pair of
horizontal frequency (HF). Thus, in total, minimum four subbands are requirednout of eight to
regenerate the original image without any loss as shown in figure 4.
3. DISCUSSION ON RESULTS
Benchmark (PGM) images [6] are taken to formulate results. IZEB is applied over gray scale
images such as Baboon, Boat, Clock, Couple, Elaine, Jet, Space, Tank, Truck and Map wit
dimension 512 x 512, few results and shown in figure 4. Figure 4.a and 4.b represents the real and
imaginary part, after forward Z
minimum subbands from real and imaginary parts that on inve
image back.
If we apply IZEB on cover images and calculate MSE and PSNR from reconstructed images at
destination it is seen that MSE is zero and that of PSNR becomes
are transmitted. Other bands are computed at destination as given in table 1.
a. Real Part
Baboon
b. Imaginary
Part Baboon
a. Real Part
Clock
b. Imaginary
Part Clock
a. Real Part
Elaine
b. Imaginary
Part
Elaine
Fig. 3. Threshold image representing frequency coefficients after FZT.
Out of eight, four subbands are sufficient to regenerate lossless image at destination, thus
minimizing the energy and the bandwidth to near
be reduced further by reducing the subbands with little loss. Statistical calculations for amount of
loss with different numbers of subbands sent to destination, calculated through IZT are shown in
table 1 and in figure 5. It is clear from the table 1 that single band LF is enough to regenerate the
image but LF needs maximum of 10 bit representation for every coefficient and rest of the bands
are need 7 to 8 bit representations for coefficient.
Computer Science & Information Technology (CS & IT)
most six subbands are required at destination to regenerate the lossless image, due to the entire
zero value imaginary part for lower frequency (LF) and vertical frequency (VF) (in this case due
). Two more subbands can reduce without any loss due to the complex conjugate pair of
horizontal frequency (HF). Thus, in total, minimum four subbands are requirednout of eight to
inal image without any loss as shown in figure 4.
ESULTS
Benchmark (PGM) images [6] are taken to formulate results. IZEB is applied over gray scale
images such as Baboon, Boat, Clock, Couple, Elaine, Jet, Space, Tank, Truck and Map wit
dimension 512 x 512, few results and shown in figure 4. Figure 4.a and 4.b represents the real and
imaginary part, after forward Z-transformation respectively, and figure 4.c represents the four
minimum subbands from real and imaginary parts that on inverse transform able to generate
If we apply IZEB on cover images and calculate MSE and PSNR from reconstructed images at
destination it is seen that MSE is zero and that of PSNR becomes ∞ (infinity) when four bands
are computed at destination as given in table 1.
b. Imaginary
Baboon
c. Minimum
subbands for
Baboon
a. Real Part
Boat
b. Imaginary
Part Boat
c. Minimum
subbands for
Boat
b. Imaginary
Clock
c. Minimum
subbands for
Clock
a. Real Part
Couple
b. Imaginary
Part Clouple
c. Minimum
subbands for
Couple
b. Imaginary
Elaine
c. Minimum
subbands for
Elaine
a. Real Part
Jet
b. Imaginary
Part Jet
c. Minimum
subbands for
Jet
. Threshold image representing frequency coefficients after FZT.
Out of eight, four subbands are sufficient to regenerate lossless image at destination, thus
minimizing the energy and the bandwidth to near about half. Further energy and bandwidth can
be reduced further by reducing the subbands with little loss. Statistical calculations for amount of
loss with different numbers of subbands sent to destination, calculated through IZT are shown in
n figure 5. It is clear from the table 1 that single band LF is enough to regenerate the
image but LF needs maximum of 10 bit representation for every coefficient and rest of the bands
are need 7 to 8 bit representations for coefficient.
most six subbands are required at destination to regenerate the lossless image, due to the entire
(LF) and vertical frequency (VF) (in this case due
). Two more subbands can reduce without any loss due to the complex conjugate pair of
horizontal frequency (HF). Thus, in total, minimum four subbands are requirednout of eight to
Benchmark (PGM) images [6] are taken to formulate results. IZEB is applied over gray scale
images such as Baboon, Boat, Clock, Couple, Elaine, Jet, Space, Tank, Truck and Map with
dimension 512 x 512, few results and shown in figure 4. Figure 4.a and 4.b represents the real and
transformation respectively, and figure 4.c represents the four
rse transform able to generate
If we apply IZEB on cover images and calculate MSE and PSNR from reconstructed images at
(infinity) when four bands
inimum
subbands for
inimum
subbands for
inimum
subbands for
Out of eight, four subbands are sufficient to regenerate lossless image at destination, thus
about half. Further energy and bandwidth can
be reduced further by reducing the subbands with little loss. Statistical calculations for amount of
loss with different numbers of subbands sent to destination, calculated through IZT are shown in
n figure 5. It is clear from the table 1 that single band LF is enough to regenerate the
image but LF needs maximum of 10 bit representation for every coefficient and rest of the bands](https://image.slidesharecdn.com/csit3625-180201085041/85/IMAGE-CODING-THROUGH-ZTRANSFORM-WITH-LOW-ENERGY-AND-BANDWIDTH-IZEB-6-320.jpg)
![232 Computer Science & Information
Fig. 5. Graphical representation of energy/bandwidth in bits with various band(s)
4. CONCLUSIONS
The issue of image coding with minimum calculation and less complexity is achieved in this
proposed work. IZEB also emphasis the subband mi
only four is enough to regenerate the image without loss this lower the energy consumption and
the bandwidth too.
