selective image encryption based on pixels of interest and singular value decomposition

Digital Signal Processing 22 (2012) 648–663

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Digital Signal Processing

www.elsevier.com/locate/dsp

Selective image encryption based on pixels of interest and singular valuedecomposition

Gaurav Bhatnagar ∗, Q.M. Jonathan Wu

Department of Electrical and Computer Engineering, University of Windsor, Windsor, Ontario, N9B 3P4, Canada

a r t i c l e i n f o a b s t r a c t

Article history:Available online 23 March 2012

Keywords:Selective encryptionSpace filling curvePixels of interestNon-linear chaotic mapSingular value decomposition

In this paper, an efficient yet simple selective encryption technique is proposed based on Saw-Toothspace filling curve, pixels of interest, non-linear chaotic map and singular value decomposition. The coreidea of the proposed scheme is to scramble the pixel positions by the means of Saw-Tooth space fillingcurve followed by the selection of significant pixels using pixels of interest method. Then the diffusionprocess is done on the significant pixels using a secret image key obtained from non-linear chaotic mapand singular value decomposition. Finally, a reliable decryption process is proposed to construct originalimage from the encrypted image. The analysis and experimental results show that the proposed schemecan achieve various purposes of selective encryption and is computationally secure.

© 2012 Elsevier Inc. All rights reserved.

1. Introduction

With the ripening in the field of communication and internettechnology, multimedia transmission over networks and storage onweb servers have become a vital part of it. However, this con-venience also causes substantial decrease in multimedia security.There are several aspects in multimedia security including copy-right protection, authentication, confidentiality and access con-trol. Generally, the foremost aspect is copyright protection whichis addressed by digital watermarking which essentially embedsa mark, called watermark, into the original multimedia and ex-tracts it later whenever ownership conflict needs to be resolved[1]. On the other hand, content confidentiality and access controlare coming next in the queue and are addressed by encryptiontechniques [2,3]. The encryption is the process by which the mul-timedia is changed into another form in intelligible manner. Thisprocess is called the encryption process and gives the encrypteddata/cipher data. The process of recovering original data from en-crypted data is called decryption process and the combination ofthese is called encryption techniques. Till now, various encryp-tion techniques have been proposed and widely used, such as DES,RSA, IDEA or AES, most of which are used for text or binary data.Since, these encryption techniques are developed for textual data,therefore, are not appropriate to implement them directly on mul-timedia [3]. Because the multimedia data needs to be processed inits entirety before users can gain any insight and has high redun-dancy. Therefore, new encryption techniques need to be developed

* Corresponding author.E-mail addresses: [email protected] (G. Bhatnagar), [email protected]

(Q.M. Jonathan Wu).

1051-2004/$ – see front matter © 2012 Elsevier Inc. All rights reserved.http://dx.doi.org/10.1016/j.dsp.2012.02.005

with the consideration of structural and statistical properties ofmultimedia content. Hence, the present work focuses on protect-ing the confidentiality and achieving access control of images. Theimages are selected because at most all the advanced sensors oftoday, for example, optical cameras, millimetre wave (MMW) cam-eras, infrared cameras, x-ray imagers, radar imagers etc. providethe information in the form of images.

In image transmission, basically there are two strategies de-pending on whether encryption is done in compressed or un-compressed domain. If encryption is applied to an image beforecompression, the statistical and structural characteristics of theoriginal image could be significantly changed, resulting in muchreduced compressibility. On the other hand, if compression is ap-plied first then it reduces computational overhead but it may de-stroy the syntax of the encoded bitstream and offers less secrecy.In this work the stressed motive is the security, hence the mainconcentration is on first strategy. To overcome these situations, se-lective or partial encryption [4–12], in which only some portion ofmultimedia data are encrypted, has been introduced to multime-dia encryption. Therefore, selective encryption is useful especiallyin real-time applications or resource-limited computing environ-ments. The stressed motive of encryption techniques is to giveconfidentiality and hence the main concentration is on selectiveencryption in uncompressed domain.

The core idea behind selective encryption techniques is to iden-tify significant and insignificant regions or pixels from the imagefollowed by the encryption of significant data. This is due to thatfact that a tiny error in the significant part will cause substantialchange in the image, while modifications in the insignificant partwill not induce much effect on the image. As a result, only thesignificant part of the data needs to be encrypted which furtherreduces the computational overhead. Fig. 1 shows the difference

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G. Bhatnagar, Q.M. Jonathan Wu / Digital Signal Processing 22 (2012) 648–663 649

Fig. 1. Difference between conventional and selective encryption. (a) Conventional, (b) selective encryption scenario I, (c) selective encryption scenario II.

between conventional and selective encryption techniques. Gener-ally, the encryption is divided into two phases viz. scrambling theimage (confusion) and encrypting the scrambled image (diffusion).Confusion casts the image elements into mix-up by changing theposition of pixel in such a way that the original image is not rec-ognizable. But the original image can be obtained by performingreverse operations. Hence to make process more complicated andto enhance the security, confused image then undergoes secondphase. This phase essentially changes the pixels values in orderto break the strong relationship among the pixels and is calleddiffusion. The confusion is done by various reversible techniquesbased on magic square transform, chaos system, gray code etc. Indiffusion, the confused image is then passed through some crypto-graphic algorithms like SCAN based methods [13,14], chaos basedmethods [15–20], tree structure based methods [21,22] and othermiscellaneous methods [23]. Based on the order of confusion thereare two scenarios for selective encryption which are depicted inFig. 1(b), (c). In Fig. 1(b) the confusion is locally i.e. only appliedto the significant data where as it is globally i.e. applied to wholeimage, in second scenario shown in Fig. 1(c). According to princi-ple, the second scenario is better than the first because it will givemore modification to the image rather than first scenario.

In this paper, a novel selective image encryption technique ispresented in un-compressed domain based on the space fillingcurves (SFC), pixels of interest (POI) and singular value decom-position (SVD). The first phase i.e. confusion is done globally byproposed Saw-Tooth SFC followed by the characterization in signif-icant and insignificant parts using POI. The second phase, diffusion,is done on significant part by singular value decomposition. Forthis purpose, singular values of a random matrix, generated fromnon-linear chaotic map, are computed via SVD. Hankel matrix isthen created using computed singular values and it is again de-composed into singular values, left and right singular vectors. Now,a secret image is obtained using left and right singular vectors.Using the secret key diffusion process is accomplished and finallyencrypted image is sent to insecure network channel. The key ofproposed selective encryption technique is secret image. For thisthe initial parameters of non-linear chaotic map are stored at boththe encryption and decryption ends, acted as keys. The several ex-periments are carried out in order to show the effectiveness androbustness of the proposed encryption technique.

