Journal of Electronic Imaging 16(3), 033020 (Jul–Sep 2007)
Improved image watermarking scheme using fast
Hadamard and discrete wavelet transforms
Emad E. Abdallah
A. Ben Hamza
Prabir Bhattacharya
Concordia Institute for Information Systems Engineering
Concordia University
Montréal, Québec, Canada
Abstract. We propose a robust image watermarking scheme by
applying the fast Hadamard transform (FHT) to small blocks computed from the four discrete wavelet transform (DWT) subbands.
Different transforms have different properties that can effectively
match various aspects of the signal’s frequencies. Our approach
consists of four main steps: (1) we decomposed the original image
into four subbands, (2) the four subbands are divided into blocks; (3)
FHT is applied to each block; and (4) the singular-value decomposition (SVD) is applied to the watermark image prior to distributing
the singular values over the DC components of the transformed
blocks. The proposed technique improves the data embedding system effectively, the watermark imperceptibility, and its resistance to
a wide range of intentional attacks. The experimental results demonstrate the improved performance of the proposed method in comparison with existing techniques in terms of the watermark imperceptibility and the robustness against attacks. © 2007 SPIE and
IS&T. 关DOI: 10.1117/1.2764466兴
1 Introduction
The problem of multimedia protection for digital images
has attracted a great deal of attention in research in the past
few years. Its importance is increasing rapidly because of
the growing problem of unauthorized replication. Watermarking refers to the process of adding a hidden structure,
called a ‘watermark’, which carries information about either the owner of the cover or the recipient of the original
data object. Such a watermark can be used for several purposes including copyright protection and transactional
watermarks.1
A variety of watermarking techniques has been proposed
for digital images.2–4 These techniques can be divided into
two main categories according to the embedding domain of
the cover image: spatial domain methods and transform domain methods. The spatial domain methods are the earliest
and simplest watermarking techniques but have a low information hiding capacity, and the watermark can be easily
erased by lossy image compression. On the other hand, the
transform domain approaches.4–12 insert the watermark into
the transform coefficients of the image cover, yielding more
information embedding and more robustness against watermarking attacks.
Recently, the unitary transformations have been widely
Paper 06206RR received Nov. 22, 2006; revised manuscript received Mar.
26, 2007; accepted for publication Mar. 26, 2007; published online Aug. 3,
2007.
1017-9909/2007/16共3兲/033020/9/$25.00 © 2007 SPIE and IS&T.
Journal of Electronic Imaging
used for data embedding including the discrete cosine
transform 共DCT兲,4,5 the discrete Fourier transform 共DFT兲,
fast Hadamard transform 共FHT兲,6,7 and discrete wavelet
transform 共DWT兲.8,9 Watermarking using singular-value
decomposition 共SVD兲 and its variants has been
proposed.9–12 The main idea of these approaches is to find
the SVD of a cover image and then modify its singular
values to embed the watermark. In Ref. 9, a hybrid nonblind watermarking scheme based on the SVD and the
DWT was proposed. This method consists of decomposing
the cover image into four transformed subbands, then the
SVD is applied to each band, followed by modifying the
singular values of the transformed subbands with the singular values of the visual watermark. This modification in
all frequencies provides more robustness to different attacks. Another SVD-block-based watermarking scheme
was proposed in Ref. 11, where the watermark embedding
is done in two layers. In the first layer, the cover image is
divided into small blocks and the singular values of the
watermark are embedded in those blocks. In the second
layer, the cover image is used as a single block to embed
the whole watermark. In addition to the limited robustness
to some attacks, the main weakness of the SVD-based techniques is that the SVD produces low-rank basis functions
that do not respect the nonnegativity of the cover image.
Motivated by the need for more robustness against attacks, better visual imperceptibility, and the computational
simplicity of the FHT, we propose a robust, imperceptible
watermarking method. Our technique uses the DWT,14
FHT, and SVD. Experiments are performed to demonstrate
the potential and the much-improved performance of the
proposed method in comparison to other methods.
The remainder of this paper is organized as follows.
Section 2 is devoted to the background material. In Section
3, we briefly review some works that are closely related to
our proposed scheme. In Section 4, we introduce the proposed watermark embedding and extraction algorithms. In
Section 5, we present some experimental results to demonstrate the much-improved performance of the proposed
method in comparison with existing techniques, and also to
show its robustness against the most common attacks. Finally, we conclude in Section 6.