REFERENCES
[1] Ghoshal, Nabin, and Mandal, Jyotsna Kumar, “A Novel Technique for Image
Frequency Domain Using Discrete Fourier Transformation Technique (IAFDDFTT)”. Malaysian
Journal of Computer Science, Vol 21 (No. 1). pp. 24
Kuala Lumpur, Malyasia, 2008.
[2] Madhumita Sengupta, J. K. Mandal, “Authentication of Images through Non Convoluted DCT
(AINCDCT)”, first International Conference on Communication and Industrial Application (ICCIA
2011), Dec, 26-28, 2011, IEEE, pp
[3] Madhumita Sengupta, J. K. Mandal, Nabin Ghoshal, “An authetication technique in frequency
domain through wavelet transform (ATFDWT), Advances in Modelling Signal Processing and
Pattern Recognition(AMSE), vol
[4] J. K. Mandal, Madhumita Sengupta, “Stegano
Fidelity (STMDF)”, IEEE, Second International Conference on Emerging Applications of
Information Technology (EAIT 2011), Print ISBN: 978
Issue Date: 19-20 Feb. 2011 p
[5] J. K. Mandal, Madhumita Sengupta, “Authentication /Secret Message Transformation Through
Wavelet Transform based Subband Image Coding (WTSIC)”, IEEE, ISED
978-0-7695-4294-2, Bhubaneswar, India, Print ISBN: 978
DOI 10.1109/ISED.2010.50.
[6] Allan G. Weber, The USC-SIPI Image Database: Version 5, Original release: October 1997, Signal
and Image Processing Institute, University of Southern California, Department of Electrical
Engineering. http://sipi.usc.edu/database/ (accessed on 25th January, 2012).
[7] Kutter M , Petitcolas F A P. A fair benchmark for image watermarking systems, Electronic Imaging
99. Security and Watermarking of Multimedia Contents. vol. 3657. Sans Jose, CA, US
1999. The International Society for Optical Engineering. pp 226
Computer Science & Information Technology (CS & IT)
. Graphical representation of energy/bandwidth in bits with various band(s)
The issue of image coding with minimum calculation and less complexity is achieved in this
proposed work. IZEB also emphasis the subband minimization technique, out of eight subbands
only four is enough to regenerate the image without loss this lower the energy consumption and
Ghoshal, Nabin, and Mandal, Jyotsna Kumar, “A Novel Technique for Image Authentication in
Frequency Domain Using Discrete Fourier Transformation Technique (IAFDDFTT)”. Malaysian
Journal of Computer Science, Vol 21 (No. 1). pp. 24-32. ISSN 0127-9084, University of Malaya,
Kuala Lumpur, Malyasia, 2008.
K. Mandal, “Authentication of Images through Non Convoluted DCT
(AINCDCT)”, first International Conference on Communication and Industrial Application (ICCIA
28, 2011, IEEE, pp- 1-4, DOI: 10.1109/ICCIndA.2011.6146672.
, J. K. Mandal, Nabin Ghoshal, “An authetication technique in frequency
domain through wavelet transform (ATFDWT), Advances in Modelling Signal Processing and
Pattern Recognition(AMSE), vol-54, Issue 2, 20011.
J. K. Mandal, Madhumita Sengupta, “Steganographic Technique Based on Minimum Deviation of
Fidelity (STMDF)”, IEEE, Second International Conference on Emerging Applications of
Information Technology (EAIT 2011), Print ISBN: 978-1-4244-9683-9, DOI: 10.1109/EAIT.2011.24,
20 Feb. 2011 pp- 298 - 301.
J. K. Mandal, Madhumita Sengupta, “Authentication /Secret Message Transformation Through
Wavelet Transform based Subband Image Coding (WTSIC)”, IEEE, ISED- 2010, pp 225
2, Bhubaneswar, India, Print ISBN: 978-1-4244-8979-4, Dec, 20th
SIPI Image Database: Version 5, Original release: October 1997, Signal
and Image Processing Institute, University of Southern California, Department of Electrical
eering. http://sipi.usc.edu/database/ (accessed on 25th January, 2012).