The rest of this paper is organized as follows: in Section 2,we briefly introduce the mathematical preliminaries i.e. non-linearchaotic map and singular value decomposition. Sections 3 and 4

describe the proposed Saw-Tooth space filling curve and pixels ofinterest method. Section 5 describes the proposed selective en-cryption scheme. Several experimental results are demonstrated inSection 6 to show the performance of the proposed scheme andfinally we conclude the present work in Section 6.

2. Mathematical preliminaries

2.1. Non-linear chaotic map

Chaos theory is a research area in mathematics, physics andphilosophy which essentially tells the behavior of dynamical sys-tems. Usually, chaos refers to a very specific kind of unpredictabil-ity i.e. deterministic behavior which is very sensitive to its initialconditions [24]. The meaning of used word deterministic is thatthe future behavior of system is fully determined by its initial con-ditions, sometimes with random element involved. As we knowthat the dynamical system refers to the mathematical formaliza-tion for any fixed rule. Generally, this mathematical formulation isdenoted by a map which gives the behavior of the system and iscoined as chaotic map. In this subsection, the introduction to non-linear chaotic one-dimensional maps is given.

Mathematically, chaotic map is the mapping of a space ontoitself i.e.

f : S → S, S ⊂R (1)

where the set S will be either [0,1] or [−1,1], in most of cases.The map is written in the form of recurrence relation as

xt+1 = f (xt,μ), t = 0,1,1, . . . (2)

where x0 ∈ S acts as the initial value for the map. This map iscalled non-linear if f is a non-linear function of x. With x0, a se-quence is obtained by iterating the map i.e.

x0 ⇒ x1 = f (x0,μ) ⇒ x2 = f (x1,μ) = f(

f (x0,μ),μ)

⇒ x3 = f (x2,μ) = f(

f (x1,μ),μ)

= f(

f(

f (x0,μ),μ),μ

), . . . (3)

Here for any x0 ∈ S , the sequence x0, x1, x2, . . . is called the orbitor trajectory generated by x0. The main goal of dynamical systemis to understand the nature of all orbits and to identify the set oforbits which are periodic, eventually periodic, asymptotic etc. In

650 G. Bhatnagar, Q.M. Jonathan Wu / Digital Signal Processing 22 (2012) 648–663

Fig. 2. Lyapunov exponent curve for (a) logistic map, (b) generalized logistic map.

other words, we want to understand what happens if t → ∞. Oneof the simple and frequently used chaotic maps is logistic map thatdescribes population growth over time and is described by

xt+1 = μxt(1 − xt) (4)

where t = 1,2, . . . ,n, 0 �μ� 4. When 3.5699 �μ� 4, the map isin the chaotic state. Besides its chaotic behavior, it has some draw-backs which cease its performance like non-uniform behavior andblank windows in chaotic region. In order to overcome these is-sues, the logistic map is generalized by making it highly non-linear[25,26]. The generalized logistic map is defined mathematically as

xt+1 = 4μxt(1 − xt)

1 + 4(μ2 − 1)xt(1 − xt)(5)

where t = 1,2, . . . ,n, −1.7 � μ � −0.3598. When −0.6795 � μ �−0.4324, the map is in the chaotic state. In order to judge theeffectiveness and chaoticity, the Lyapunov exponent (LE) is usedas the measure. In general, positive value of Lyapunov exponent issign of chaoticity. Hence, if LE of a map is positive for all valuesof μ then the map is in chaos for whole domain. The effectivenessof generalized logistic map is discussed here in terms of LE. InFig. 2, the LE curve for all values of μ is depicted for logistic andits generalized one. If the LE curve of logistic map is consideredthen the positive value of LE occurs on μ = 3.5699 hence this isthe start point of chaotic area which ends at μ = 4 and this areais coined as chaotic region for logistic map. In the region 3.5699 �μ� 4 there are many areas where the LE is either zero or negative.Hence, there are some areas in chaotic region where logistic mapis not in chaos and it also justifies the non-uniform behavior andblank window of chaotic region. On the contrary, the chaotic regionfor generalized logistic map starts at μ = −0.6795 and ends atμ = −0.4324 and there is no non-chaotic area in chaotic regionwhich further proves the uniform behavior of generalized logisticmap. Hence, the generalized version of logistic map rectifies all thelimitations of logistic map.

2.2. Singular value decomposition

In linear algebra, the singular value decomposition (SVD) [27] isan important factorization of a rectangular real or complex matrixwith many applications in signal/image processing and statistics.

The SVD for square matrices was discovered independently by Bel-trami in 1873 and Jordan in 1874, and extended to rectangularmatrices by Eckart and Young in the 1930s. Let A be a general real(complex) matrix of order m ×n. The singular value decomposition(SVD) of X is the factorization

X = U ∗ S ∗ V T (6)

where U and V are orthogonal (unitary) and S = diag(σ1, σ2, . . . ,

σr ), where σi , i = 1(1)r are the singular values of the matrix Xwith r = min(m,n) and satisfying σ1 � σ2 � · · · � σr . The first rcolumns of V are the right singular vectors and the first r columnsof U are the left singular vectors.

Using of SVD in digital image processing has some advantages.First, the size of the matrices for SVD transformation is not fixed. Itcan be a square or rectangle. Secondly, singular values in a digitalimage are less affected if general image processing is performed.Finally, singular values contain intrinsic algebraic image properties.All the properties of SVD are summarized as follows:

• Stability: When a small perturbation is added to the matrix,large variance of its singular values does not occur.

• Singular values represent algebraic properties of an image.• To some extent, singular values possess algebraic and geomet-

ric invariance.• Rotation: given an image I and its rotated (with arbitrary an-

gle) Ir , both have the same singular values.• Translation: given an image I and its translated It , both have

the same singular values.• Scaling: given an image I and its scale I s , if I has the singular

values σi , then I s has the singular values σi ∗√LR LC where LR

and LC are the scaling factor of rows and columns respectively.If rows (columns) are mutually scaled, I s has the singular val-ues σi ∗ √

LR (σi ∗ √LC ).

