IEEE TRANSACTIONS ON INFORMATION FORENSICS AND …€¦ · Low-Complexity Features for JPEG Steganalysis Using Undecimated DCT Vojtˇ ech Holub and Jessica Fridrich, Member, IEEE

IEEE TRANSACTIONS ON INFORMATION FORENSICS AND SECURITY, VOL. 10, NO. 2, FEBRUARY 2015 219

Low-Complexity Features for JPEG SteganalysisUsing Undecimated DCT

Vojtech Holub and Jessica Fridrich, Member, IEEE

Abstract— This paper introduces a novel feature set forsteganalysis of JPEG images. The features are engineered asfirst-order statistics of quantized noise residuals obtained fromthe decompressed JPEG image using 64 kernels of the discretecosine transform (DCT) (the so-called undecimated DCT). Thisapproach can be interpreted as a projection model in theJPEG domain, forming thus a counterpart to the projectionspatial rich model. The most appealing aspect of this proposedsteganalysis feature set is its low computational complexity, lowerdimensionality in comparison with other rich models, and acompetitive performance with respect to previously proposedJPEG domain steganalysis features.

Index Terms— Image, steganalysis, JPEG, DCT, features.

I. INTRODUCTION

STEGANALYSIS of JPEG images is an active and highlyrelevant research topic due to the ubiquitous presence of

JPEG images on social networks, image sharing portals, andin Internet traffic in general. There exist numerous stegano-graphic algorithms specifically designed for the JPEG domain.Such tools range from easy-to-use applications incorporat-ing quite simplistic data hiding methods to advanced toolsdesigned to avoid detection by a sophisticated adversary.According to the information provided by Wetstone Technolo-gies, Inc, a company that keeps an up-to-date comprehensivelist of all software applications capable of hiding data inelectronic files, as of March 2014 a total of 349 applicationsthat hide data in JPEG images were available for download. 1

Historically, two different approaches to steganalysis havebeen developed. One can start by adopting a model for thestatistical distribution of DCT coefficients in a JPEG fileand design the detector using tools of statistical hypothesistesting [7], [30], [34]. In the second, much more commonapproach, a representation of the image (a feature) is iden-tified that reacts sensitively to embedding but does not varymuch due to image content. For some simple steganographicmethods that introduce easily identifiable artifacts, such asJsteg, it is often possible to identify a scalar feature – anestimate of the payload length [4], [19], [31]–[33].

Manuscript received April 5, 2014; revised August 25, 2014; acceptedOctober 15, 2014. Date of publication October 23, 2014; date of currentversion December 29, 2014. The work was supported by the Air Force Officeof Scientific Research, Arlington, VA, USA, under Grant FA9950-12-1-0124.The associate editor coordinating the review of this manuscript and approvingit for publication was Prof. Hitoshi Kiya.

The authors are with the Department of Electrical and Computer Engi-neering, Binghamton University, Binghamton, NY 13902 USA (e-mail:[email protected]; [email protected]).

Digital Object Identifier 10.1109/TIFS.2014.23649181Personal communication by Chet Hosmer, CEO of Wetstone Tech.

More sophisticated embedding algorithms usually requirehigher-dimensional feature representation to obtain more accu-rate detection. In this case, the detector is typically builtusing machine learning through supervised training duringwhich the classifier is presented with features of cover as wellas stego images. Alternatively, the classifier can be trainedthat recognizes only cover images and marks all outliers assuspected stego images [26], [28]. Recently, Ker and Pevnýproposed to shift the focus from identifying stego imagesto identifying “guilty actors,” e.g., Facebook users, usingunsupervised clustering over actors in the feature space [17].Irrespectively of the chosen detection philosophy, the mostimportant component of the detectors is the feature space –their detection accuracy is directly tied to the ability of thefeatures to capture the steganographic embedding changes.

