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base/
Scalable Coding of Encrypted Images
Xinpeng Zhang , Member, IEEE , Guorui Feng, Yanli Ren, and
Zhenxing Qian
Abstract—This paper proposes a novel scheme of scalable coding forencrypted images. In the encryption phase, the original pixel values are
masked by a modulo-256 addition with pseudorandom numbers that arederived from a secret key. After decomposing the encrypted data into adownsampled subimage and several data sets with a multiple-resolutionconstruction, an encoder quantizes the subimage and the Hadamardcoefficients of each data set to reduce the data amount. Then, the data of quantized subimage and coefficients are regarded as a set of bitstreams.At the receiver side, while a subimage is decrypted to provide the roughinformation of the original content, the quantized coefficients can be usedto reconstruct the detailed content with an iteratively updating procedure.
Because of the hierarchical coding mechanism, the principal original
content with higher resolution can be reconstructed when more bitstreamsare received.
Index Terms—Hadamard transform, image compression, image encryp-tion, scalable coding.
I. INTRODUCTION
In recent years, encrypted signal processing has attracted con-
siderable research interests [1]. The discrete Fourier transform and
adaptive filtering can be implemented in the encrypted domain based
on the homomorphic properties of a cryptosystem [2], [3], and a
composite signal representation method can be used to reduce the size
of encrypted data and computation complexity [4]. In joint encryption
and data hiding, a part of significant data of a plain signal is encryptedfor content protection, and the remaining data are used to carry
the additional message for copyright protection [5], [6]. With some
Manuscript received July 26, 2011; revised October 29, 2011 and December15, 2011; accepted January 26, 2012. Date of publication February 13, 2012;date of current version May 11, 2012. This work was supported in part by theNational Natural Science Foundation of China under Grant 61073190, Grant61103181, and Grant 60832010, and in part by the Alexander von HumboldtFoundation. The associate editor coordinating the review of this manuscript andapproving it for publication was Dr. Anthony Vetro.
The authors are withSchool of Communicationand Information Engineering,Shanghai University, Shanghai 200072, China (e-mail: xzhang@shu.edu.cn).
Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TIP.2012.2187671
1057-7149/$31.00 © 2012 IEEE
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buyer–seller protocols [7], [8], the fingerprint data are embedded into
an encrypted version of digital multimedia to ensure that the seller
cannot know the buyer’s watermarked version while the buyer cannot
obtain the original product.
A numberof works on compressing encrypted imageshavebeen also
presented. When a sender encrypts an original image for privacy pro-
tection, a channel provider without the knowledge of a cryptographic
key and original content may tend to reduce the data amount due to thelimited channel resource. In [9], the compression of encrypted data is
investigated with the theory of source coding with side information at
the decoder, and it is pointed out that the performance of compressing
encrypted data may be as good as that of compressing nonencrypted
data in theory. Two practical approaches are also given in [9]. In the
first one, the original binary image is encrypted by adding a pseudo-
random string, and the encrypted data are compressed by finding the
syndromes of low-density parity-check (LDPC) channel code. In the
second one, the original Gaussian sequence is encrypted by adding
an independent identically distributed Gaussian sequence, and the en-
crypted data are quantized and compressed as the syndromes of trellis
code. While Schonberg et al. [10] study the compression of encrypted
data for memoryless and hidden Markov sources using LDPC codes,
Lazzeretti and Barni [11] present several lossless compression methods
for encrypted gray and color images by employing LDPC codes into
various bit planes. In [12], the encrypted image is decomposed in a
progressive manner, and the data in most significant planes are com-
pressed using rate-compatible punctured turbo codes. Based on local
statistics of a low-resolution version of the image, the original content
can be perfectly reconstructed. By extending the statistical models to
video, some algorithms for compressing encrypted video are presented
in [13]. In most of aforementioned schemes, the syndrome of channel
code is exploited to generate the compressed data in a lossless manner.