• Transpose: given an image I and its transposed I T , both havethe same singular values.

• Flip: given an image I and its row and column flipped Ir f andIc f , both have the same singular values.

3. Saw-Tooth space filling curve

A space filling curve (SFC) [28] is a continuous scan that tra-verses every pixel of an image exactly once. SFCs are attractive to


Fig. 3. (a) Raster, (b) ZIG-ZAG space filling curve.

Fig. 4. Saw-Tooth curve.

many image–space algorithms which are based on the spatial co-herence of nearby pixels. After scanning, the resulting sequence ofpixels is processed as required by the particular application andto obtain the image after processing, the (possibly modified) pixel-sequence is placed back in a frame along the same SFC. In litera-ture, there are three famous SFC, namely raster, ZIG-ZAG and Hilbertspace filling curves. The traversing of resolution 8 × 8 using raster,ZIG-ZAG and Hilbert space filling curves are depicted in Figs. 3(a),(b), (c) respectively. The raster SFC is a standard scanning method,which traverses an image row by row whereas ZIG-ZAG SFC tra-verses an image in the neighbor of leading diagonal. Among thesethe most popular and frequently used SFC is the Hilbert SFC [29,30] which traverses an image of size 2m × 2m while never main-taining the same direction for more than three consecutive points.Once the curve has strayed three points in a straight line it turnsaround and comes back. The raster SFC is very simple in structurebut it is not able to give degradation to the image whereas theZIG-ZAG and Hilbert SFC give degradation to the image but havea complicated structure and work only for square images. Theselimitations of existing SFC necessitate introduction of a new SFCfilling curve which is quite simple in structure and also applicablefor rectangular images.

In this paper, a new space filling curve is introduced, namelySaw-Tooth space filling curve which is based on Saw-Tooth curve.Mathematically, Saw-Tooth curve is defined as:

y = a

(1 − x

T

), 0 < x < T (7)

where a gives height to the curve and T is the period of the curve.The graphical representation of the Saw-Tooth curve is given in thefollowing figure (Fig. 4).

Saw-Tooth space filling curve can be defined based on the abovementioned Saw-Tooth curve. Here a and T are equal to the heightand width of the image respectively. The basic element of the pro-posed map is made up of 3 line segments or in other words basicelement is a right angled triangle. Hence, the Saw-Tooth SFC is acurve that visits every point in a right angled triangular grid andtraverses every point/pixel once. The saw-tooth patterns for reso-lution 2 × 2, 3 × 2, 3 × 4 and 4 × 4 are shown in Figs. 5(a), (b), (c)and (d) respectively. Figs. 5(a), (b), (c), (d) assure that the values

of a and T are just equal to the height and width of the reso-lution. Further, the initial definition of Saw-Tooth SFC is extendedby imposing conditions on the value a. For this purpose, one canmake Saw-Tooth SFC height dependent, meaning that the right an-gled triangular grid involves how much height of the resolution.More precisely, the height of any resolution can be viewed as thecombination of the several rows. For instance, if the value of ais taken to be k (< height) then right angled triangular grid tra-verses whole space in the portion of k × T × p, where p = height/kand interprets the number by which height is partitioned. Thereis only one restriction for the extended definition and that is kmust be a divisor of the height. This restriction complies in orderto avoid the underflow condition. Fig. 6 shows the extended def-inition of the Saw-Tooth SFC on 8 × 8 resolution. Fig. 6(a) showsthe initial Saw-Tooth pattern whereas Figs. 6(b), (c) show the ex-tended Saw-Tooth pattern when k = 2 and k = 4 respectively. Thevisual assessment of the proposed Saw-Tooth SFC is given in Fig. 7.From the figure, it is clear that the extended version is giving farbetter results in terms of the degradation i.e. extended version isable to destroy the correlation among image pixels and shuffle theimage perfectly. Since, one cannot judge the overview of the orig-inal Lady image for k = 32,16,8 whereas for other values of kthe overview of Lady image can be judged partially or completely.Hence, visually best results are obtained when k = 32,16,8. Forthis conclusion, a set of original images is shown to some humanobservers. The set contains original Lady image and other standardgray-scale images of size 256 × 256 namely Mandril, Cameraman,Lena and Barbara. Finally, the scanned version of all these stan-dard images, with different values of k and SFC, are mixed andrandomly shown and observers are asked to judge the name oforiginal image. The most of observers failed to judge the name ofimage correctly when k = 32,16,8. The total number of human ob-servers included in the experiment are 50, students of Departmentof E&CE, University of Windsor and Department of Mathematics,IIT Roorkee. Apart from this experiment, numerical analysis is alsodone to calculate the degradation in the scanned images. For thispurpose, the PSNR, SD, UQI and SSIM are calculated between theoriginal images and their scanned versions. Basically, the abilityof the proposed Saw-Tooth SFC in terms of degradation is beingtried to judge. The mathematical definitions and interpretation ofthese measures can be seen in Appendix A. As it is evident thathigher values of SD whereas lower values of PSNR, UQI and SSIMshow the better degradation in the image. Fig. 8 shows the varia-tion in these measures for different SFC for different images. It isclear that the proposed SFC is able to distort images and the per-formance of the proposed Saw-Tooth SFC is far better than HilbertSFC which is the best among existing SFC, when k = 32,16,8. Al-though numerically Saw-Tooth SFC performs well but visually theresults are not impressive when k = 2,64,128 i.e. scanned imagegives the overview of the original image. Therefore, these valuesmust not be taken. Generally, if the image size is M × N whereM = 2Q 1 and N = 2Q 2 with two integers Q 1, Q 2 greater than orequal to 1, then, three values of k must not be chosen according


Fig. 5. Saw-Tooth pattern of resolution. (a) 2 × 2, (b) 3 × 2, (c) 3 × 4, (d) 4 × 4.

Fig. 6. (a) Initial Saw-Tooth pattern; extended Saw-Tooth pattern (b) when k = 2, (c) k = 4 of resolution 8 × 8.