Selected examples of popular feature sets proposed fordetection of steganography in JPEG images are the historicallyfirst image quality metric features [1], first-order statistics ofwavelet coefficients [8], Markov features formed by sampleintra-block conditional probabilities [29], inter- and intra-block co-occurrences of DCT coefficients [6], the PEV featurevector [27], inter and intra-block co-occurrences calibratedby difference and ratio [23], and the JPEG RichModel (JRM) [20]. Among the more general techniques thatwere identified as improving the detection performance is thecalibration by difference and Cartesian calibration [18], [23].By inspecting the literature on features for steganalysis, onecan observe a general trend – the features’ dimensionalityis increasing, a phenomenon elicited by developments insteganography. More sophisticated steganographic schemesavoid introducing easily detectable artifacts and moreinformation is needed to obtain better detection. To addressthe increased complexity of detector training, simplermachine learning tools were proposed that better scale w.r.t.feature dimensionality, such as the FLD-ensemble [21] orthe perceptron [25]. Even with more efficient classifiers,however, the obstacle that may prevent practical deploymentof high-dimensional features is the time needed to extract thefeature [3], [13], [16], [22].

In this article, we propose a novel feature set forJPEG steganalysis, which enjoys low complexity, relativelysmall dimension, yet provides competitive detection perfor-mance across all tested JPEG steganographic algorithms.The features are built as histograms of residuals obtainedusing the basis patterns used in the DCT. The featureextraction thus requires computing mere 64 convolutions ofthe decompressed JPEG image with 64 8 × 8 kernels and

1556-6013 © 2014 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

220 IEEE TRANSACTIONS ON INFORMATION FORENSICS AND SECURITY, VOL. 10, NO. 2, FEBRUARY 2015

forming histograms. The features can also be interpretedin the DCT domain, where their construction resemblesthe PSRM with non-random orthonormal projection vectors.Symmetries of these patterns are used to further compactify thefeatures and make them better populated. The proposedfeatures are called DCTR features (Discrete Cosine TransformResidual).

In the next section, we introduce the undecimated DCT,which is the first step in computing the DCTR features.Here, we explain the essential properties of the undecimatedDCT and point out its relationship to calibration and otherprevious art. The complete description of the proposed DCTRfeature set as well as experiments aimed at determiningthe free parameters appear in Section III. In Section IV,we report the detection accuracy of the DCTR feature seton selected JPEG domain steganographic algorithms. Theresults are contrasted with the performance obtained usingcurrent state-of-the-art rich feature sets, including the JPEGRich Model and the Projection Spatial Rich Model. Thepaper is concluded in Section V, where we discuss futuredirections.

A condensed version of this paper was submitted tothe IEEE Workshop on Information Security and Foren-sics (WIFS) 2014.

II. UNDECIMATED DCT

In this section, we describe the undecimated DCT andstudy its properties relevant for building the DCTR featureset in the next section. Since the vast majority of stegano-graphic schemes embed data only in the luminance component,we limit the scope of this paper to grayscale JPEG images.For easier exposition, we will also assume that the size of allimages is a multiple of 8.

A. Description

Given an M × N grayscale image X ∈ RM×N , the undeci-

mated DCT is defined as a set of 64 convolutions with 64 DCTbasis patterns B(k,l):

U(X) = {U(k,l)|0 ≤ k, l ≤ 7}U(k,l) = X � B(k,l), (1)

where U(k,l) ∈ R(M−7)×(N−7) and ‘�’ denotes a convolution

without padding. The DCT basis patterns are 8 × 8 matrices,B(k,l) = (B(k,l)

mn ), 0 ≤ m, n ≤ 7:

B(k,l)mn = wkwl

4cos

πk(2m + 1)

16cos

πl(2n + 1)

16, (2)

and w0 = 1/√

2, wk = 1 for k > 0.When the image is stored in the JPEG format, before

computing its undecimated DCT it is first decompressed tothe spatial domain without quantizing the pixel values to{0, . . . , 255} to avoid any loss of information.

For better readability, from now on we will reserve theindices i, j and k, l to index DCT modes (spatial frequencies);they will always be in the range 0 ≤ i, j, k, l ≤ 7.

1) Relationship to Prior Art: The undecimated DCT hasalready found applications in steganalysis. The concept of cali-bration, for the first time introduced in the targeted quantitativeattack on the F5 algorithm [9], formally consists of computingthe undecimated DTC, subsampling it on an 8×8 grid shiftedby four pixels in each direction, and computing a referencefeature vector from the subsampled and quantized signal.Liu [23] made use of the entire transform by computing 63inter- and intra-block 2D co-occurrences from all possibleJPEG grid shifts and averaging them to form a more powerfulreference feature that was used for calibration by differenceand by ratio. In contrast, in this paper we avoid using theundecimated DCT to form a reference feature, and, insteadkeep the statistics collected from all shifts separated.