Kutter M , Petitcolas F A P. A fair benchmark for image watermarking systems, Electronic Imaging
99. Security and Watermarking of Multimedia Contents. vol. 3657. Sans Jose, CA, US
1999. The International Society for Optical Engineering. pp 226-239,1999.
. Graphical representation of energy/bandwidth in bits with various band(s)
The issue of image coding with minimum calculation and less complexity is achieved in this
nimization technique, out of eight subbands
only four is enough to regenerate the image without loss this lower the energy consumption and
Authentication in
Frequency Domain Using Discrete Fourier Transformation Technique (IAFDDFTT)”. Malaysian
9084, University of Malaya,
K. Mandal, “Authentication of Images through Non Convoluted DCT
(AINCDCT)”, first International Conference on Communication and Industrial Application (ICCIA
, J. K. Mandal, Nabin Ghoshal, “An authetication technique in frequency
domain through wavelet transform (ATFDWT), Advances in Modelling Signal Processing and
graphic Technique Based on Minimum Deviation of
Fidelity (STMDF)”, IEEE, Second International Conference on Emerging Applications of
9, DOI: 10.1109/EAIT.2011.24,
J. K. Mandal, Madhumita Sengupta, “Authentication /Secret Message Transformation Through
2010, pp 225-229, ISBN
4, Dec, 20th -22nd, 2010,
SIPI Image Database: Version 5, Original release: October 1997, Signal
and Image Processing Institute, University of Southern California, Department of Electrical
Kutter M , Petitcolas F A P. A fair benchmark for image watermarking systems, Electronic Imaging
99. Security and Watermarking of Multimedia Contents. vol. 3657. Sans Jose, CA, USA. January](https://image.slidesharecdn.com/csit3625-180201085041/85/IMAGE-CODING-THROUGH-ZTRANSFORM-WITH-LOW-ENERGY-AND-BANDWIDTH-IZEB-8-320.jpg)
IMAGE CODING THROUGH ZTRANSFORM WITH LOW ENERGY AND BANDWIDTH (IZEB)
- 1. Jan Zizka (Eds) : CCSIT, SIPP, AISC, PDCTA - 2013 pp. 225–232, 2013. © CS & IT-CSCP 2013 DOI : 10.5121/csit.2013.3625 IMAGE CODING THROUGH Z- TRANSFORM WITH LOW ENERGY AND BANDWIDTH (IZEB) Madhumita Sengupta, J. K. Mandal Department of Computer Science and Engineering University of Kalyani Kalyani, Nadia, Pin. 741235, West Bengal, India madhumita.sngpt@gmail.com, jkm.cse@gmail.com ABSTRACT In this paper a Z-transform based image coding technique has been proposed. The techniques uses energy efficient and low bandwidth based invisible data embedding with a minimal computational complexity. In this technique near about half the bandwidth is required compared to the traditional Z–transform while transmitting the multimedia content such as images through network. KEYWORDS Z-Transform; Frequency Domain; Mean Square Error (MSE); Peak Signal to Noise Ratio (PSNR); Image Fidelity (IF). 1. INTRODUCTION In today’s digital world there is an enormous increase in the amount of multimedia content over internet such as image, video and audio materials. Such materials are traverse through wire and unwired medium in a carryon fashion. Small digital device such as tablet, capsule, mobile, PDA’s and many others are not yet small in physical senses due to their processing power and memory capacity. But the problem of sharing large amount of multimedia contents over internet creates a concern among researchers regarding the bandwidth utilization. In case of static spectrum assignment for mobile or radio networks bandwidth becomes a major concern. Digital data are organized in a structure, traverse through network in a bit form of spatial data or frequency components. A single flip of bit is enough to destroy single pixel, but in case of frequency domain signal/images are first converted from spatial domain which decreases the probability of pixel value changes.