Furthermore, several methods for lossy compressing encrypted im-
ages have been developed. In [14], a compressive sensing mechanism
is introduced to achieve the lossy compression of encrypted images,
and a basis pursuit algorithm is used to enable joint decompressionand decryption. In [15], the original gray image is encrypted by pixel
permutation; then, the encrypted data are compressed by discarding the
excessively rough and fine information of coefficients generated from
orthogonal transform. When having the compressed data and the per-
mutation way, a receiver can reconstruct the principal content of the
original image by retrieving the values of coefficients. However, the
rate–distortion performance in [14] is low, and there is a leakage of
statistical information in [15] since only the pixel positions are shuf-
fled and the pixel values are not masked in the encryption phase.
This paper proposes a novel scheme of scalable coding for encrypted
gray images. Although there have been a lot of works on scalable
coding of unencrypted images/videos [16], [17], the scalable coding
of encrypted data has not been reported. In the encryption phase of
the proposed scheme, the pixel values are completely concealed so
that an attacker cannot obtain any statistical information of an original
image. Then, the encrypted data are decomposed into several parts,
and each part is compressed as a bitstream. At the receiver side with
the cryptographic key, the principal content with higher resolution can
be reconstructed when more bitstreams are received.
II. PROPOSED SCALABLE CODING SCHEME
In the proposed scheme, a series of pseudorandom numbers derived
from a secret key are used to encrypt the original pixel values. After
decomposing the encrypted data into a subimage and several data
sets with a multiple-resolution construction, an encoder quantizes
the subimage and the Hadamard coefficients of each data set toeffectively reduce the data amount. Then, the quantized subimage
and coefficients are regarded as a set of bitstreams. When having the
encoded bitstreams and the secret key, a decoder can first obtain an
approximate image by decrypting the quantized subimage and then
reconstructing the detailed content using the quantized coefficients
with the aid of spatial correlation in natural images. Because of the
hierarchical coding mechanism, the principal original content with
higher resolution can be reconstructed when more bitstreams are
received.
A. Image Encryption
Assume that the original image is in an uncompressed format and
that the pixel values are within [0, 255], and denote the numbers of
rows and columns as and and the pixel number as
. Therefore, the bit amount of the original image is . The
content owner generates a pseudorandom bit sequence with a length
of . Here, we assume the content owner and the decoder has the
same pseudorandomnumber generator (PRNG)and a sharedsecret key
used as the seed of the PRNG. Then, the content owner divides the
pseudorandom bit sequence into pieces, each of which containing 8
bits, and converts each piece as an integer number within [0, 255]. An
encrypted image is produced by a one-by-one addition modulo 256 asfollows:
(1)
where represents the gray values of pixels at positions ,
represents the pseudorandom numbers within [0, 255] gener-
ated by the PRNG, and represents the encrypted pixel values.
Fig. 1 gives an original image Lena and its encrypted version.
Clearly, the encrypted pixel values are pseudorandom
numbers since values are pseudorandom numbers. It is well
known that there is no probability polynomial time (PPT) algorithm to
distinguish a pseudorandom number sequence and a random numbersequence until now. Therefore, any PPT adversary cannot distinguish
an encrypted pixel sequence and a random number sequence. That
is to say, the image encryption algorithm that we have proposed is
semantically secure against any PPT adversary.
B. Encrypted Image Encoding
Although an encoder does not know the secret key and the original
content, he can still compress the encrypted data as a set of bitstreams.
The detailed encoding procedure is as follows.
First, the encoder decomposes the encrypted image into a series of
subimages and data sets with a multiple-resolution construction. The
subimage at the th level is generated by downsampling
the subimage at the th level as follows:
(2)
where is just the encrypted image and is the number of decom-
position levels. In addition, the encrypted pixelsthat belong to but
do not belong to form data set as follows:
(3)
That means each is decomposed into and , and
the data amount of is three times of that of . After the
multiple-level decomposition, the encrypted image is reorganized as, and .