B. Properties

First, notice that when subsampling the convolution U(i, j ) =X � B(i, j ) on the grid G8×8 = {0, 7, 15, . . . , M − 9} ×{0, 7, 15, . . . , N − 9} (circles in Figure 1 on the left), oneobtains all unquantized values of DCT coefficients for DCTmode (i, j) that form the input into the JPEG representationof X.

We will now take a look at how the values of theundecimated DCT U(X) are affected by changing one DCTcoefficient of the JPEG representation of X. Suppose onemodifies a DCT coefficient in mode (k, l) in the JPEG filecorresponding to (m, n) ∈ G8×8. This change will affect all8 × 8 pixels in the corresponding block and an entire 15 × 15neighborhood of values in U(i, j ) centered at (m, n) ∈ G8×8.In particular, the values will be modified by what we call the“unit response”

R(i, j )(k,l) = B(i, j ) ⊗ B(k,l), (3)

where ⊗ denotes the full cross-correlation. While this unitresponse is not symmetrical, its absolute values are symmet-rical by both axes: |R(i, j )(k,l)

a,b | = |R(i, j )(k,l)−a,b |, |R(i, j )(k,l)

a,b | =|R(i, j )(k,l)

a,−b | for all 0 ≤ a, b ≤ 7 when indexing R ∈ R15×15

with indices in {−7, . . . ,−1, 0, 1, . . . , 7}.Figure 2 shows two examples of unit responses. Note that

the value at the center (0, 0) is zero for the response on theleft and 1 for the response on the right. This central valueequals to 1 only when i = k and j = l.

We now take a closer look at how a particular valueu ∈ U(i, j ) is computed. First, we identify the four neighborsfrom the grid G8×8 that are closest to u (follow Figure 1where the location of u is marked by a triangle). We willcapture the position of u w.r.t. to its four closest neighborsfrom G8×8 using relative coordinates. With respect to theupper left neighbor (A), u is at position (a, b), 0 ≤ a, b,≤ 7((a, b) = (3, 2) in Figure 1). The relative positions w.r.t. theother three neighbors (B–D) are, correspondingly, (a, b − 8),(a − 8, b), and (a − 8, b − 8). Also recall that the elementsof U(i, j ) collected across all (i, j), 0 ≤ i, j ≤ 7, at A, formall non-quantized DCT coefficients corresponding to the 8×8block A (see, again Figure 1).

Arranging the DCT coefficients from the neighboring blocksA–D into 8×8 matrices Akl , Bkl , Ckl and Dkl , where k and l

HOLUB AND FRIDRICH: LOW-COMPLEXITY FEATURES FOR JPEG STEGANALYSIS 221

Fig. 1. Left: Dots correspond to elements of U(i, j) = X � B(i, j), circles correspond to grid points from G8×8 (DCT coefficients in the JPEG representationof X). The triangle is an element u ∈ U(i, j) with relative coordinates (a, b) = (3, 2) w.r.t. its upper left neighbor (A) from G8×8. Right: JPEG representationof X when replacing each 8 × 8 pixel block with a block of quantized DCT coefficients.

denote the horizontal and vertical spatial frequencies in the8 × 8 DCT block, respectively, u ∈ U(i, j ) can be expressed as

u =7∑

k=0

7∑

l=0

Qkl

[Akl R(i, j )(k,l)

a,b + Bkl R(i, j )(k,l)a,b−8

+ Ckl R(i, j )(k,l)a−8,b + Dkl R(i, j )(k,l)

a−8,b−8

], (4)

where the subscripts in R(i, j )(k,l)a,b capture the position of u w.r.t.

its upper left neighbor and Qkl is the quantization step of the(k, l)-th DCT mode. This can be written as a projection of256 dequantized DCT coefficients from four adjacent blocks

from the JPEG file with a projection vector p(i, j )a,b

u =

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

Q00 A00...

Q77 A77Q00 B00

...Q77 B77

...Q00 D00

...Q77 D77

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

T

·

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

R(i, j )(1,1)a,b

...

R(i, j )(8,8)a,b

R(i, j )(1,1)a−8,b

...

R(i, j )(8,8)a−8,b

...

R(i, j )(1,1)a−8,b−8

...