- 2. 226 Computer Science & Information Many transformation techniques such as DFT[1], DCT[2], DWT[3] and others are already implemented in digital world and widely used in today’s technique such data compression and many more. This paper proposed a frequency domain based technique where the digital content such as image is converted form spatial domain to Z major to generate 2x2 real and 2x2 imaginary matrices through less amount of computation and without the trigonometric complexity. This paper also emphasis on transmitting the image over network where half of the Z domain coefficients are sufficient to regenerate the or Various parametric tests are performed based on the original and reconstructed image at destination which shows that both the images are identical in terms of MSE as zero, PSNR as infinity and that of image fidelity as one. Section 2 of this paper deals with the proposed technique through five sub sections. Results and discussions are outlined in section 3, conclusions are drawn in section 4 and references are given at end. 2. THE TECHNIQUE The IZEB technique is divided into three major 2.1, inverse Z-Transform given in section 2.2, and that of section 2.5 elaborate the procedure of bandwidth minimization. Traditional and fast Z example in section 2.3 and 2.4 respectively. 2.1 Forward Z-Transformation Z-transform in signal processing converts a discrete time domain signal which is a sequence of real or complex numbers into a complex frequency domain representation. Z defined in two ways, unilaterally or bilaterally. In bilateral or two sided Z-transform of discrete time signal x[n] is the formal power series X( defined by eq(1). Where n is an integer and z is, in general, a complex number. Alternatively, in cases where x[n] is defined only for transform is defined by eq(2). Where r is the magnitude of ᵶ, j is the imaginary unit, and by substituting the value of ᵶ in eq(2). Computer Science & Information Technology (CS & IT) Many transformation techniques such as DFT[1], DCT[2], DWT[3] and others are already implemented in digital world and widely used in today’s technique such as steganography[4], data compression and many more. This paper proposed a frequency domain based technique where the digital content such as image is converted form spatial domain to Z-domain using a 2x2 sliding window based mask in a row ate 2x2 real and 2x2 imaginary matrices through less amount of computation and without the trigonometric complexity. This paper also emphasis on transmitting the image over network where half of the Z domain coefficients are sufficient to regenerate the original image. Various parametric tests are performed based on the original and reconstructed image at destination which shows that both the images are identical in terms of MSE as zero, PSNR as infinity and that of image fidelity as one. his paper deals with the proposed technique through five sub sections. Results and discussions are outlined in section 3, conclusions are drawn in section 4 and references are given The IZEB technique is divided into three major tasks. Forward Z-transform as describe in section Transform given in section 2.2, and that of section 2.5 elaborate the procedure of bandwidth minimization. Traditional and fast Z-transform calculations are also given with n 2.3 and 2.4 respectively. Transformation transform in signal processing converts a discrete time domain signal which is a sequence of real or complex numbers into a complex frequency domain representation. Z-transform can be two ways, unilaterally or bilaterally. transform of discrete time signal x[n] is the formal power series X( (1) Where n is an integer and z is, in general, a complex number. x[n] is defined only for n ≥0, the single sided or (2) , j is the imaginary unit, and ω is the angle in radians. We get eq(3) in eq(2). Many transformation techniques such as DFT[1], DCT[2], DWT[3] and others are already as steganography[4], This paper proposed a frequency domain based technique where the digital content such as image domain using a 2x2 sliding window based mask in a row ate 2x2 real and 2x2 imaginary matrices through less amount of computation and without the trigonometric complexity. This paper also emphasis on transmitting the image over iginal image. Various parametric tests are performed based on the original and reconstructed image at destination which shows that both the images are identical in terms of MSE as zero, PSNR as his paper deals with the proposed technique through five sub sections. Results and discussions are outlined in section 3, conclusions are drawn in section 4 and references are given transform as describe in section Transform given in section 2.2, and that of section 2.5 elaborate the procedure of transform calculations are also given with transform in signal processing converts a discrete time domain signal which is a sequence of transform can be transform of discrete time signal x[n] is the formal power series X(ᵶ) or unilateral Z- is the angle in radians. We get eq(3)