Fig. 7. Visual assessment of proposed Saw-Tooth SFC. (a) Original Lady image; scanned Lady image with (b) raster SFC, (c) ZIG-ZAG SFC, (d) Hilbert SFC, (e) initial Saw-ToothSFC; extended Saw-Tooth SFC when (f) k = 128, (g) k = 64, (h) k = 32, (i) k = 16, (j) k = 8, (k) k = 4, (l) k = 2.

to human visual perception and are given by k = 2, M/2 = 2Q 1−1

and M/22 = 2Q 1−2. Hence, for M = 64 k = 2,32,64; for M = 256k = 2,64,128; for M = 512 k = 2,128,256 might not be chosen.For example, Fig. 9 shows the scanned version of Lady image whenit is of size 1024 × 1024. Figs. 9(b), (c), (j) give the overview of theLady image hence k = 2,256,512 have not been chosen.

4. Pixels of interest: A pixels selection criterion

Finding the regions from an image, attracting human attention,is one of the interesting areas in image processing and is coinedas the Region of Interest (ROI) detection in images [31]. Usuallyfor this purpose an image is divided into non-overlapping blocks


Fig. 8. Numerical analysis of distortion produced by different SFC. (a) PSNR (in dB), (b) spectral distortion, (c) universal image quality, (d) structural similarity index measure(R = Raster SFC, Z = ZIG-ZAG SFC, H = Hilbert SFC, S1 = Initial Saw-Tooth SFC, Extended Saw-Tooth SFC, S2 when k = 2, S3 when k = 4, S4 when k = 8, S5 when k = 16,S6 when k = 32, S7 when k = 64, S8 when k = 128).

Fig. 9. Visual assessment of proposed Saw-Tooth SFC on original Lady image of size 1024 × 1024; scanned Lady image with (a) initial Saw-Tooth SFC; extended Saw-ToothSFC when (b) k = 512, (c) k = 256, (d) k = 128, (e) k = 64, (f) k = 32, (g) k = 16, (h) k = 8, (i) k = 4, (j) k = 2.


of equal size followed by the computation of ROI score. ROI scorerepresent the level of interest that a human eye could have for theblock.

On the same idea, we suggest a criterion of finding the pixels ofan image which attracts the human attention and coin it as Pixelsof Interest (POI). Similar to ROI score, POI score represents the levelof interest that a human eye could have for a particular pixel. Thefive influencing parameters are considered to estimate POI scoreare intensity, contrast, location, edginess and texture. According tohuman visual system (HVS), the pixels which are closer to midintensity of image are the most sensitive to the human eye. Forcontrast, a pixel which has high level of contrast with respect to itssurrounding pixels, is perceptually important and attracts humanattention. For location, usually the central and its surrounding pix-els are of perceptually more importance than other pixels. A pixelwhich lies in prominent edges catches human attention. Finally, forthe fifth parameter texture, the pixels lying in the textured areasattract human eyes. Therefore, pixels lying in flat regions must beneglected.

In order to obtain POI score, the S × S local surrounding ofeach pixel, say region S , is considered. For the boundary pixels,wrapping method is used. The mathematical definitions of thesefive parameters are given as

1. Intensity score: Intensity score (POII ) is calculated by compar-ing average luminance/intensity of the whole image with thatof S . If Iavg and Inbd are average intensity of the whole imageand region S then the POII is given as

POII = |Iavf − Inbd| (8)

2. Contrast score: As it is mentioned before that a pixel whichhas the highest level of contrast with respect to its surround-ing pixels is perceptually important. For surrounding pixels,the pixels lie in 8-neighborhood (D8) are considered. Let Isur

are average sum intensities of local surrounding of D8 pixelsthen the contrast score (POIC ) is given as

POIC = |Isur − Inbd| (9)

3. Location score: The location score (POIL ) of each pixel is mea-sured by calculating the ratio of number of pixels in region Sthat are lying in the center-quarter of the image. This is dueto the fact that the viewer’s eyes are aimed at the center, usu-ally 25% of the screen used for displaying [31]. Mathematically,POIL is given as

POIL = nC

nA(10)

where nC is the number of pixels lying in the central quarterof S whereas nA is the total number of pixels in S .

4. Edginess score: The edginess score (POIE ) is the total numberof edge pixels in the region S . In order to calculate this score,the edges in the original image are computed using cannyedge detector followed by the counting number of pixels lyingon the edge in S . The easiest way is to calculate the numberof 1’s in the local surrounding of the i jth pixel in the edgeimage.

5. Texture score: The texture score (POIT ) is obtained by theconventional method of using variance. For this purpose, thevariance of S is calculated. Generally, the high variance showsthat the pixel is not in the flat areas.

After getting each score for all pixels, some weight is given foreach score according to their importance for a given problem. Forexample, in the case of image comparison where location scoreis not too useful because some pixels of region S would be in

quarter center and moreover this parameter is the same for two ormore images. Hence, the weights are decided by the definition ofthe problem. First, these scores are normalized in the range (0,1)

followed by the combination of these five scores by consideringthe weights. Let us suppose the weights for each score are wi :i = 1,2,3,4,5 then the optimal POI score is given by

POI = w1POII + w2POIC + w3POIL + w4POIE + w5POIT (11)

As far as encryption is concerned, the chances for each pixel get-ting selected are unpredictable and equally the same. Therefore,we rely completely on the human visual system and treat eachscore of equivalent weightage. Hence, the optimal POI score givenby Eq. (11) is reduced to

POI = POII + POIC + POIL + POIE + POIT (12)

The calculated optimal POI is then sorted and the pixels having thehighest value are regarded as the perceptually most important pix-els and are selected. If in the original image all pixels are replacedby the corresponding POI score, one can get the POI matrix of theimage. The main reason for making POI matrix is to see the effectof the varying values of the local surrounding neighborhood i.e.the value of S on the POI score. The visual results for different val-ues of S are depicted in Fig. 10 on Lady image. From the figure it isclear that for small values of S , the contribution of some score (e.g.edginess score) is greater over the other score. But as the value ofS increases the contribution of the other score increases gradu-ally. Therefore, either proper weight functions or greater value ofS must be considered for the best results.

5. Proposed technique

In this section, some motivating factors in the design of our ap-proach to selective image encryption are discussed. The proposedtechnique uses a color image and gives an encrypted image whichcan be decrypted later for various purposes. Without loss of gen-erality, assume that G represents the original color image of sizeM × N . The proposed technique comprises of three phases. In thefirst phase, the RGB color channels are transformed into secretcolor space (say SEC) using reversible integer transform (RIT) fol-lowed by the second phase i.e. the encryption of each secret colorchannel. For encryption, first each channel undergoes confusion ofpixels via proposed Saw-Tooth SFC followed by the characteriza-tion of significant pixels considering POI. Finally, diffusion is doneon the selected significant pixels using non-linear chaotic map andsingular value decomposition. The whole process is summarized asfollows.