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Fig. 1. (a) Original image Lena and (b) its encrypted version.
For the subimage , the encoder quantizes each value using a
step as follows:
(4)
where the operator takes an integer toward minus infinity and
(5)
Here, is an integer shared by the encoder and the decoder, and its
value will be discussed later. Clearly
(6)
Then, thedata of areconverted into a bitstream, which is denoted
as BG. The bit amount of BG is
(7)
For each data set , the encoder permutes and
divides encrypted pixels in it into groups, each of which con-
taining pixels . In this way, the
pixels in the same group scatter in the entire image. The permutationway is shared by the encoder and the decoder, and the values of
will be discussed later. Denote the encrypted pixels of the th group
as , and perform the
Hadamard transform in each group as follows:
......
(8)
where is a Hadamard matrix made up of 1 or 1. That
implies the matrix meets
(9)
where is a transpose of , is an identity matrix, and
must be a multiple of 4. For each coefficient , calculate
(10)
where
round (11)
and round finds the nearest integer. In (10), the remainder of
modulo 256 is quantized as integer , and the quantization stepsare approximately proportional to square roots of . Then,
at different levels are converted into bitstreams, which are denoted as
BS . Since
(12)
and the number of at the th level is , the bit amount of
BS is
(13)
The encoder transmits the bitstreams with an order of
BG BS BS BS . If the channel bandwidth is
limited, the latter bitstreams may be abandoned. A higher resolution
image can be reconstructed when more bitstreams are obtained at the
receiver side. Here, the total compression ratio , which is a ratio
between the amount of the encoded data and the encrypted image
data, is
(14)
C. Image Reconstruction
With the bitstreams and the secret key, a receiver can reconstruct the
principal content of the original image, and the resolution of the re-
constructed image is dependent on the number of received bitstreams.
While BG provides the rough information of the original content, BS
can be used to reconstruct the detailed content with an iteratively up-dating procedure. The image reconstruction procedure is as follows.
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When having the bitstream BG, the decoder may obtain the values
of and decrypts them as a subimage, i.e.,
(15)
where are derived from the secret key.
If the bitstreams BS are also received, an image
with a size of will be reconstructed. First,
upsample the subimage by factor to yield an
image as follows:
(16)
and estimate the values of other pixels according to the pixel values in
(16) using a bilinear interpolation method. Then, the interpolated pixels
are reorganized as data sets with multiple-resolution construction, and
the data in each set are permuted and divided into a series of groups.
Denote the interpolated pixel values of the th group at the th level as
and their
corresponding original pixel values as .
The errors of interpolated values are
(17)
Define the encrypted values of as
(18)
where are pseudorandom numbers derived from the secret key
and corresponding to . Then
(19)
We also define
......
(20)
where is a Hadamard matrix made up of 1 or 1.
Since only the addition and subtraction are involved in the Hadamard
transform
......
.
..(21)
That means the transform of errors in the plain domain is equivalent
to the transform of errors in the encrypted domain with the modular
arithmetic. Denoting
......
(22)
we have
(23)
With the bitstreams BS , the values of can
be retrieved, which provide the information of . Therefore, the
receiver may use an iterative procedure to progressively improve the
quality of the reconstructed image by updating the pixel values ac-
cording to . The detailed procedure is as follows.
1) For each group , calculate
and using (18) and (22).
2) Calculate
(24)
if
if (25)
are the differences between the values consistent with the
corresponding and . Then, considering as
an estimate of , modify the pixel values of each group as
follows:
..
.
..
.
..
.
(26)
and enforce the modified pixel values into [0, 255] as follows:
if
if
if
(27)
3) Calculate the average energy of difference due to the modification
as follows:
(28)
If is not less than a given threshold of 0.10, for each pixel ,
after putting it back to the position in the image and regarding the av-
erage value of its four neighbor pixels as its new value , go to
Step 1. Otherwise, terminate the iteration, and output the image as a
final reconstructed result.