- 3. Computer Science & Information Technology (CS & IT) On applying eq (3) for forward transformation on 2x2 mask of cover image in a row major order, four frequency component generates such as lower, horizontal, vertical and complex conjugate pair of horizontal frequency as shown in figure 1.a which is similar frequency coefficient in lower to higher frequency bands are complex number in the format of ‘a + j b’. Separation of real and imaginary parts is shown in figure 1.b and 1.c respectively. Lower Frequency (LF) Horizontal frequency (HF) Vertical frequency (VF) Complex conjugate pair of (HF) (a) Z-coefficient quadrants of complex value ‘a + j b’ Fig. 1. Structural representation of forward Z 2.2 Inverse Z-Transformation Every transform technique exists with pair of equations, forward and inverse. The inverse Z transform can be obtained by eq(4). where C is a counter clockwise closed path encircling the origin and entirely in the convergence (ROC). A special case of this contour integral occurs when C is the unit circle. The inverse Z-transform simplifies to eq (5). The original gray scale image as shown in figure 2.a ‘Map.pgm’ on forward Z transf generates four real and four imaginary subband as shown in figure 2.b and 2.d respectively. The information on the bands are emphasis and imaginary parts respectively. Threshold increases present in the band. Inverse Z transform applied on real and imaginary parts generate lossless image back with a MSE[7] as zero and that of PSNR[7] is infinity. Computer Science & Information Technology (CS & IT) (3) On applying eq (3) for forward transformation on 2x2 mask of cover image in a row major order, four frequency component generates such as lower, horizontal, vertical and complex conjugate pair of horizontal frequency as shown in figure 1.a which is similar to subband coding[5]. Every frequency coefficient in lower to higher frequency bands are complex number in the format of ‘a b’. Separation of real and imaginary parts is shown in figure 1.b and 1.c respectively. Real part of LF Real part of HF Imaginary part of LF Imaginary part of HF Real part of VF Real part of HF Imaginary part of VF Negation of Imaginary part of HF quadrants (b) Real part of all frequency component ‘a’ (c) Imaginary part of all frequency component ‘j b’ Structural representation of forward Z-Transform (FZT) coefficients Transformation Every transform technique exists with pair of equations, forward and inverse. The inverse Z transform can be obtained by eq(4). (4) where C is a counter clockwise closed path encircling the origin and entirely in the convergence (ROC). A special case of this contour integral occurs when C is the unit circle. The transform simplifies to eq (5). (5) The original gray scale image as shown in figure 2.a ‘Map.pgm’ on forward Z transf generates four real and four imaginary subband as shown in figure 2.b and 2.d respectively. The information on the bands are emphasis through threshold as shown in figure 2.c and 2.e for real and imaginary parts respectively. Threshold increases the brightness of the small information present in the band. Inverse Z transform applied on real and imaginary parts generate lossless image back with a MSE[7] as zero and that of PSNR[7] is infinity. 227 On applying eq (3) for forward transformation on 2x2 mask of cover image in a row major order, four frequency component generates such as lower, horizontal, vertical and complex conjugate to subband coding[5]. Every frequency coefficient in lower to higher frequency bands are complex number in the format of ‘a b’. Separation of real and imaginary parts is shown in figure 1.b and 1.c respectively. Imaginary part of HF Negation of Imaginary part of HF b’ Every transform technique exists with pair of equations, forward and inverse. The inverse Z- where C is a counter clockwise closed path encircling the origin and entirely in the region of convergence (ROC). A special case of this contour integral occurs when C is the unit circle. The The original gray scale image as shown in figure 2.a ‘Map.pgm’ on forward Z transform (FZT) generates four real and four imaginary subband as shown in figure 2.b and 2.d respectively. The through threshold as shown in figure 2.c and 2.e for real the brightness of the small information present in the band. Inverse Z transform applied on real and imaginary parts generate lossless
- 4. 228 Computer Science & Information Fig. 2. Forward Z transform followed by inverse 2.3 Traditional Z Transform In a specific case of Z-transform where we consider value of magnitude of angular frequency ω € {[0, π/2, π mask of 2 x 2) is represented in figure 3(a), and the equation is given in eq (6). The elaborate form of eq (6) is shown in eq (7) where C X00 X01 X10 X11 (a) Single mask Fig. 2. Single mask representation of Image and its components after forward Z Different angular frequency ω coefficients values Cv = Rlm + jIlm Here ω depends on the value of lm. 2.4 Fast Z-Transformation Z-Transform can also be done with minimizing the computation and without the use of trigonometry, only by applying addition and subtraction Computer Science & Information Technology (CS & IT) Fig. 2. Forward Z transform followed by inverse transform over image Traditional Z Transform transform where we consider value of magnitude of ᵶ (r = 1), and the π/2, π, 3π/2]}. The forward Z-transformation for the vector X (single is represented in figure 3(a), and the equation is given in eq (6). The elaborate form of eq (6) is shown in eq (7) where Cv is the coefficient value. R00 R01 I00 I01 R10 R11 I10 I11 (a) Single mask (b) Real value after FZT (c) Imaginary value after FZT . Single mask representation of Image and its components after forward Z-Transformation (FZT). (6) (7) ω € {[0, π/2, π, 3π/2]} is taken and with r = 1 the complex lm are calculated by equation set eq(8). (8) depends on the value of lm. m 0 1 l 0 ω = 0 ω = π/2 1 ω = π ω = 3π/2 Transform can also be done with minimizing the computation and without the use of trigonometry, only by applying addition and subtraction. (r = 1), and the transformation for the vector X (single is represented in figure 3(a), and the equation is given in eq (6). The elaborate (c) Imaginary value after FZT Transformation (FZT). /2]} is taken and with r = 1 the complex Transform can also be done with minimizing the computation and without the use of