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Fig. 2 Illustration of the SVD approximation.
Fig. 1 An example of the fast Hadamard transform 共a兲 original image, 共b兲 transformed image.
2 Background
2.1 Discrete Wavelet Transform (DWT)
The DWT provides a number of powerful image processing
algorithms including noise reduction, edge detection, and
compression. The DWT is computed by successive lowpass and high-pass filtering of the discrete time-domain signal. Its significance is in the manner it connects the
continuous-time multiresolution to discrete-time filters. At
each level, the high-pass filter produces detailed information, while the low-pass filter associated with scaling function produces coarse approximations. To use DWT for image processing, we use a 2-D version of the analysis and
synthesis filter banks. In the 2-D case, the 1-D analysis
filter bank is first applied to the columns of the image and
then applied to the rows. If the image has m rows and m
columns, then, after applying the 2-D analysis filter bank,
we obtain four subband images 共LL, LH, HL, and HH兲,
each having m / 2 rows and m / 2 columns.
2.2 Fast Hadamard Transform (FHT)
The 2-D Hadamard transform has been used with great success for image compression and image watermarking. Unlike the other well-known transforms, such as the DFT and
the DCT, the elements of the basis vectors of the Hadamard
transform take only the binary values +1 and −1. Hence,
the FHT is well suited for digital image processing applications where computational simplicity is required.
Let C be the original image of size m ⫻ m. The 2-D
Hadamard transform of C is given by Ĉ = 共1 / m兲HCH,
where H is a Hadamard matrix of order m = 2k 共k is an
integer兲, and with entries 兵−1 , + 1其. The Hadamard matrix
H has mutually orthogonal rows or columns and satisfies
HHT = mIm, where Im is the identity matrix. Hence, the
original image may be recovered using C = 共1 / m兲HĈH.
Figure 1 shows an example of a 2-D image with its FHT
result. Furthermore, the Hadamard matrix of order m may
be generated from the Hadamard matrix of order m / 2 using
the Kronecker product property Hm = H2 丢 Hm/2, where
H2 =
冋
+1 +1
+1 −1
册
is the Hadamard matrix of order m = 2. Consequently H4
becomes
Journal of Electronic Imaging
H4 =
冋
+ H2 + H2
+ H2 − H2
册
=
冤
+1 +1 +1
1
+1 −1 +1 −1
+1 +1 −1 −1
+1 −1 −1 +1
冥
.
2.3 Singular-Value Decomposition (SVD)
The SVD of an image C of size m ⫻ m is given by C
= U⌺V⬘, where U is an orthogonal matrix 共U⬘U = I兲, ⌺
= diag共␭i兲 is a diagonal matrix of singular values ␭i, 1 ⱕ i
ⱕ m, arranged in decreasing order, and V is an orthogonal
matrix 共V⬘V = I兲, as depicted in Fig. 2. The columns of U
are the left singular vectors, whereas the columns of V are
the right singular vectors of the image C. The matrix V⬘
denotes the transpose of V.
3 Related Work
In this section, we will review four representative methods
for digital image watermarking that are closely related to
our proposed approach. We briefly discuss their mathematical foundations and algorithmic methodologies as well as
their limitations.
3.1 SVD Watermarking Scheme
Let C be a cover image of size m ⫻ m and W be a watermark image of size n ⫻ n with n ⱕ m. The SVD of the cover
image and the watermark image are given by C = U⌺V⬘ and
W = Uw⌺wVw⬘ , respectively, where ⌺w = diag共␭wi兲 is a diagonal matrix of singular values 共SVs兲 of the visual watermark.
The SVD watermark embedding algorithm is given by
␭di = ␭i + ␣␭wi,
1 ⱕ i ⱕ n,
where ␭di denotes the distorted SVs of the watermarked
image, and ␣ is a constant scaling factor. Hence, the watermarked image is given by M = U⌺dV⬘, where ⌺d = diag共␭di 兲.
We may extract the SVs of the visual watermark using
␭ˆ wi = 共␭di − ␭i兲 / ␣. Consequently, the extracted watermark Ŵ
is given by Ŵ = U ⌺̂ V⬘ , where ⌺̂ = diag共␭ˆ 兲.
w
w w
w
wi
In the SVD-block-based watermarking scheme,11 the
cover image C is divided into blocks of size ᐉ ⫻ ᐉ and the
SVD of each block L is given by L = Uᐉ⌺ᐉVᐉ⬘. The SVs of
the visual watermark W are embedded into each block of
the cover image by modifying the largest singular value of
each block.