R(i, j )(8,8)a−8,b−8

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

︸ ︷︷ ︸p(i, j)

a,b

. (5)

It is proved in Appendix that the projection vectorsform an orthonormal system satisfying for all (a, b), (i, j),

Fig. 2. Examples of two unit responses scaled so that medium graycorresponds to zero.

and (k, l)

p(i, j )Ta,b · p(k,l)

a,b = δ(i, j ),(k,l), (6)

where δ is the Kronecker delta. Projection vectors that aretoo correlated (in the extreme case, linearly dependent) wouldlead to undesirable redundancy (near duplication) of fea-ture elements. Orthonormal (uncorrelated) projection vectorsincrease features’ diversity and provide better dimensionality-to-detection ratio.

The projection vectors also satisfy the following symmetry∣∣∣p(i, j )

a,b

∣∣∣ =∣∣∣p(i, j )

a,b−8

∣∣∣ =∣∣∣p(i, j )

a−8,b

∣∣∣ =∣∣∣p(i, j )

a−8,b−8

∣∣∣ (7)

for all i, j and a, b when interpreting the arithmetic operationson indices as mod 8.

III. DCTR FEATURES

The DCTR features are built by quantizing the absolutevalues of all elements in the undecimated DCT and collecting

222 IEEE TRANSACTIONS ON INFORMATION FORENSICS AND SECURITY, VOL. 10, NO. 2, FEBRUARY 2015

TABLE I

HISTOGRAMS ha,b TO BE MERGED ARE LABELED WITH THE SAME

LETTER. ALL 64 HISTOGRAMS CAN THUS BE MERGED INTO 25.

LIGHT SHADING DENOTES MERGING OF FOUR HISTOGRAMS,

MEDIUM SHADING TWO HISTOGRAMS, AND DARK

SHADING DENOTES NO MERGING

the first-order statistic separately for each mode (k, l) and eachrelative position (a, b), 0 ≤ a, b ≤ 7. Formally, for each(k, l) we define the matrix2 U(k,l)

a,b ∈ R(M−8)/8×(N−8)/8 as a

submatrix of U(k,l) with elements whose relative coordinatesw.r.t. the upper left neighbor in the grid G8×8 are (a, b). Thus,

each U(k,l) = ∪7a,b=0U(k,l)

a,b and U(k,l)a,b ∩ U(k,l)

a′,b′ = ∅ whenever(a, b) = (a′, b′). The feature vector is formed by normalizedhistograms for 0 ≤ k, l ≤ 7, 0 ≤ a, b ≤ 7:

h(k,l)a,b (r) = 1

∣∣U(k,l)a,b

∣∣∑

u∈U(k,l)a,b

[QT (|u|/q) = r ], (8)

where QT is a quantizer with integer centroids {0, 1, . . . , T },q is the quantization step, and [P] is the Iverson bracketequal to 0 when the statement P is false and 1 when P istrue. We note that q could potentially depend on a, b as wellas the DCT mode indices k, l, and the JPEG quality factor(see Section III-D for more discussions).

Because U(k,l) = X � B(k,l) and the sum of all ele-ments of B(k,l) is zero (they are DCT modes (2)) eachU(k,l) is an output of a high-pass filter applied to X. Fornatural images X, the distribution of u ∈ U(k,l)

a,b will thusbe approximately symmetrical and centered at 0 for all a, b,

which allows us to work with absolute values of u ∈ U(k,l)a,b

giving the features a lower dimension and making them betterpopulated.

Due to the symmetries of projection vectors (7), it ispossible to further decrease the feature dimensionality byadding together the histograms corresponding to indices (a, b),(a, 8−b), (8−a, b), and (8−a, 8−b) under the condition thatthese indices stay within {0, . . . , 7}× {0, . . . , 7} (see Table I).Note that for (a, b) ∈ {1, 2, 3, 5, 6, 7}2, we merge fourhistograms. When exactly one element of (a, b) is in {0, 4},only two histograms are merged, and when both a and bare in {0, 4} there is only one histogram. Thus, the totaldimensionality of the symmetrized feature vector is 64 ×(36/4 + 24/2 + 4) × (T + 1) = 1600 × (T + 1).

2Since U(k,l) ∈ R(M−7)×(N−7), the height (width) of U(k,l)

a,b is larger byone when a = 0 (b = 0).

In the rest of this section, we provide experimental evi-dence that working with absolute values and symmetrizingthe features indeed improves the detection accuracy. We alsoexperimentally determine the proper values of the threshold Tand the quantization step q , and evaluate the performance ofdifferent parts of the DCTR feature vector w.r.t. the DCT modeindices k, l.