In the iterative procedure, while the decrypted pixels are
used to give an initial estimation of other pixels, the values of in
bitstreams BS provide more detailed information to produce the final
reconstructed result with satisfactory quality. In Step 2, by estimating
according to , the pixel values are modified to lower
the reconstruction errors. If the image is uneven and is big, theabsolute value of actual may be more than 128 due to the
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Fig. 2. Reconstructed Lena using BS , BS BS , BS BS BS , and BS BS BS BS . The values of PSNR in (a), (b), (c), and(d) when regarding the corresponding downsampled versions of original Lena as references are 38.4, 34, 37.1, and 38.4 dB.
TABLE ICOMPRESSION RATIOS, PSNR IN RECONSTRUCTED RESULTS AND ITERATION NUMBERS WITH DIFFERENT
WHEN , , , AND WERE USED FOR LENA AND MAN
error accumulation in a group, so that in (25) may be not close
to . To avoid this case, we let decrease with a increasing
since the spatial correlation in a subimage with lower resolution is
weaker. For instance, , , and for .
Furthermore, in Step 3, thevalueof each pixel is assigned as theaverage
of its four neighbors to further approach its original value. Although
the estimate of a certain pixel may be very different from its original
value, the updating operation in Step 3 can effectively lower the error
on the pixel since its neighbors are probably modified well. At last,we terminate the iterative procedure when the reconstruction quality
is not improved further. Here, the small threshold of 0.10 ensures the
convergence of iterative procedure.
III. EXPERIMENTAL RESULTS AND DISCUSSION
Two test images Lena and Man that are sized 512 512 were used
as the original images in the experiment. We let and encoded
the encrypted images using , , , and
to produce the bitstreams BG, BS , BS , and BS . In
this case, the total compression ratio . Fig. 2 gives the
reconstructed Lena using BG , BG BS , BG BS BS
and BG BS BS BS , respectively. Reconstructed results
with higher resolution were obtained when more bitstreams were used.
When regarding the corresponding downsampled versions of originalimages as reference, the values of PSNR in reconstructed results are
denoted as PSNR , PSNR , PSNR , and PSNR . While the PSNR
values of Lena are 38.4, 34, 37.1, and 38.4 dB, those of Man are 38.4,
31.9, 33.9, and 37.1 dB. In addition, the iterative updating procedure
significantly improved the reconstruction quality. For example, while
PSNR in an interpolated 512 512, Lena is 23.9 dB; this value in the
final reconstructed image is 38.4 dB with a gain of 14.5 dB.
Table I lists the compression ratios; the PSNR in reconstructed re-
sults and the numbers of iterations with respect to different when
, , , and were used for im-ages Lena and Man. All the encryption, encoding and reconstruction
procedures were finished in several seconds by a personal computer.
When the value of is larger, the compression ratio is higher, and the
reconstruction quality are better since provide more detailed in-
formation. As there is less texture/edge content in Lena than Man, the
quality of reconstructed Lena is better than that of Man. In addition,
the larger corresponds to the lower compression ratio and more
detailed . When we changed from (4, 8, 12)
to (4, 12, 32), the compression ratio decreased from 0.318 to 0.283,
and the value of PSNR in reconstructed Lena and Man were 37.8
and 35.2 dB, respectively. Compared with the results in Table I, the
new -PSNR performance of Lena is better, whereas that of Man
is worse. The reason is that Lena is smoother than Man. For Lena, the
larger was helpful to uniformly distribute the errors on pixels intothe Hadamard coefficients, and most of still fell into [ 128,
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Fig. 3. Performance of the proposed scheme with different .
Fig. 4. Performance comparison of several compression methods.