3.2 Block-Based FHT-SVD Scheme
The cover image C is divided into blocks of size ᐉ ⫻ ᐉ, and
the FHT of each block L is given by L̂ = 共1 / ᐉ 兲HLH, where
H is a Hadamard matrix of order ᐉ = 2k 共k is an integer兲. The
SVs of the visual watermark W are embedded into each
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Fig. 3 Watermark embedding algorithm.
block of the cover image by modifying the DC components
of the transformed blocks. The block-based FHT-SVD watermark embedding algorithm is given by
dwi = di0 + ␤i␭wi ,
where i ranges from 1 to the number of block, and di0 and
dwi are the original and the modified DC components, respectively. The scaling factor ␤i is chosen in relation to the
DC components where ␤i = ␣b / n, where ␣ is a constant, b
is the block number, and n is the watermark image width.
Hence, we may extract the SVs of the visual watermark
using ␭ˆ wi = 共dwi − di0兲 / ␤i. Therefore, the extracted watermark
Ŵ is given by Ŵ = U ⌺̂ V⬘ , where ⌺̂ = diag共␭ˆ 兲.
w
w w
w
wi
3.3 DWT-SVD Watermarking Scheme
The cover image C is decomposed into four subbands: the
approximation coefficient LL, and the detailed coefficients
HL, LH, and HH. The SVD is applied to each subband
Ck 苸 兵LL , HL , LH , HH其 of the cover image Ck = Ukc⌺kcVc⬘k,
k = 1 , 2 , 3 , 4, and ␭ki , 1 ⱕ i ⱕ m / 2, are the SVs of ⌺kc. The
SVD of the watermark image of size m / 2 ⫻ m / 2 is applied
to W = Uw⌺wVw⬘ , where ⌺w = diag共␭wi兲. The SVs of the cover
image in each subband are modified with the SVs of the
watermark as follows:
k
␭dk
i = ␭i + ␣k␭wi,
1 ⱕ i ⱕ m/2, 1 ⱕ k ⱕ 4,
where ␭dk
i denotes the distorted SVs of a subband of the
watermarked image. Hence, the four modified subbands are
obtained by M k = Ukc⌺kdVc⬘k where ⌺kd = diag共␭dk
i 兲, 1 ⱕ k ⱕ 4.
Then, the inverse DWT is applied using the four sets of the
modified DWT coefficients to produce the watermarked
image. The algorithm is invertible, and the watermark can
be extracted from the watermarked image.
Journal of Electronic Imaging
4 Proposed Watermarking Method
In this section, we provide the main steps of the proposed
watermark embedding and extraction algorithms, which are
also illustrated in the block diagrams shown in Figs. 3 and
4.
In Refs. 9 and 13, the authors prove experimentally that
embedding the watermark in the low- and high-frequency
components increases the robustness against attacks. For
example, embedding in low-frequency components increases the robustness to the attacks that have lowfrequency characteristics like filtering, lossy compression,
and geometric distortions. However, embedding in the
middle- and high-frequency components is typically less
robust against low-pass filtering and small geometric deformations of the image, but are extremely robust to noise
adding, contrast adjustment, gamma correction, and histogram manipulations.
Therefore, our goal of the proposed approach may be
described as applying multiple transforms to the cover image to embed the watermark many times in all the frequencies, which provides better robustness against attacks, amplifies the difficulty of destroying the watermark from all
the frequencies, and provides a high visual quality of the
watermarked image. The use of these multiple transforms is
motivated by the facts that the DWT is a powerful analysis
tool for multiresolution image representation in scalable
lossless coding and the FHT has significant advantage in
shorter processing time and ease of hardware implementation. Denote by C the cover image of size m ⫻ m and by W
the watermark image of size n ⫻ n with 2n = m.
4.1 Watermark Embedding Algorithm
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1. Apply DWT to the cover image C to obtain 4 subbands 共LL, LH, HL, HH兲.
2. Divide each subband into blocks of size ᐉ ⫻ ᐉ.
3. To each block L, apply the FHT: L̂ = 共1 / ᐉ 兲HBH,
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Fig. 4 Watermark extraction algorithm.
where H is the Hadamard matrix of order ᐉ.