A. Experimental Setup

All experiments in this section are carried out on BOSSbase1.01 [2] containing 10,000 grayscale 512 × 512 images.All detectors were trained as binary classifiers imple-mented using the FLD ensemble [21] with default settingsavailable from http://dde.binghamton.edu/download/ensemble.As described in the original publication [21], the ensembleby default minimizes the total classification error probabilityunder equal priors PE. The random subspace dimensionalityand the number of base learners is found by minimizingthe out-of-bag (OOB) estimate of the testing error, EOOB,on bootstrap samples of the training set. We also use EOOBto report the detection performance since it is an unbiasedestimate of the testing error on unseen data [5]. For experi-ments in Sections III-B–III-E, the steganographic method wasJ-UNIWARD at 0.4 bit per non-zero AC DCT coefficient(bpnzAC) with JPEG quality factor 75. We selected thissteganographic method as an example of a state-of-the-art datahiding method for the JPEG domain.

B. Symmetrization Validation

In this section, we experimentally validate the feature sym-metrization. We denote by EOOB(X) the OOB error obtainedwhen using features X . The histograms concatenated over theDCT mode indices will be denoted as

ha,b =7∨

k,l=0

h(k,l)a,b . (9)

For every combination of indices a, b, c, d ∈ {0, . . . , 7}2,we computed three types of error (the symbol ‘&’ meansfeature concatenation):

1) ESinglea,b � EOOB(ha,b)

2) EConcat(a,b),(c,d) � EOOB(ha,b ∨ hc,d )

3) EMerged(a,b),(c,d) � EOOB(ha,b + hc,d )

to see the individual performance of the features across therelative indices (a, b) as well as the impact of concatenatingand merging the features on detectability. In the followingexperiments, we fixed q = 4 and T = 4. This gave eachfeature ha,b the dimensionality of 64 × (T + 1) = 320 (thenumber of JPEG modes, 64, times the number of quantizationbins T + 1 = 5).

Table II informs us about the individual performance offeatures ha,b. Despite the rather low dimensionality of 320,every ha,b achieves a decent detection rate by itself (c.f.,Figure 4 in Section IV).

The next experiment was aimed at assessing the loss ofdetection accuracy when merging histograms corresponding

HOLUB AND FRIDRICH: LOW-COMPLEXITY FEATURES FOR JPEG STEGANALYSIS 223

TABLE II

ESINGLEa,b IS THE DETECTION OOB ERROR WHEN

STEGANALYZING WITH ha,b

TABLE III

EMERGED(a,b),(c,d) − ECONCAT

(a,b),(c,d) FOR (a, b) AS A FUNCTION OF (c, d)

TABLE IV

EOOB(h(k,l)) AS A FUNCTION OF k, l

to different relative coordinates as opposed to concatenatingthem. When this drop of accuracy is approximately zero, bothfeature sets can be merged. Table III shows the detectiondrop EMerged

(a,b),(c,d) − EConcat(a,b),(c,d) when merging h1,2 with hc,d

as a function of c, d . The results clearly show which featuresshould be merged; they are also consistent with the symmetriesanalyzed in Section II-B.

C. Mode Performance Analysis

In this section, we analyze the performance of the DCTRfeatures by DCT modes when steganalyzing with the mergerh(k,l) �

∑7a,b=0 h(k,l)

a,b of dimension 25 × (T + 1) = 125.Table I explains why the total number of histograms can bereduced from 64 to 25 by merging histograms for differentshifts a, b. Interestingly, as Table IV shows, for J-UNIWARD

Fig. 3. The effect of feature quantization without normalization (top charts)and with normalization (bottom charts) on detection accuracy.

the histograms corresponding to high frequency modes providethe same or better distinguishing power than those of lowfrequencies.

D. Feature Quantization and Normalization

In this section, we investigate the effect of quantization andfeature normalization on the detection performance.

We carried out experiments for two quality factors,75 and 95, and studied the effect of the quantization step qon detection accuracy (the two top charts in Figure 3).Additionally, we also investigated whether it is advanta-geous, prior to quantization, to normalize the features bythe DCT mode quantization step, Qkl , and by scaling U(k,l)

to a zero mean and unit variance (the two bottom chartsin Figure 3).