128], so that the quality of reconstructed result was better. For Man,
the excessively large caused more with absolute values
bigger than 128, leading to a lower reconstruction quality. Fig. 3 gives
the -PSNR curves with different values of . When an encrypted
image is decomposed within more levels, more data are involved in
quantization and compression; therefore, the -PSNR performance
is better, and more iterations for image reconstruction are required. It is
also shown that the performance improvement is not significant when
using a higher more than 3.We also compare the proposed scheme with the previous methods
and unencrypted JPEG compression in Fig. 4. Because it is difficult
to completely remove the spatial data redundancy by the operations in
the encrypted domain, the rate–distortion performance of the proposed
scheme is significantly lower than that of JPEG compression. On the
other hand, the proposed scheme outperforms the method in [15]. With
the method in [15], the original image is encrypted by pixel permu-
tation, which implies an attacker without the knowledge of the secret
key can know the original histogram from an encrypted image. In this
proposed scheme, the original values of all pixels are encrypted by a
modulo-256 addition with pseudorandom numbers, leading to semantic
security. That means the attacker cannot obtain the original histogram
from an encrypted image. In addition, the method in [15] does not sup-
port the function of scalable coding. Liu et al. [12] proposed a losslesscompression method for encrypted images in a bit-plane based fashion.
By discarding the encrypted data in the lowest bit planes, the method in
[12] can be extended to achieve lossy compression. The performance
of the extended method, which is also given in Fig. 4, is better than
that of the proposed scheme. However, a decoder with higher compu-
tation complexity and the decoder’s feedback for sending rate of each
bit plane are required in the method extended from [12]. That means
the proposed scheme is more suitable for real-time decompression and
some scenarios without feedback channel.
IV. CONCLUSION
This paper has proposed a novel scheme of scalable coding for
encrypted images. The original image is encrypted by a modulo-256
addition with pseudorandom numbers, and the encoded bitstreams
are made up of a quantized encrypted subimage and the quantized
remainders of Hadamard coefficients. At the receiver side, while the
subimage is decrypted to produce an approximate image, the quantized
data of Hadamard coefficients can provide more detailed information
for image reconstruction. Since the bitstreams are generated with a
multiple-resolution construction, the principal content with higher
resolution can be obtained when more bitstreams are received. The
lossy compression and scalable coding for encrypted image with better
performance deserves further investigation in the future.
ACKNOWLEDGMENT
The authors would like to thank the anonymous reviewers for their
valuable comments.
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Onboard Low-Complexity Compression of
Solar Stereo Images
Shuang Wang, Lijuan Cui, Samuel Cheng, Lina Stankovic, andVladimir Stankovic
Abstract—We propose an adaptive distributed compression solutionusing particle filtering that tracks correlation, as well as performingdisparity estimation, at the decoder side. The proposed algorithm istested on the stereo solar images captured by the twin satellites system
of NASA’s Solar TErrestrial RElations Observatory (STEREO) project.Our experimental results show improved compression performance w.r.t.
to a benchmark compression scheme, accurate correlation estimation byour proposed particle-based belief propagation algorithm, and significantpeak signal-to-noise ratio improvement over traditional separate bit-planedecoding without dynamic correlation and disparity estimation.
Index Terms—Distributed source coding, image compression, multiview
imaging, remote sensing.
I. INTRODUCTION
Onboard data processing has been a challenging task in remote
sensing applications due to severe computational limitations of
onboard equipment. This is especially the case in deep-space ap-
plications where mission spacecraft are collecting a vast amount of
images. In such emerging applications, efficient low-complexity image
compression is a must. While conventional solutions such as JPEG
have been used in many prior missions, the demand for increasing
image volume and resolution, as well as increased space resolution
Manuscript received July 17, 2011; revised November 28, 2011; acceptedJanuary 17, 2012. Date of publication February 13, 2012; date of current ver-sion May 11, 2012. This work was supported in part by the National ScienceFoundation underGrantCCF 1117886. This paper waspresented in part at IEEEInternational Conference on Image Processing (ICIP-2011), Brussels, Belgium,September2011. The associateeditor coordinating the review of thismanuscriptand approving it for publication was Prof. Brian D. Rigling.