4. Apply SVD to the watermark image: W = Uw⌺wVw⬘ ,
where ⌺w is a diagonal matrix of SVs.
5. Modify the DC components of the transformed
blocks L̂ using dwi = di0 + ␤␭wi, where di0 and dwi are the
original and the modified DC components, respectively, ␭wi are the SVs of W, and ␤ = ␣b / n, where ␣ is
a constant and b is the block number. Add the original
DC components to the secret key if the original covers are not available for the extraction algorithm.
6. Apply the inverse fast Hadamard transform 共IFHT兲 to
all the blocks to produce the watermarked transformed subbands.
7. Apply the inverse discrete wavelet transform 共IDWT兲
using the four watermarked subbands from Step 6 to
produce the watermarked image.
4.2 Watermark Extraction Algorithm
1. Apply the first three steps of the embedding algorithm to the watermarked image.
2. For each subband, extract the SVs from each block
using ␭wi = 共dwi − di0兲 / 共␤兲, where di0 and dwi are the
original and the watermarked DC components, respectively, and ␤ = ␣b / n where ␣ is a constant and b
is the block number. Note that the di0 are saved in the
secret key during the embedding stage.
3. Construct the four watermarks images using the SVs
extracted from the four subbands: Ŵk = Uw⌺̂kVw⬘ ,
Journal of Electronic Imaging
where Uw and Vw are the left and right singular vectors of W, respectively, and ⌺̂k is the extracted matrix
of SVs for each subband k 共1 ⱕ k ⱕ 4兲.
5
Experimental Results
In this section, we perform a number of experiments using
a variety of gray-scale images to show the effectiveness of
our proposed scheme. We conducted the experiments to test
the imperceptibility of the watermark and robustness
against attacks. Comparisons between our proposed scheme
and other transformed watermarking techniques were also
conducted. The images used in the experiments are of size
512⫻ 512 for the cover images and of size 256⫻ 256 for
the watermark images. The LL subband has the lowestfrequency components of the cover image and the highest
wavelet coefficients 共highest magnitude兲. The scaling factor
is chosen according to the wavelet coefficients of each subband. The HL, LH, and HH subbands have very similar
wavelet coefficients values; for simplicity, we used one
scaling factor for all middle- and high-frequency subbands.
Figure 5 shows the wavelet coefficients values for all
four subbands. In particular, the wavelet coefficients of the
LL subband are the highest among all the coefficients of the
other bands.
We used a strength factor ␤ = ␣b / n defined in terms of
the block position b, the subband width n, and the constant
scaling factor ␣ set to 0.7 for the LL subband and 0.2 for all
the other subbands. The scaling factors ␣ are chosen ex-
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Fig. 5 DWT coefficients of all the four subbands of the guy image: 共a兲 LL subband; 共b兲 LH subband; 共c兲
HL subband; and 共d兲 HH subband.
perimentally to repel as much attack as possible and also to
obtain a watermarked image with invisible degradation to
the human observer.
We split the subbands of size 256⫻ 256 into blocks of
size 16⫻ 16. In this case, the resulting number of blocks is
256. Hence, we may embed the watermark image once in
each subband. Figures 6共a兲 and 6共c兲 depict an example of
the cover image Guy and the watermark image Peppers,
respectively. The watermarked image Guy and one of the
extracted watermarks from the four subbands are shown in
Figs. 6共b兲 and 6共d兲, respectively.
5.1 Robustness
To verify the robustness of our proposed method, we applied different attacks to the watermarked image. The attacks include JPEG compression, Gaussian noise, multiplicative noise, Gaussian filter, deblurring with undersized
point-spread function 共PSF兲, deblurring with oversized
PSF, gamma correction, histogram equalization, cropping,
rescaling, sharpening, contrast adjustment, brightness
change, motion blurring, and morphological opening. It is
worth pointing out that the morphological opening may be
Journal of Electronic Imaging
Fig. 6 共a兲 Original image; 共b兲 watermarked image; 共c兲 visual watermark; and 共d兲 one of the four extracted watermarks from the four
subbands.