5.1. RGB to SEC conversion

It is evident that the R, G and B channels of an image are highlydependent on each other whereas for a good encryption techniquethis relation must be broken before encryption so that the ro-bustness of the technique increases. Usually, the RGB channels aretransformed into independent transform like HSI, YCbCr etc. Butthe main problem with these existing independent transforms isnon-perfect reconstruction. Hence, for a perfect color space con-version, there must be a transformation which maps integers tointegers. For this purpose and to enhance security, RGB channelsare first transformed into three secret independent channels (SECchannel) using reversible integer transform (RIT) and then the en-cryption is done either in all or any of S, E and C channels.

The general linear transform Y = A X is said to be reversibleinteger transform if transform matrix A is elementary reversiblematrix (ERM) [32]. The most common example of ERM is the


Fig. 10. Visual assessment of POI with varying values of S . (a) Original Lady image; POI image when (b) S = 3, (c) S = 5, (d) S = 7, (e) S = 9, (f) S = 11, (g) S = 13, (h) S = 15.

lower or upper triangular matrices having integer factors on di-agonal and usually coined as triangular ERM (TERM). Using anyof the TERM i.e. upper or lower one can map integers to inte-gers. In the proposed work, lower TERM is used to create secretchannels (SEC) from conventional RGB channel. Let us supposeA = {aij: aij = 0 if i < j}i, j=1,2,3 is an upper TERM then RGB spaceis converted into SEC space as⎡⎣ S

E

C

⎤⎦ =⎛⎝ A

⎡⎣ R

G

B

⎤⎦⎞⎠⇒

⎡⎣ S

E

C

⎤⎦ =⎡⎣

⎛⎜⎝a11 0 0

a21 a22 0

a31 a32 a33

⎞⎟⎠⎡⎣ R

G

B

⎤⎦⎤⎦ (13)

and the RGB channel is again obtained from SEC channel as⎡⎣ R

G

B

⎤⎦ =⎛⎝ A−1

⎡⎣ S

E

C

⎤⎦⎞⎠

⇒⎡⎣ R

G

B

⎤⎦ =⎛⎜⎝

⎡⎢⎢⎣1

a110 0

−a21a22

1a22

0

−a31a33

−a32a33

1a33

⎤⎥⎥⎦⎡⎣ S

E

C

⎤⎦⎞⎟⎠ (14)

where (r) denotes the rounding arithmetic for any real number r.Since, the definition of TERM says that there are integers factorson diagonal that do not change the magnitude, the output are in-tegers if the input are integers. Rounding arithmetic is used due to{aij: i > j} which are not an integer factor that does change themagnitude, the output are not integers if the input are integers.Moreover, if the off-diagonal elements of A are all zero then thereversible integer transform reduces to the scaling of each colorspace by a given factor. The usage of above mentioned RIT is de-picted in Fig. 11 when {aij: i > j} are either an integer factor ornon-integer factor or zero.

5.2. Encryption process

The encryption process takes an image to be protected as inputand gives an encrypted image whose size is the same as that ofinput image as output. The complete encryption process, illustratedin Fig. 12, consists of the following steps.

1. The RGB color channels of original image G are first convertedin SEC color channel denoted by F .

2. Confusion: Scramble pixel positions of F using k-order Saw-Tooth SFC, which is denoted by Fs .

3. Characterize significant and in-significant pixels in Fs usingpixels of interest (POI) method.

4. Select P % significant pixels, stack into an array of size P1 ×P2, P1 < P2 and name it as Significant Matrix, denoted by F .

5. By adopting K0 and μ as the keys, iterate generalized lo-gistic map (Eq. (5)) to generate MN values of the sequenceK = {Ki: i = 1,2, . . . , MN}.

6. Map the obtained sequence K into an integer sequence I ={Ii: 0 � Ii � 255, i = 1,2, . . . , MN} as follows

Ii =(

255 × Ki + 1

2

)(15)

where (r) is the arithmetic rounding operation of real num-ber r.

7. The last P % values of integer sequence I are selected andarranged into a random matrix (Z ) of size P1 × P2, P1 < P2followed by SVD on it.

Z = U Z S Z V TZ (16)

8. Obtain a Hankel matrix (H Z ) with the help of singular valuesof Z i.e. S Z = {σ j: j = 1,2, . . . , r(= min(P1, P2))} and givenby

H Z =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

σ1 σ2 σ3 · · · σr−1 σr

σ2 σ3 σ4 · · · σr 0

σ3 σ4 σ5 · · · 0 0...

......

. . ....

...

σr−1 σr 0 · · · 0 0

σr 0 0 · · · 0 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦(17)

9. Perform SVD on the obtained Hankel matrix.

H Z = U H Z S H Z V TH Z

(18)

10. Obtain the matrix key (K) by U H Z and V H Z as

K = U H Z V TH (19)

Z


Fig. 11. Example of RGB to SEC and SEC to RGB channel conversion when (a) all {aij : i > j} have integer factor, (b) all {aij : i > j} have not integer factor, (c) all off-diagonalelements are zero.

Fig. 12. Block diagram of proposed encryption process.

where the matrix key K is an orthogonal matrix i.e. KKT = I .Since, it is the multiplication of two orthogonal matrices.

11. Diffusion: Change the pixel values of significant matrix F usingmatrix key K to get the encrypted significant matrix F e as

F e = K F (20)

12. Map encrypted significant coefficients onto their original posi-tions.

13. SEC color channels are converted back to RGB color channelsto get encrypted image Ge .

The proposed selective encryption technique gives flexibility to theuser in terms of percentage of pixels (P ). Obviously, if P = 100then the selective encryption is converted into full encryption.While selecting P % significant pixels, P % is used. The floor func-tion is used to deal with overflow condition. Suppose, M = 256and N = 128 and 45% of significant pixels have to be selectedfrom these M × N = 32 768 pixels. The 45% of 32 768 is 14745.6.Since, we are dealing with pixels which must be a positive inte-ger. Therefore, this situation is called the overflow situation andfloor function takes care of it. Hence, in this case, the selected 45%pixels are 14745.6 = 14 745.

