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3108 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 21, NO. 6, JUNE 2012 REFERENCES [1] E. J. Candès, J. Romberg, and T. Tao , “Robust uncertainty princ iples: Exactsignal recon struc tionfrom high ly inco mple te frequ encyinforma - tion,” IEEE Trans. Inf. Theory, vol. 52, no. 2, pp. 489–509, Feb. 2006. [2] D. L. Donoho, “Compressed sensing ,” IEEE Trans. Inf. Theory, vol. 52, no. 4, pp. 1289–1306, Apr. 2006. [3] R. G. Baraniuk, V. Cevher, M. Duarte, and C. Heg de, “Model-based compressiv e sensing, IEEE Trans. Inf. Theory, vol. 56, no. 4, pp. 1982–2001, Apr. 2010. [4] L. He and L. Carin, “Exploiting structure in wavelet-based Bayesi an compressive sensing,” IEEE Trans. Signal Process. , vol. 57, no. 9, pp. 3488–3497, Sep. 2009. [5] J. Huang, D. Metaxas, and T. Zhang , “Learning with structured spa r- sity,” in ACM Int. Conf. Proc. Ser., 2009, vol. 382, pp. 417–424. [6] S. Mun and J. E. Fowler, “Block comp ressed sensing of images usin g directional transforms,” in Proc. IEEE ICIP, 2009, pp. 3021–3024. [7] X. Wu, X. Zhang, and J. Wang, “Model-guided adapti ve recovery of compressive sensing,” in Proc. Data Compression Conf. , Snowbird, UT, 2009, pp. 123–132. [8] P. J. Garrigues, “Sparse coding mod els of natural images: Algorithms for efcient inference and learning of higher-order structure,” Ph.D. dissertation, Univ. California, Berkeley , CA, 2009. [9] Y. Kim, M. S. Nadar, and A. Bilgin, “Exploiting wavel et-domain de- pendencies in compressed sensing,” in Proc. Data Compression Conf., Snowbird, UT, 2010, p. 536. [10] S. G. Mallat  , A Wavelet T our of Signal Processing: The Sparse Way. Amsterdam, The Netherlands: Elsevier, 2009. [11] M. J. Wain wright and E. P. Simoncelli, “Scale mixtures of Gaussian s and the statistics of natural images,” Adv. Neural Inf. Process. Syst., vol. 12, no. 1, pp. 855–861, 2000. [12 ] S. G. Chang , B. Yu, andM. V ett erli, “Sp atiall y ada pti vewave letthres h- oldi ng with cont ext mode lingfor imag e deno isin g,  IEEE Tran s. Image Process., vol. 9, no. 9, pp. 1522–1531, Sep. 2000. [13] J. M. Shapiro, “Embed ded image coding using zerotrees of wav elet coefcients,” IEEE Tr ans. Sign al Pr ocess., vo l. 41, no. 12 , pp . 3445–3462, Dec. 1993. [14] A. Said and W. A. Pearlman, “A new, fast, and efcient image codec based on set partitioning in hierarchical trees,” IEEE Trans. Circuits Syst. Video Technol., vol. 6, no. 3, pp. 243–250, Jun. 1996. [15] D. S. Taub man and M. W. Marcellin  , JPEG2000: Image Compression Fundamentals, Standards, and Practice. Boston, MA: Kluwer, 2002. [16] Y. M. Lu andM. N. Do,“Samplin g sig nal s froma unionof sub spa ces ,  IEEE Signal Process. Mag., vol. 25, no. 2, pp. 41–47, Mar. 2008. [17] T. Blumensa th and M. E. Dav ies, “Sampli ng theorems for sign als from the union of nite-dimensional linear subspaces,” IEEE Trans.  Inf. Theory, vol. 55, no. 4, pp. 1872–1882, Apr. 2009. [18] C. Hegde, M. F. Duarte, and V. Cev her, “Compr essi ve sensing re- cov ery of spik e train s usin g a struc tured sparsity model, prese nted at the Signal Processing Adaptive Sparse Structured Representations Conf., Saint-Malo, France, 2009, Paper EPFL-CONF-151471. [19] M. N. Do and C. N. H. La, “Tree-bas ed majorize-maximize alg orithm for compressed sensing with sparse-tree prior,” Proc. IEEE Int. Work- shopon Comp utat iona l Adva nces in Mult i-Sen sor Ada ptiveProces sing (CAMPSAP 2007), pp. 129–132, 2007. [20] E. J. Candès, M. B. Wakin , and S. P. Boyd, “Enhancing spars ity by reweighted minimization,” J. Fourier Anal. Appl., vol. 14, no. 5, pp. 877–905, 2008. [21] I. Daubechies, R. DeV ore, M. Forn asier,and C. S. Günt ürk, “Itera tiv ely reweighted least squares minimization for sparse recovery ,” Commun. Pure Appl. Math., vol. 63, no. 1, pp. 1–38, Jan. 2010. [22] T. Blumensath and M. E. Davies, “Iterative thresh olding for sparse ap- proximations,” J. Fourier Anal. Appl., vol. 14, no. 5/6, pp. 629–654, 2008. [23] J. Portilla, V. Strela, M. J. Wain