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Fig. 7 Illustration of the watermarked Guy image with different attacks.
used as an attack to destroy the watermark by separating
the touching objects in an image. Figure 7 shows the watermarked images with different kinds of attacks, and Fig. 8
depicts their corresponding best extracted watermarks. For
each attack, we extracted four watermarks from the four
subbands, and then we selected the best watermark that had
the highest correlation coefficient with the original watermark. The correlation coefficient ␳ between the original
watermark image W and the extracted watermark Ŵ is defined as
兺i,j=1 共Wij − W̄兲共Ŵij − Ŵ¯兲
冑共兺
n
i,j=1
共Wij − W̄兲
2
兲冉
兺i,j=1 共Ŵij − Ŵ¯兲2
n
冊
,
¯
where W̄ and Ŵ are the mean values of W and Ŵ, respectively. The label below each image in Fig. 8 shows the best
extracted watermark subband and the correlation coefficient
between the original and the best extracted watermark.
5.2 Invisibility
To measure the perceptual quality of the watermarked images, we calculate the peak signal-to-noise ratio 共PSNR兲,
which is used to estimate the quality of the watermarked
images in comparison with the original ones. The PSNR15
is defined as follows:
Journal of Electronic Imaging
冉冑 冊
MAXi
MSE
,
where MAXi = max兵Ĉij , 1 ⱕ i , j ⱕ m其, and the MSE is the
mean-squared error between the cover image C and the
watermarked image Ĉ:
m
m
1
MSE = 2 兺 兺 储 Cij − Ĉij 储2 .
m i=1 j=1
The PSNR experimental results as shown in Figs. 9 and 10
and Table 1 clearly indicate that the proposed method gives
high visual quality of the reconstructed image, and hence it
guarantees the watermark’s imperceptibility.
n
␳=
PSNR = 20 log10
5.3 Comparisons with Existing Techniques
We conducted several experiments to compare the robustness of the proposed method with related existing techniques, in particular with the pure SVD watermarking
scheme,10 DWT with SVD scheme,9 block-based
FHT-SVD,6 and block-based SVD.11 Figure 11 depicts the
correlation coefficient comparisons between our proposed
method and four different schemes with different attacks. In
these comparisons, we used the Peppers, Guy, Lena, and
Liftingbody as a cover images and the Cameraman, Peppers, Letter G, and MRI as watermark images. The results
obtained for all the attacks clearly indicate that our proposed method performs the best in terms of robustness
against the attacks.
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Fig. 8 Best extracted watermarks for different attacks. 共a兲 Gaussian noise ␴ = 0.3; 共b兲 multiplicative
uniform noise ␴ = 0.05; 共c兲 salt and pepper noise 4%; 共d兲 low-pass filter 关5 ⫻ 5兴; 共e兲 deblurring with
oversized point-spread function 共PSF兲; 共f兲 deblurring with undersized PSF; 共g兲 cropping 50%; 共h兲
brightness, −128; 共i兲 histogram equalization; 共j兲 motion blurring 45°; 共k兲 morphological opening; 共l兲
sharpening; 共m兲 rescaling 512− 256− 512; 共n兲 JPEG compression Q = 30%; 共o兲 gamma correction 0.6.
The negative correlation coefficients indicate that the extracted watermark image looks like the picture in the negative film 共the lighter areas appear dark, and vice versa兲. For
some watermark images like the letter G, the extracted watermark could be considered as detected in both cases: a
white letter G on a uniform black background 共high positive correlation兲 or a black letter G on a white background
共high negative correlation兲.
Table 1 lists the PSNRs of the proposed method and the
other existing methods for the same test images. Our algorithm clearly outperforms all other methods in terms of the
visual quality of the reconstructed watermarked image.
This better performance is, in fact, consistent with a variety
of images used for experimentation.
Fig. 9 PSNR between different cover images and their corresponding watermarked images with different strength factors. For the LH,
HL, and HH subbands, ␣ is fixed⫽0.2, and for the LL subband, we
used the following values: 共a兲 ␣ = 0.8; 共b兲 ␣ = 0.7; 共c兲 ␣ = 0.6; 共d兲 ␣
= 0.5; 共e兲 ␣ = 0.4; 共f兲 ␣ = 0.3; 共g兲 ␣ = 0.2; 共h兲 ␣ = 0.1.
Fig. 10 PSNR between different cover images and their corresponding watermarked images with different strength factors. For
the LL subband, ␣ is fixed ⫽ 0.7, and for the LH, HL, and HH subbands, we used the following values: 共a兲 ␣ = 0.85; 共b兲 ␣ = 0.5; 共c兲 ␣
= 0.4; 共d兲 ␣ = 0.3; 共e兲 ␣ = 0.2; 共f兲 ␣ = 0.1; 共g兲 ␣ = 0.05; 共h兲 ␣ = 0.01.