- 5. Computer Science & Information Technology (CS & IT) Algorithm for forward Z-transformation Input: X00, X01, X10 and X11. Output: R00, R01, R10, R11, I00, I01 Method: Perform arithmetic calculation for fast forward Z Calculate R00 and I00 , R10 and I respectively. , Equation set 10 shows a relation as ax + by = c the value of r used. Two unknown x and y needs to be calculated by two equations. R11= R01 Algorithm for inverse Z-transformation In case of inverse transform the calculation will be followed through set of equations eq(12) and eq(13). Input : R00, R01, R10, R11, I00, I01, I Output: X00, X01, X10 and X11. Method: Perform arithmetic calculation by equation set (12) and (13) for fast inverse Z transformation. Calculate X00 , X01 by eq (12), and X 2.5 Bandwidth minimization/Transmission efficiency A 2 x 2 mask of spatial data on FZT generates two 2 x 2 matrixes containing real and imaginary frequency coefficient values, which means, information need to traverse after FZT will become double in terms of data. On analysis of eight subbands, four real based on the temporary environment created for z transform, it can be clearly elucidate that at Computer Science & Information Technology (CS & IT) transformation 01, I10, and I11. Perform arithmetic calculation for fast forward Z-transform. and I10 , R01 and I01 , R11 and I11 by equation set (9), (10) and (11) (9) , , (10) Equation set 10 shows a relation as ax + by = c1 and ax – by = c2, where ‘a’ and ‘b’ are depend on the value of r used. Two unknown x and y needs to be calculated by two equations. 01 and I11 = – I01 (11) transformation In case of inverse transform the calculation will be followed through set of equations eq(12) and , I10, and I11. Perform arithmetic calculation by equation set (12) and (13) for fast inverse Z by eq (12), and X10 and X11 by eq (13). , , 2.5 Bandwidth minimization/Transmission efficiency A 2 x 2 mask of spatial data on FZT generates two 2 x 2 matrixes containing real and imaginary frequency coefficient values, which means, information need to traverse after FZT will become double in terms of data. On analysis of eight subbands, four real and four imaginary valued, based on the temporary environment created for z transform, it can be clearly elucidate that at 229 by equation set (9), (10) and (11) , where ‘a’ and ‘b’ are depend on In case of inverse transform the calculation will be followed through set of equations eq(12) and Perform arithmetic calculation by equation set (12) and (13) for fast inverse Z- (12) (13) A 2 x 2 mask of spatial data on FZT generates two 2 x 2 matrixes containing real and imaginary frequency coefficient values, which means, information need to traverse after FZT will become and four imaginary valued, based on the temporary environment created for z transform, it can be clearly elucidate that at
- 6. 230 Computer Science & Information most six subbands are required at destination to regenerate the lossless image, due to the entire zero value imaginary part for lower to ω). Two more subbands can reduce without any loss due to the complex conjugate pair of horizontal frequency (HF). Thus, in total, minimum four subbands are requirednout of eight to regenerate the original image without any loss as shown in figure 4. 3. DISCUSSION ON RESULTS Benchmark (PGM) images [6] are taken to formulate results. IZEB is applied over gray scale images such as Baboon, Boat, Clock, Couple, Elaine, Jet, Space, Tank, Truck and Map wit dimension 512 x 512, few results and shown in figure 4. Figure 4.a and 4.b represents the real and imaginary part, after forward Z minimum subbands from real and imaginary parts that on inve image back. If we apply IZEB on cover images and calculate MSE and PSNR from reconstructed images at destination it is seen that MSE is zero and that of PSNR becomes are transmitted. Other bands are computed at destination as given in table 1. a. Real Part Baboon b. Imaginary Part Baboon a. Real Part Clock b. Imaginary Part Clock a. Real Part Elaine b. Imaginary Part Elaine Fig. 3. Threshold image representing frequency coefficients after FZT. Out of eight, four subbands are sufficient to regenerate lossless image at destination, thus minimizing the energy and the bandwidth to near be reduced further by reducing the subbands with little loss. Statistical calculations for amount of loss with different numbers of subbands sent to destination, calculated through IZT are shown in table 1 and in figure 5. It is clear from the table 1 that single band LF is enough to regenerate the image but LF needs maximum of 10 bit representation for every coefficient and rest of the bands are need 7 to 8 bit representations for coefficient. Computer Science & Information Technology (CS & IT) most six subbands are required at destination to regenerate the lossless image, due to the entire zero value imaginary part for lower frequency (LF) and vertical frequency (VF) (in this case due ). Two more subbands can reduce without any loss due to the complex conjugate pair of horizontal frequency (HF). Thus, in total, minimum four subbands are requirednout of eight to inal image without any loss as shown in figure 4. ESULTS Benchmark (PGM) images [6] are taken to formulate results. IZEB is applied over gray scale images such as Baboon, Boat, Clock, Couple, Elaine, Jet, Space, Tank, Truck and Map wit dimension 512 x 512, few results and shown in figure 4. Figure 4.a and 4.b represents the real and imaginary part, after forward Z-transformation respectively, and figure 4.c represents the four minimum subbands from real and imaginary parts that on inverse transform able to generate If we apply IZEB on cover images and calculate MSE and PSNR from reconstructed images at destination it is seen that MSE is zero and that of PSNR becomes ∞ (infinity) when four bands are computed at destination as given in table 1. b. Imaginary Baboon c. Minimum subbands for Baboon a. Real Part Boat b. Imaginary Part Boat c. Minimum subbands for Boat b. Imaginary Clock c. Minimum subbands for Clock a. Real Part Couple b. Imaginary Part Clouple c. Minimum subbands for Couple b. Imaginary Elaine c. Minimum subbands for Elaine a. Real Part Jet b. Imaginary Part Jet c. Minimum subbands for Jet . Threshold image representing frequency coefficients after FZT. Out of eight, four subbands are sufficient to regenerate lossless image at destination, thus minimizing the energy and the bandwidth to near about half. Further energy and bandwidth can be reduced further by reducing the subbands with little loss. Statistical calculations for amount of loss with different numbers of subbands sent to destination, calculated through IZT are shown in n figure 5. It is clear from the table 1 that single band LF is enough to regenerate the image but LF needs maximum of 10 bit representation for every coefficient and rest of the bands are need 7 to 8 bit representations for coefficient. most six subbands are required at destination to regenerate the lossless image, due to the entire (LF) and vertical frequency (VF) (in this case due ). Two more subbands can reduce without any loss due to the complex conjugate pair of horizontal frequency (HF). Thus, in total, minimum four subbands are requirednout of eight to Benchmark (PGM) images [6] are taken to formulate results. IZEB is applied over gray scale images such as Baboon, Boat, Clock, Couple, Elaine, Jet, Space, Tank, Truck and Map with dimension 512 x 512, few results and shown in figure 4. Figure 4.a and 4.b represents the real and transformation respectively, and figure 4.c represents the four rse transform able to generate If we apply IZEB on cover images and calculate MSE and PSNR from reconstructed images at (infinity) when four bands inimum subbands for inimum subbands for inimum subbands for Out of eight, four subbands are sufficient to regenerate lossless image at destination, thus about half. Further energy and bandwidth can be reduced further by reducing the subbands with little loss. Statistical calculations for amount of loss with different numbers of subbands sent to destination, calculated through IZT are shown in n figure 5. It is clear from the table 1 that single band LF is enough to regenerate the image but LF needs maximum of 10 bit representation for every coefficient and rest of the bands
- 7. Computer Science & Information Technology (CS & IT) Table 1. Statistical Data of Loss with further reduced number of subbands Cover Image MSE/ PSNR (db) Number 1| LF Clock .pgm MSE 26.827541 PSNR 33.844995 Elaine .pgm MSE 43.429588 PSNR 31.752946 Space .pgm MSE 12.659260 PSNR 37.106720 Tank. pgm MSE 39.988754 PSNR 32.111425 Truck .pgm MSE 39.040760 PSNR 32.215621 Fig. 4. Comparison in terms of Energy/bandwidth verses PSNR (Table 1) In case of original image of 512 x 512 in dimension, to send spatial data, transmission medium need to send 512*512*8 = 2097152 bits. FZT complex coefficients take 512*512*2*8 = 4194304 bits. For LF band 256*256*10 = 655360 bits required, for rest of the bands 8 bit per coeff required. All the computational results are given in table II and visual interpretation of the same is shown in figure 6. Table 2. Number of bits nedded to be send for transmission of 512 x 512 dimension image Slno Number & 1. Original Image in spatial domain 2. Complete Z 3. 4. 5. 6. 7. Computer Science & Information Technology (CS & IT) . Statistical Data of Loss with further reduced number of subbands umber and Name of Band used to reconstruct through IZT 2| LF & HF 2| LF & VF 3| LF, HF & VF 3| LF, HF & imgHF 4| LF,HF, VF & imgHF 26.827541 29.483864 41.385128 44.007004 27.010376 0.000 33.844995 33.434960 31.962361 31.695586 33.815497 ∞ 43.429588 58.092613 62.663673 77.373775 47.226242 0.000 31.752946 30.489594 30.160645 29.244866 31.388970 ∞ 12.659260 14.525913 20.651455 22.501396 13.147259 0.000 37.106720 36.509369 34.981297 34.608709 36.942452 ∞ 39.988754 53.691200 53.740608 67.420712 43.637718 0.000 32.111425 30.831772 30.827778 29.842870 31.732183 ∞ 39.040760 56.560173 48.327442 66.218987 44.382500 0.000 32.215621 30.605696 31.288866 29.920978 31.658686 ∞ . Comparison in terms of Energy/bandwidth verses PSNR (Table 1) image of 512 x 512 in dimension, to send spatial data, transmission medium need to send 512*512*8 = 2097152 bits. FZT complex coefficients take 512*512*2*8 = 4194304 bits. For LF band 256*256*10 = 655360 bits required, for rest of the bands 8 bit per coeff required. All the computational results are given in table II and visual interpretation of the same is . Number of bits nedded to be send for transmission of 512 x 512 dimension image Number & name of subbands Number of bits Original Image in spatial domain 2097152 Complete Z-transform coefficient 4194304 4| LF,HF,VF & imgHF 2228224 3| LF, HF & VF 1703936 2| LF & HF 1179648 2| LF & VF 1179648 1| LF 655360 231 4| LF,HF, VF & imgHF 0.000 ∞ 0.000 ∞ 0.000 ∞ 0.000 ∞ 0.000 ∞ image of 512 x 512 in dimension, to send spatial data, transmission medium need to send 512*512*8 = 2097152 bits. FZT complex coefficients take 512*512*2*8 = 4194304 bits. For LF band 256*256*10 = 655360 bits required, for rest of the bands 8 bit per coefficient is required. All the computational results are given in table II and visual interpretation of the same is . Number of bits nedded to be send for transmission of 512 x 512 dimension image