wright, and E. P. Simoncelli, “Image denoising using scale mixtures of Gaussians in the wavelet domain,”  IEEE Tran s. Image Proces s., vol. 12, no. 11, pp. 1338–1351, Nov. 2003. [24] Compressiv e Sensing Resources [Online]. Available: http://dsp.rice. edu/cs [25] R. Garg and R. Khandekar, “Gradient descen t with sparsication: An iterative algorithm for sparse recovery with restricted isometry prop- erty,” in Proc. 26th Annu. Int. Conf. Mach. Learn., 2009, pp. 337–344. [26] D. L. Dono ho and I. M. Johnst one, “Ideal spatial adaptation by wa velet shrinkage,” Biometrika , vol. 81, no. 3, pp. 425–455, Aug. 1994. [27] T. T. Do, T. D. Tran, and L. Gan, “Fast comp ressi ve sampli ng with structural ly random matri ces, in Pr oc. IEEE ICAS SP, 2008, pp. 3369–3372. [28] L. Gan, T. T. Do, and T. D. Tran , “Fast compres siv e imaging usin g scrambled block Hadamard ensemble,” in Proc. Eur. Signal Process. Conf. (EUSIPCO), Lausanne, Switzerland, 2008. [29] E. J. Candès, J. Romberg, and T. T ao, “Stable signal recovery from in- complete and inaccurate measurements,” Commun. Pure Appl. Math., vol. 59, no. 8, pp. 1207–1223, Aug. 2006. [30] l Magic Toolbox [Online]. Available: http://users.ece.gatech.edu/ ~justin/l1magic/ [31] NEST A toolbox [Online]. Available: http://acm.caltech.edu/~nesta/ [32] Spa rsi fy toolbox [On lin e]. Available: htt p:/ /www.personal. soton.ac.uk/tb1m08/ [33] USC-S IPI imag e data base [Onl ine]. Availab le: http ://si pi.us c.edu/data- base/ Scalable Coding of Encrypted Images Xinpeng Zhang  , Member , IEEE , Guoru i Feng, Y anli Ren, and Zhenxing Qian  Abstract—This paper proposes a novel scheme of scalable coding for encrypted images. In the encryption phase, the original pixel values are masked by a modulo-256 addition with pseudorandom numbers that are derived from a secret key. After decomposing the encrypted data into a downsampled subimage and several data sets with a multiple-resolution const ructi on, an enco der quant izes the subimage and the Hada mard coefcients of each data set to reduce the data amount. Then, the data of quantized subimage and coefcients are regarded as a set of bitstreams. At the receiver side, while a subimage is decrypted to provide the rough information of the original content, the quantized coefcients can be used to reconstruct the detailed content with an iteratively updating procedure. Becau se of the hiera rchi cal codin g mecha nism, the princ ipal origina l content with higher resolution can be reconstructed when more bitstreams are received.  Index Terms—Hadamard transform, image compression, image encryp- tion, scalable coding. I. INTRODUCTION In rece nt year s, encr ypted signal proce ssing has attracted con- siderable research interests [1]. The discrete Fourier transform and adaptive ltering can be implemented in the encrypted domain based on the homomorphic prope rti es of a cryptosys tem [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 signicant data of a plain signal is encrypted for content protec tio n, and the re mai ning data are used to car ry the additional message for copyright protection [5], [6]. With some Manuscript received July 26, 2011; revised October 29, 2011 and December 15, 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 the National Natural Science Foundation of China under Grant 61073190, Grant 61103181, and Grant 60832010, and in part by the Alexander von Humboldt Foundation. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Anthony Vetro. The auth ors are withSchool of Communic ationand Infor mati on Engineering, Shanghai University, Shanghai 200072, China (e-mail: [email protected]). Color versions of one or more of the gures in this paper are available online at http://ieeexplo re.ieee.org. Digital Object Identier 10.1109/TIP.2012.218 7671 1057-7149/$31.00 © 2012 IEEE http://ieeexploreprojects.blogspot.com
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3108 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 21, NO. 6, JUNE 2012