Journal of Electronic Imaging
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Fig. 11 Correlation coefficient comparison results between the proposed approach and other methods
for different cover and watermark images. 共a兲 Gaussian noise ␴ = 0.3; 共b兲 multiplicative uniform noise
␴ = 0.04; 共c兲 additive uniform noise ␴ = 0.4; 共d兲 salt and pepper’s 4%; 共e兲 JPEG compression Q = 30%;
共f兲 low-pass filter 关5 ⫻ 5兴; 共g兲 gamma correction 0.6; 共h兲 histogram equalization; 共i兲 sharpening; 共j兲
rescaling 512− 256− 512; 共K兲 corroding 50% right; 共l兲 brightness, −128; 共m兲 motion blurring 45°; 共n兲
morphological opening; 共o兲 deblurring with undersized PSF; 共p兲 deblurring with oversized PSF; 共q兲
deblurring with initial PSF.
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Table 1 PSNR comparison results. The boldface numbers indicate
the best PSNR for each example.
Watermarking
Technique
Peppers
Cameraman
Guy
Peppers
Lena
Letter G
Liftingbody
MRI
Proposed method
46.93
44.45
57.61
48.06
DWT+ SVD9
25.55
25.29
29.63
30.25
25.58
25.33
29.66
30.28
Block SVD11
43.78
42.31
55.47
45.92
Block FHT6
44.44
42.42
56.58
47.2
Pure SVD
10
5.4 Computational Complexity
The computational complexity of embedding a watermark
of size m / 2 ⫻ m / 2 into a cover image of size m ⫻ m is
obtained by the calculation of the DWT applied to the entire cover image and also by the calculation of the FHT to
small blocks of size 共ᐉ ⫻ ᐉ 兲, where ᐉ m. The DWT and
FHT costs are given by O共m log m兲 and O共ᐉ log ᐉ 兲, respectively. Therefore, the most expensive process of our
proposed scheme is the computation of the SVD of the
watermarked image, which is given by O共共m / 2兲3兲.
6 Conclusion
In this paper, we proposed a simple and robust image watermarking methodology for embedding a watermarked in
the transform domain. The key idea is to encode the SVs of
the watermarked image after applying the FHT to small
blocks computed from the four DWT subbands. The performance of the proposed method was evaluated through extensive experiments that clearly showed a better visual imperceptibility and an excellent resiliency against a wide
range of attacks.
Acknowledgments
The authors would like to thank the anonymous reviewers
for helpful and very insightful comments. This work was
supported in part by NSERC and by the Canada Research
Chair Foundation.
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Emad E. Abdallah received his BS degree
in computer science from Yarmouk University, Jordan, and his MS degree in computer science from the University of Jordan
in 2000 and 2004, respectively. He is currently pursuing his PhD degree in computer science at Concordia University,
Montreal, Canada. His research interests
include multimedia security and pattern
recognition.
A. Ben Hamza received his PhD degree in
electrical engineering from North Carolina
State University in 2003, where he worked
on computational imaging, 3-D object recognition, and information theory. He is currently an assistant professor at the Concordia Institute for Information Systems
Engineering 共CIISE兲 at Concordia University, Montreal, Canada. Prior to joining CIISE, he was a postdoctoral research associate at Duke University in North Carolina,
affiliated with both the Department of Electrical and Computer Engineering and the Fitzpatrick Center for Photonics and Communications Systems. His current research interests include multivariate
process control, computer graphics, multimedia security, and multisensor data processing.
Prabir Bhattacharya is currently a full professor at the Concordia Institute for Information Systems Engineering, Concordia
University, Montreal, Canada, where he
holds a Canada Research Chair, Tier 1.
From 1986 to 1999, he worked at the Department of Computer Science and Engineering, University of Nebraska–Lincoln,
USA. From 1999 to 2004, he worked as a
principal scientist at the Panasonic Information and Networking Technologies Lab
in Princeton, New Jersey. He received a DPhil from the University of
Oxford, UK, in 1979 and completed his undergraduate studies at the
University of Delhi, India. He is a fellow of the IEEE, the IAPR, and
the IMA. He is currently serving as the associate editor-in-chief of
the IEEE Transactions on Systems, Man and Cybernetics, Part B
共Cybernetics兲. Also, he is an associate editor of three other journals.
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