S. Wang, L. Cui, and S. Cheng are with School of Electrical and ComputerEngineering, The University of Oklahomaat Tulsa, Tulsa, OK 74135-2512 USA(e-mail: shuangwang@ou.edu; lj.cui@ou.edu; samuel.cheng@ou.edu).
L. Stankovic and V. Stankovic are with Department of Electronic and Elec-trical Engineering, University of Strathclyde, Glasgow G1 1XW, U.K. (e-mail:lina.stankovic@eee.strath.ac.uk; vladimir.stankovic@eee.strath.ac.uk).
Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TIP.2012.2187669
and wide-swath imaging, calls for larger coding efficiency at reduced
encoding complexity.
NASA’s Solar TErrestrial RElations Observatory (STEREO) is pro-
viding groundbreaking images of the Sun using two space-based obser-
vatories.1 These images aim to reveal the processes in the solar surface
(photosphere) through the transition region into the corona and provide
the 3-D structure of coronal mass ejections (CMEs).
A variety of image compression tools are currently used in deep-space missions, ranging from Rice and lossy wavelet-based compres-
sion tools (used in PICARD mission by CNES 2009), discrete cosine
transform (DCT) + scalar quantization + Huffman coding (Clementine,
NASA 1994), and ICER (a low-complexity wavelet-based progressive
compression algorithm used in Mars mission, NASA 2003) to (12-bit)
JPEG-baseline (Trace NASA1998, Solar-B JAXA2006) [1]. The key
characteristics of these algorithms are relatively low encoding power
consumption, coding efficiency, and error resilience features. Note that
all current missions, including STEREO, use 2-D monoview image
compression trading off computational cost and compression perfor-
mance. Since STEREO images are essentially multiview images, with
high interview correlation, current compression tools do not provide an
optimum approach. In this paper, we propose a distributed multiview
image compression (DMIC) scheme for such emerging remote sensing
setups.
When an encoder can access images from multiple views, a joint
coding scheme [2] achieves higher compression performance than
schemes with separate coding. However, due to the limited computing
and communication power of space imaging systems, it is not feasible
to perform high-complexity power-hungry onboard joint encoding of
captured solar images. Although, intuitively, this restriction of separate
encoding seems to compromise the compression performance, dis-
tributed source coding (DSC) theory [3], [4] proves that independent
encoding can be designed as efficiently as joint encoding as long as
joint decoding is allowed.
The proposed DMIC image codec is characterized by low-com-
plexity image encoding and relatively morecomplex decoding meant tobe performed on theground. Theproposedschemeextends ourprevious
work [5], where a joint bit-plane decoder is described, which integrates
particle filtering with standard belief propagation (BP) decoding to
perform inference on a single joint 2-D factor graph. In [5], the pro-
posed decoding method is used in the context of monoview coding of
natural video based on DCT-based distributed video coding (DVC) [6].
In this paper, we extend the scheme to multiview image compression
to further reduce the complexity we work in the pixel domain. The key
contributions of this paper can be summarized as follows.
• An adaptive distributed multiview image decoding scheme, which
can estimate the blockwise correlation and disparity change be-
tween two correlated images.
• A BP decoder with integrated particle filtering to estimate block-
wise correlation changes in the pixel domain. This extends our
previous work [5], [7] from 1-D correlation estimation to 2-D and
from time-varying correlation estimation to spatially varying cor-
relation estimation.
• A joint bit-plane decoder (as compared with the traditional sep-
arate bit-plane decoder [8]), which allows the estimation of the
correlation and the disparity between two pixels directly rather
than just the correlation between a corresponding pair of bits of
the pixels as in [5].
We test our lossy DMIC setup with grayscalestereo solar images ob-
tained from NASA’s STEREO mission to demonstrate high compres-
1[Online.] Available: http://www.nasa.gov/mission_pages/stereo/mis-sion/index.html
1057-7149/$31.00 © 2012 IEEE
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