REFERENCES

[1] E. J. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles:Exactsignal reconstructionfrom highly incomplete frequencyinforma-tion,” IEEE Trans. Inf. Theory, vol. 52, no. 2, pp. 489–509, Feb. 2006.

[2] D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory, vol.52, no. 4, pp. 1289–1306, Apr. 2006.

[3] R. G. Baraniuk, V. Cevher, M. Duarte, and C. Hegde, “Model-based

compressive sensing,” IEEE Trans. Inf. Theory, vol. 56, no. 4, pp.1982–2001, Apr. 2010.[4] L. He and L. Carin, “Exploiting structure in wavelet-based Bayesian

compressive sensing,” IEEE Trans. Signal Process., vol. 57, no. 9, pp.3488–3497, Sep. 2009.

[5] J. Huang, D. Metaxas, and T. Zhang, “Learning with structured spar-sity,” in ACM Int. Conf. Proc. Ser., 2009, vol. 382, pp. 417–424.

[6] S. Mun and J. E. Fowler, “Block compressed sensing of images usingdirectional transforms,” in Proc. IEEE ICIP, 2009, pp. 3021–3024.

[7] X. Wu, X. Zhang, and J. Wang, “Model-guided adaptive recovery of compressive sensing,” in Proc. Data Compression Conf., Snowbird,UT, 2009, pp. 123–132.

[8] P. J. Garrigues, “Sparse coding models of natural images: Algorithmsfor efficient inference and learning of higher-order structure,” Ph.D.dissertation, Univ. California, Berkeley, CA, 2009.

[9] Y. Kim, M. S. Nadar, and A. Bilgin, “Exploiting wavelet-domain de-pendencies in compressed sensing,” in Proc. Data Compression Conf.,

Snowbird, UT, 2010, p. 536.[10] S. G. Mallat , A Wavelet Tour of Signal Processing: The Sparse Way.

Amsterdam, The Netherlands: Elsevier, 2009.[11] M. J. Wainwright and E. P. Simoncelli, “Scale mixtures of Gaussians

and the statistics of natural images,” Adv. Neural Inf. Process. Syst.,vol. 12, no. 1, pp. 855–861, 2000.

[12] S. G. Chang, B. Yu, andM. Vetterli, “Spatially adaptivewaveletthresh-olding with context modelingfor image denoising,” IEEE Trans. ImageProcess., vol. 9, no. 9, pp. 1522–1531, Sep. 2000.

[13] J. M. Shapiro, “Embedded image coding using zerotrees of waveletcoefficients,” IEEE Trans. Signal Process., vol. 41, no. 12, pp.3445–3462, Dec. 1993.

[14] A. Said and W. A. Pearlman, “A new, fast, and efficient image codecbased on set partitioning in hierarchical trees,” IEEE Trans. Circuits

Syst. Video Technol., vol. 6, no. 3, pp. 243–250, Jun. 1996.[15] D. S. Taubman and M. W. Marcellin , JPEG2000: Image Compression

Fundamentals, Standards, and Practice. Boston, MA: Kluwer, 2002.[16] Y. M. Lu andM. N. Do,“Sampling signals froma unionof subspaces,” IEEE Signal Process. Mag., vol. 25, no. 2, pp. 41–47, Mar. 2008.