5.3. Decryption process

The stressed motive of the decryption process is to obtain theimage as perfectly as possible from the encrypted image. Owingthe values of: Saw-Tooth SFC order (k), initial conditions for thegeneralized logistic map (K0) along with the positions of signifi-cant pixels, matrix key and encrypted image at receiver end, de-cryption is performed perfectly. The decryption process is depictedin Fig. 13 and is summarized as follows.

1. The RGB color channels of encrypted image Ge are first con-verted in SEC color channel denoted by F e .

2. Obtain encrypted significant matrix F e from F e by selectingP % pixels redbased on the saved original positions of signif-icant pixels.

3. By adopting keys K0 and μ, step 5 to step 10 of encryptionprocess are performed to get matrix key K.

4. Inverse diffusion: Obtain the decrypted significant matrix fromF e with the help of matrix key as

F d = inv(K ) F e = K T F e (21)

5. Map decrypted pixels from F d to their original positions to getthe decrypted confused image F s .


Fig. 13. Block diagram of proposed decryption process.

Fig. 14. (a) Original Lena image; encrypted image with (b) P = 10%, (c) P = 20%, (d) P = 30%, (e) P = 40%, (f) P = 50%, (g) P = 60%, (h) P = 70%, (i) P = 80%, (j) P = 90%,(k) P = 100%.

6. Inverse confusion: Inverse k-order Saw-Tooth SFC is performedon F s to get (F d).

7. SEC color channels are converted back to RGB color channelsto get the decrypted image Gd .

6. Experiments and security analysis

The performance of the proposed selective encryption tech-nique is demonstrated using MATLAB platform. A number of ex-periments are performed on color Lena image, which is used asoriginal image having size 256 × 256. In the proposed technique,three parameters are used as the keys, these parameters are k, μand K0. The first key is used as the order of Saw-Tooth SFC and istaken as k = 8. On the other hand, K0 and μ are used as the pa-rameters for generalized logistic map to create matrix key wherethe values are taken as μ = −0.5469 and K0 = 0.8147 are usedas the initial parameters for the generalized logistic map. The en-crypted and decrypted images using above mentioned keys areshown in Figs. 14 and 15 respectively. Figures show the encrypted

and decrypted Lena images with the varying values of P . First 10%pixels of Lena image are selected for encryption which is furtherincreased by 10% until images are encrypted up to 100%.

The heart of the security of the proposed encryption techniqueis K0, which is used as the initial seed for generalized logistic mapand further responsible for encryption and decryption processes.Therefore, the initial seed must be obtained very carefully. In thepresent work, a systematic secret rule is opted instead of randomselection to ensure the enhanced security. The key K0 is generatedby considering statistical features of the experimental image. It isassumed that the generating process is secret and only authorizedusers have access to it. The complete process for generating initialseed K0 consists of the following steps.

• Divide the original image into two parts: G = G1 + G2, whereG∗ denotes half of the image.

• Calculate the mean of both parts denoted by mG1 and mG2

respectively.


Fig. 15. Decrypted image with (a) P = 10%, (b) P = 20%, (c) P = 30%, (d) P = 40%, (e) P = 50%, (f) P = 60%, (g) P = 70%, (h) P = 80%, (i) P = 90%, (j) P = 100%.

• Subtract these means and divide by the total number of graylevels in the image to get normalized difference mean (Th), i.e.,

Th = |mG1 − mG2 |/(2L − 1

), 0 � Th � 1 (22)

• The key for generalized logistic map is calculated as

K0 = 1 − (Th × 218) mod 1 (23)

From Eq. (23), it is clear that the value of K0 lies from 0 to 1.The normalized difference mean is amplified by a scaling factor218 to digitize it and modulo 1 arithmetic is used to ensure that0 � K0 � 1. Digitization is not a unique process. However, in manycases one can identify “a natural way” in doing this as we do. Here,changing the value of scaling factor will result in alterations inthe digitization of Th and hence alterations in the initial seed K0.In our experiments, the value of initial seed is come out to beK0 = 0.8147.

Security is a major issue of encryption techniques. A good en-cryption technique should be robust against all kinds of cryptana-lytic, statistical and brute-force attacks. In this section, a completeinvestigation is made on the security of the proposed encryptiontechnique such as sensitivity analysis, statistical analysis, numericanalysis etc. to prove that the proposed encryption technique issecure against the most common attacks.

6.1. Key sensitivity analysis

According to the Kirchhoff’s principle, the information systemshould be secure even if everything about the system, except thekey, is publicly available. Hence, keys play a very vital role inthe security of information system. According to the principle, theslight change in the keys never gives the perfect decryption fora good security. For this purpose, the key sensitivity of the pro-posed technique is validated. In the proposed technique, three keys(k, μ and K0) are used. Among these, key k is an integer of theform 2b: b > 0. In order to check sensitivity with respect to k,the nearest values are considered i.e. k = 4 and k = 16. The re-spective results are shown in Fig. 16(a), (b) along with the varyingpower of P . It is clear that after changing the value of k one cannotget the correct decrypted image. Hence, the proposed techniqueis quite sensitive to k. Similarly, Figs. 16(c)–(f) and (g)–(j) showthe decrypted images with slight change in K0 and μ for P = 50%and P = 100% respectively. The changes are made in the way suchthat older values (K0 = −0.5469,μ = 0.8147) and newer values(K0 = −0.54688,μ = 0.81471) are approximately the same. It isclear from the figures that a slight modification in the keys will

result in non-perfect decryption. Hence, the proposed technique ishighly sensitive to the keys.