- 8. 232 Computer Science & Information Fig. 5. Graphical representation of energy/bandwidth in bits with various band(s) 4. CONCLUSIONS The issue of image coding with minimum calculation and less complexity is achieved in this proposed work. IZEB also emphasis the subband mi only four is enough to regenerate the image without loss this lower the energy consumption and the bandwidth too. REFERENCES [1] Ghoshal, Nabin, and Mandal, Jyotsna Kumar, “A Novel Technique for Image Frequency Domain Using Discrete Fourier Transformation Technique (IAFDDFTT)”. Malaysian Journal of Computer Science, Vol 21 (No. 1). pp. 24 Kuala Lumpur, Malyasia, 2008. [2] Madhumita Sengupta, J. K. Mandal, “Authentication of Images through Non Convoluted DCT (AINCDCT)”, first International Conference on Communication and Industrial Application (ICCIA 2011), Dec, 26-28, 2011, IEEE, pp [3] Madhumita Sengupta, J. K. Mandal, Nabin Ghoshal, “An authetication technique in frequency domain through wavelet transform (ATFDWT), Advances in Modelling Signal Processing and Pattern Recognition(AMSE), vol [4] J. K. Mandal, Madhumita Sengupta, “Stegano Fidelity (STMDF)”, IEEE, Second International Conference on Emerging Applications of Information Technology (EAIT 2011), Print ISBN: 978 Issue Date: 19-20 Feb. 2011 p [5] J. K. Mandal, Madhumita Sengupta, “Authentication /Secret Message Transformation Through Wavelet Transform based Subband Image Coding (WTSIC)”, IEEE, ISED 978-0-7695-4294-2, Bhubaneswar, India, Print ISBN: 978 DOI 10.1109/ISED.2010.50. [6] Allan G. Weber, The USC-SIPI Image Database: Version 5, Original release: October 1997, Signal and Image Processing Institute, University of Southern California, Department of Electrical Engineering. http://sipi.usc.edu/database/ (accessed on 25th January, 2012). [7] Kutter M , Petitcolas F A P. A fair benchmark for image watermarking systems, Electronic Imaging 99. Security and Watermarking of Multimedia Contents. vol. 3657. Sans Jose, CA, US 1999. The International Society for Optical Engineering. pp 226 Computer Science & Information Technology (CS & IT) . Graphical representation of energy/bandwidth in bits with various band(s) The issue of image coding with minimum calculation and less complexity is achieved in this proposed work. IZEB also emphasis the subband minimization technique, out of eight subbands only four is enough to regenerate the image without loss this lower the energy consumption and Ghoshal, Nabin, and Mandal, Jyotsna Kumar, “A Novel Technique for Image Authentication in Frequency Domain Using Discrete Fourier Transformation Technique (IAFDDFTT)”. Malaysian Journal of Computer Science, Vol 21 (No. 1). pp. 24-32. ISSN 0127-9084, University of Malaya, Kuala Lumpur, Malyasia, 2008. K. 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Mandal, Madhumita Sengupta, “Authentication /Secret Message Transformation Through Wavelet Transform based Subband Image Coding (WTSIC)”, IEEE, ISED- 2010, pp 225 2, Bhubaneswar, India, Print ISBN: 978-1-4244-8979-4, Dec, 20th SIPI Image Database: Version 5, Original release: October 1997, Signal and Image Processing Institute, University of Southern California, Department of Electrical eering. http://sipi.usc.edu/database/ (accessed on 25th January, 2012). Kutter M , Petitcolas F A P. A fair benchmark for image watermarking systems, Electronic Imaging 99. Security and Watermarking of Multimedia Contents. vol. 3657. Sans Jose, CA, US 1999. The International Society for Optical Engineering. pp 226-239,1999. . Graphical representation of energy/bandwidth in bits with various band(s) The issue of image coding with minimum calculation and less complexity is achieved in this nimization technique, out of eight subbands only four is enough to regenerate the image without loss this lower the energy consumption and Authentication in Frequency Domain Using Discrete Fourier Transformation Technique (IAFDDFTT)”. Malaysian 9084, University of Malaya, K. Mandal, “Authentication of Images through Non Convoluted DCT (AINCDCT)”, first International Conference on Communication and Industrial Application (ICCIA , J. K. Mandal, Nabin Ghoshal, “An authetication technique in frequency domain through wavelet transform (ATFDWT), Advances in Modelling Signal Processing and graphic Technique Based on Minimum Deviation of Fidelity (STMDF)”, IEEE, Second International Conference on Emerging Applications of 9, DOI: 10.1109/EAIT.2011.24, J. K. Mandal, Madhumita Sengupta, “Authentication /Secret Message Transformation Through 2010, pp 225-229, ISBN 4, Dec, 20th -22nd, 2010, SIPI Image Database: Version 5, Original release: October 1997, Signal and Image Processing Institute, University of Southern California, Department of Electrical Kutter M , Petitcolas F A P. A fair benchmark for image watermarking systems, Electronic Imaging 99. Security and Watermarking of Multimedia Contents. vol. 3657. Sans Jose, CA, USA. January