[17] T. Blumensath and M. E. Davies, “Sampling theorems for signalsfrom the union of finite-dimensional linear subspaces,” IEEE Trans.

 Inf. Theory, vol. 55, no. 4, pp. 1872–1882, Apr. 2009.[18] C. Hegde, M. F. Duarte, and V. Cevher, “Compressive sensing re-

covery of spike trains using a structured sparsity model,” presentedat the Signal Processing Adaptive Sparse Structured RepresentationsConf., Saint-Malo, France, 2009, Paper EPFL-CONF-151471.

[19] M. N. Do and C. N. H. La, “Tree-based majorize-maximize algorithmfor compressed sensing with sparse-tree prior,” Proc. IEEE Int. Work-

shopon Computational Advances in Multi-Sensor AdaptiveProcessing(CAMPSAP 2007), pp. 129–132, 2007.

[20] E. J. Candès, M. B. Wakin, and S. P. Boyd, “Enhancing sparsity byreweighted minimization,” J. Fourier Anal. Appl., vol. 14, no. 5, pp.877–905, 2008.

[21] I. Daubechies, R. DeVore, M. Fornasier,and C. S. Güntürk, “Iterativelyreweighted least squares minimization for sparse recovery,” Commun.Pure Appl. Math., vol. 63, no. 1, pp. 1–38, Jan. 2010.

[22] T. Blumensath and M. E. Davies, “Iterative thresholding for sparse ap-proximations,” J. Fourier Anal. Appl., vol. 14, no. 5/6, pp. 629–654,2008.

[23] J. Portilla, V. Strela, M. J. Wainwright, and E. P. Simoncelli, “Imagedenoising using scale mixtures of Gaussians in the wavelet domain,”

 IEEE Trans. Image Process., vol. 12, no. 11, pp. 1338–1351, Nov.2003.

[24] Compressive Sensing Resources [Online]. Available: http://dsp.rice.edu/cs

[25] R. Garg and R. Khandekar, “Gradient descent with sparsification: Aniterative algorithm for sparse recovery with restricted isometry prop-erty,” in Proc. 26th Annu. Int. Conf. Mach. Learn., 2009, pp. 337–344.

[26] D. L. Donoho and I. M. Johnstone, “Ideal spatial adaptation by wavelet

shrinkage,”Biometrika

, vol. 81, no. 3, pp. 425–455, Aug. 1994.

[27] T. T. Do, T. D. Tran, and L. Gan, “Fast compressive sampling withstructurally random matrices,” in Proc. IEEE ICASSP, 2008, pp.3369–3372.

[28] L. Gan, T. T. Do, and T. D. Tran, “Fast compressive imaging usingscrambled block Hadamard ensemble,” in Proc. Eur. Signal Process.Conf. (EUSIPCO), Lausanne, Switzerland, 2008.

[29] E. J. Candès, J. Romberg, and T. Tao, “Stable signal recovery from in-complete and inaccurate measurements,” Commun. Pure Appl. Math.,

vol. 59, no. 8, pp. 1207–1223, Aug. 2006.[30] l Magic Toolbox [Online]. Available: http://users.ece.gatech.edu/ ~justin/l1magic/ 

[31] NESTA toolbox [Online]. Available: http://acm.caltech.edu/~nesta/ [32] Sparsify toolbox [Online]. Available: http://www.personal.

soton.ac.uk/tb1m08/ [33] USC-SIPI image database [Online]. Available: http://sipi.usc.edu/data-

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: [email protected]).

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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IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 21, NO. 6, JUNE 2012 3109

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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3110 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 21, NO. 6, JUNE 2012

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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IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 21, NO. 6, JUNE 2012 3111

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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3112 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 21, NO. 6, JUNE 2012

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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IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 21, NO. 6, JUNE 2012 3113

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.

REFERENCES

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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: [email protected]; [email protected]; [email protected]).

L. Stankovic and V. Stankovic are with Department of Electronic and Elec-trical Engineering, University of Strathclyde, Glasgow G1 1XW, U.K. (e-mail:[email protected]; [email protected]).

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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