6.2. Statistical analysis

Another method to evaluate encryption technique is statisti-cal analysis. This analysis is composed of two terms 1) Histogramanalysis 2) Correlation analysis. According to the first term, for agood encryption technique, there is uniform change in the imagehistogram after encryption. Since, the proposed scheme is selec-tive encryption. Hence, whenever the value of P is less there isless change in histogram but as the value of P increases the his-togram changes gradually and becomes uniform when P = 100%.Fig. 17 shows the variation in histogram of encrypted images withthe varying power of P . The second term says that a good en-cryption technique must break the correlation among the adjacentimage pixels. For this purpose, the correlation between two ad-jacent pixels is calculated and it is said to be good encryption ifcorrelation comes to be as far from 1. We randomly select 2500pairs of two adjacent pixels horizontally, vertically and diagonallyfollowed by correlation coefficient computation as

rx,y = cov(x, y)√D(x)

√D(y)

(24)

where

cov(x, y) = 1

N

N∑i=1

(xi − 1

N

N∑j=1

x j

)(yi − 1

N

N∑j=1

y j

)

D(x) = 1

N

N∑i=1

(xi − 1

N

N∑j=1

x j

)2

D(y) = 1

N

N∑i=1

(yi − 1

N

N∑j=1

y j

)2

The above definition of correlation coefficient is for gray-scale im-age. In order to use the above definition for color images, averagecorrelation coefficient of R , G and B color channel is used. Hence,Eq. (24) changes to

rRGBx,y = 1

3

∑θ∈{R,G,B}

cov(xθ , yθ )√D(xθ )

√D(yθ )

(25)

The value of rRGBx,y lies between [−1,1]. If the value of rRGB

x,y is equalto 1 then the adjacent pixels are highly correlated and they are un-correlated whenever rRGB

x,y is equal to zero whereas a high negative


Fig. 16. Decrypted images with wrong keys. (a) Wrong k (k = 4) with P = 50%, (b) wrong k (k = 16) with P = 50%, (c) wrong K0 (K0 = −0.54688) with P = 50%, (d) wrongμ (μ = 0.54688) with P = 50%, (e) wrong K0 (K0 = −0.54688) and μ (μ = 0.54688) with P = 50%, (f) all keys wrong (K0 = −0.54688, μ = 0.54688 and k = 16) withP = 50%, (g) wrong K0 (K0 = −0.54688) with P = 100%, (h) wrong μ (μ = 0.54688) with P = 100%, (i) wrong K0 (K0 = −0.54688) and μ (μ = 0.54688) with P = 100%, (j)all keys wrong (K0 = −0.54688, μ = 0.54688 and k = 16) with P = 100%.

Table 1Correlation coefficients of two adjacent pixels in original and encrypted image.

Image Correlation coefficients in

Horizontal Vertical Diagonal

Original 0.9816 0.9929 0.9723Encrypted with P = 10% 0.6116 0.6288 0.6687Encrypted with P = 20% 0.5082 0.5154 0.5963Encrypted with P = 30% 0.3804 0.3413 0.4619Encrypted with P = 40% 0.3177 0.2639 0.3941Encrypted with P = 50% 0.2048 0.1615 0.2439Encrypted with P = 60% 0.1123 0.0861 0.1278Encrypted with P = 70% 0.0971 0.0778 0.1032Encrypted with P = 80% 0.0525 0.0398 0.0669Encrypted with P = 90% 0.0058 −0.0331 0.0088Encrypted with P = 100% −0.0139 −0.0022 0.0087

correlation indicates a high correlation in the case of inversion ofone of the series. Fig. 18 shows the correlation distribution of twohorizontally adjacent pixels in the original and encrypted imagefor all values of P . If the figure is observed then one can see theadjacent pixels are highly correlated and as the encryption takesplace then correlation reduces gradually and reaches to zero whenimage is encrypted 100%. The same phenomena can be observedfrom Table 1 in which the correlation coefficients values are de-picted in all directions. Hence, the proposed technique is able tobreak the high correlation among the pixels.

6.3. Gray value analysis

In gray value analysis, the gray difference in the image beforeand after encryption is calculated considering neighbor pixels in

each image. The gray difference of a pixel with a neighbor pixel iscalculated as

GDij =∑

i′ j′(xij − xi′ j′)2

8,

where(i′, j′

) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩(i − 1, j), (i + 1, j)

(i − 1, j − 1), (i − 1, j + 1)

(i, j − 1), (i, j + 1)

(i + 1, j − 1), (i + 1, j + 1)

(26)

where xij denotes the gray value in position (i, j) and replicationis used while calculating gray differences for boundary pixels. Theabove obtained matrix of gray differences (GD) is normalized in[0,1], in order to get normalized gray difference for every pixel as

GDij = GDij − minGD

maxGD −minGD(27)

where minGD and maxGD are the minimum and maximum gray dif-ference present in the gray difference matrix GD. After computingthe normalized gray difference for every pixel, the average neigh-borhood gray difference (ANNGD) of the whole image is calculatedas

ANGD =∑M

i=1∑N

j=1 GNij

MN(28)

Based on the average neighborhood gray difference, the gray valuedegree (GVD) is defined as

GVD =∣∣∣∣ANGD(I) − ANGD(I)˜

∣∣∣∣ (29)
ANGD(I) + ANGD(I)


Fig. 17. Histogram of (a) original image; encrypted image with (b) P = 10%, (c) P = 20%, (d) P = 30%, (e) P = 40%, (f) P = 50%, (g) P = 60%, (h) P = 70%, (i) P = 80%,(j) P = 90%, (k) P = 100%.

where I and I are the original and encrypted/decrypted im-ages respectively. The value of GVD lies between [0,1]. If thevalue of GVD is one then there is a huge variation in the aver-age neighborhood gray difference i.e. the gray values are signif-icantly changed after encryption/decryption. On the contrary, ifGVD is zero then there is no variation in the average neighbor-hood gray difference and therefore the gray values remain un-altered after encryption/decryption. Hence, gray value degree isas close as possible to 1 for good encryption whereas as closeas possible to 0 for perfect decryption. Fig. 19 shows the gray

value degree for encrypted and decrypted images with differentP . From Fig. 19(a), it is clear that as the value of P increasesthe gray value significantly increases and reaches its peak whenP = 100. Although, for the lower values of P the GVD is alsogreater than 0.92 which is enough close to 1. Hence for ev-ery value of P , the proposed encryption technique perfectly en-crypts the images. Similarly, the ability of perfect decryption canbe viewed in Fig. 19(b) where GVD are approximately zero forall P .


Fig. 18. Correlation distribution of two horizontally adjacent pixels in (a) original image; encrypted image with (b) P = 10%, (c) P = 20%, (d) P = 30%, (e) P = 40%, (f) P = 50%,(g) P = 60%, (h) P = 70%, (i) P = 80%, (j) P = 90%, (k) P = 100%.

6.4. Numerical analysis

Numerical analysis includes the values of the objective met-rics. A metric which provides more efficient test methods and issuitable for computer simulations is called objective metric. PSNR,SD, UQI and SSIM are used as the objective metrics to evaluatethe proposed technique (mathematical definitions can be seen inAppendix A). Table 2 shows the values of objective metrics be-tween original-encrypted and original-decrypted images with cor-rect keys. From the table, it is clear that the proposed techniqueperfectly encrypts and decrypts the image.

7. Conclusions

This paper proposes a simple yet efficient selective encryptiontechnique that encrypts an image partially using Saw-Tooth spacefilling curve, pixels of interest, non-linear chaotic map and singu-lar value decomposition. Saw-Tooth space filling curve is used forconfusion whereas pixel of interest is used to identify significantpixels. The diffusion of significant pixels is done with the help ofnon-linear chaotic map and singular value decomposition. Somesecurity analysis is also given to demonstrate that the right com-bination of keys is important to reveal the original image. Further,


Fig. 19. Gray value analysis of proposed technique: gray value degree for (a) encrypted, (b) decrypted image with different values of P .

Table 2Numerical analysis of proposed selective encryption technique.

Image metric Encrypted image Decrypted image

PSNR SD UQI SSIM PSNR SD UQI SSIM

P = 10% 14.3719 247.2086 −0.0009 0.0344 359.9017 0.0021 0.9991 0.9997P = 20% 12.4673 388.3121 −0.0010 0.0298 356.7588 0.0077 0.9998 0.9900P = 30% 11.9571 503.1177 −0.0009 0.0261 356.0826 0.0077 0.9989 0.9993P = 40% 11.0437 636.6045 −0.0002 0.0235 354.1422 0.0043 0.9988 0.9991P = 50% 10.7442 756.0058 −0.0008 0.0204 353.568 0.0066 0.9991 0.9990P = 60% 10.4451 833.3553 −0.0003 0.0184 353.3812 0.0019 0.9990 0.9988P = 70% 10.3749 906.3500 −0.0006 0.0149 352.3227 0.0039 0.9990 0.9987P = 80% 0.3197 933.9257 −0.0011 0.0121 351.8346 0.0074 0.9991 0.9987P = 90% 10.1671 923.0396 −0.0005 0.0086 351.9437 0.0065 0.9996 0.9986P = 100% 9.5132 947.9022 −0.0004 0.0006 350.6638 0.0077 0.9997 0.9987

the proposed technique has advantages of convenient realization,less computation complexity and better security. The algorithm issuitable for any kind of color image which can be further extendedfor video. This extension can easily be done by employing pro-posed technique separately to each frame.

Acknowledgments

This work was supported in part by the Canada Chair ResearchProgram and the Natural Sciences and Engineering Research Coun-cil of Canada.

Appendix A. Degradation/similarity measures

1. Peak Signal to Noise Ratio (PSNR): The PSNR indicates the sim-ilarity between two images. The higher the value of PSNR, thegreater similarity in the images. Mathematically, PSNR is de-fined as

PSNR(x, y) = 10 log102552

1MN

∑Mi=1

∑Nj=1[xi, j − yi, j]2

(30)

where MN is the total number of pixels in the image, xi, j andyi, j are the values of the i jth pixel in original and encryptedimage.

2. Spectral Distortion (SD): SD indicates the similarity betweenspectral information of the images. The lower the value of SD,the greater similarity in the images. Mathematically, SD is de-fined as

SD(x, y) = 1

MN

M∑ N∑|xi, j − yi, j| (31)

i=1 j=1

where MN is the total number of pixels in the image, xi, j andyi, j are the values of the i jth pixel in original and encryptedimage.

3. Universal Image Quality Index (UIQ): The UIQ indicates thestructural similarity between two images. The UIQ lies be-tween [−1,1] and the value closer to 1, the greater similarityin the images. Mathematically, UIQ is defined as

UQI(x, y) = σxy

σx σy.

2μxμy

μ2x + μ2

y.

2σxσy

σ 2x + σ 2

y(32)

where μx , μy , σx , σy and σxy are the mean of x & y, variancex & y and the covariance of x and y respectively.

4. Structural Similarity Index Measure (SSIM): The SSIM is theextended version of the UIQ index. The SSIM lies between[−1,1] and the value closer to 1, the greater similarity in theimages. Mathematically, SSIM is defined as

SSIM(x, y) = (2μxμy + C1)(2σxy + C2)

(μ2x + μ2

y + C1)(σ2x + σ 2

y + C2)(33)

where C1, C2 are two constants and are used to stabilize thedivision with weak denominator.

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Gaurav Bhatnagar is the member of the Com-puter Vision and Sensing Systems Laboratory in theDepartment of Electrical and Computer Engineeringat University of Windsor, ON, Canada since 2009. Hereceived his PhD and MSc degree in Applied Mathe-matics from Indian Institute of Technology Roorkee,India, in 2010 and 2005 respectively. He has coau-thored more than 40 journals, conference proceedingsand contributed to two books in his area of interest.

His research interests include digital watermarking, encryption techniques,biometrics, image analysis, wavelet analysis and fractional transform the-ory.

Q.M. Jonathan Wu received the Ph.D. degree inelectrical engineering from the University of Wales,Wales, UK, in 1990. From 1995, he has been withthe National Research Council of Canada, Ottawa, ON,Canada, for ten years, where he became a Senior Re-search Officer and Group Leader. He is currently aFull Professor with the Department of Electrical andComputer Engineering, University of Windsor, Wind-sor, ON. He holds the Tier 1 Canada Research Chair in

Automotive Sensors and Sensing Systems. He has published more than 200peer-reviewed papers in the areas of computer vision, image processing,intelligent systems, robotics, micro-sensors and actuators, and integratedmicro-systems. His current research interests include 3-D computer vision,active video object tracking and extraction, interactive multimedia, sensoranalysis and fusion, and visual sensor networks.

Dr. Wu is an Associate Editor for the IEEE Transaction on Systems, Man,and Cybernetics (part A). He is on the editorial board of the InternationalJournal of Robotics and Automation. He is a member of the IEEE Techni-cal Committee on Robotics and Intelligent Sensing. He has served on theTechnical Program Committees and International Advisory Committees formany prestigious international conferences.

selective image encryption based on pixels of interest and singular value decomposition

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