Figure 3 shows that the effect of feature normalization isquite weak and it appears to be slightly more advantageousto not normalize the features and keep the feature designsimple. The effect of the quantization step q is, however, muchstronger. For quality factor 75 (95), the optimal quantizationsteps were 4 (0.8). Thus, we opted for the following linear fit3

to obtain the proper value of q for an arbitrary quality factor

3Coincidentally, the term in the bracket corresponds to the multiplier usedfor computing standard quantization matrices.

224 IEEE TRANSACTIONS ON INFORMATION FORENSICS AND SECURITY, VOL. 10, NO. 2, FEBRUARY 2015

Fig. 4. Detection error EOOB for J-UNIWARD for quality factors 75 and 95when steganalyzed with the proposed DCTR and other rich feature sets.

TABLE V

EOOB OF THE ENTIRE DCTR FEATURE SET WITH DIMENSIONALITY

1600 × (T + 1) AS A FUNCTION OF THE THRESHOLD T

FOR J-UNIWARD AT 0.4 BPNZAC

in the range 50 ≤ K ≤ 99:

qK = 8 ×(

2 − K

50

). (10)

E. Threshold

As Table V shows, the detection performance is quite insen-sitive to the threshold T . Although the best performance isachieved with T = 6, the gain is negligible compared tothe dimensionality increase. Thus, in this paper we optedfor T = 4 as a good compromise between performance anddetectability.

Fig. 5. Detection error EOOB for UED with ternary embedding for qualityfactors 75 and 95 when steganalyzed with the proposed DCTR and other richfeature sets.

To summarize, the final form of DCTR features includes thesymmetrization as explained in Section III, no normalization,quantization according to (10), and T = 4. This gives theDCTR set the dimensionality of 8,000.

IV. EXPERIMENTS

In this section, we subject the newly proposed DCTR featureset to tests on selected state-of-the-art JPEG steganographicschemes as well as examples of older embedding schemes.Additionally, we contrast the detection performance to previ-ously proposed feature sets. Each time a separate classifieris trained for each image source, embedding method, andpayload to see the performance differences.

Figures 4, 5 and 6 show the detection error EOOB forJ-UNIWARD [14], ternary-coded UED (Uniform Embed-ding Distortion) [12], and nsF5 [11] achieved using theproposed DCTR, the JPEG Rich Model (JRM) [20] ofdimension 22,510, the 12,753-dimensional version of theSpatial Rich Model called SRMQ1 [10], the merger of JRMand SRMQ1 abbreviated as JSRM (dimension 35,263), andthe 12,870 dimensional Projection Spatial Rich Model with

HOLUB AND FRIDRICH: LOW-COMPLEXITY FEATURES FOR JPEG STEGANALYSIS 225

Fig. 6. Detection error EOOB for nsF5 for quality factors 75 and 95 whensteganalyzed with the proposed DCTR and other rich feature sets.

quantization step 3 specially designed for the JPEG domain(PSRMQ3) [13]. When interpreting the results, one needsto take into account the fact that the DCTR has by far thelowest dimensionality and computational complexity of alltested feature sets.

The most significant improvement is seen for J-UNIWARD,even though it remains very difficult to detect. Despite its com-pactness and a significantly lower computational complexity,the DCTR set is the best performer for the higher quality factorand provides about the same level of detection as PSRMQ3 forquality factor 75. For the ternary UED, the DCTR is the bestperformer for the higher JPEG quality factor for all but thelargest tested payload. For quality factor 75, the much larger35,263-dimensional JSRM gives a slightly better detection.The DCTR also provides quite competitive detection for nsF5.The detection accuracy is roughly at the same level as for the22,510-dimensional JRM.

The DCTR feature set is also performing quite wellagainst the state-of-the-art side-informed JPEG algorithmSI-UNIWARD [14] (Figure 7). On the other hand,

Fig. 7. Detection error EOOB for the side-informed SI-UNIWARD for qualityfactors 75 and 95 when steganalyzed with the proposed DCTR and other richfeature sets. Note the different scale of the y axis.

JSRM and JRM are better suited to detect NPQ [15] (Figure 8).This is likely because NPQ introduces (weak) embeddingartifacts into the statistics of JPEG coefficients that are easierto detect by the JRM, whose features are entirely built asco-occurrences of JPEG coefficients. We also point out thesaturation of the detection error below 0.5 for quality factor95 and small payloads for both schemes. This phenomenon,which was explained in [14], is caused by the tendency of bothalgorithms to place embedding changes into four specific DCTcoefficients.

In Table VI, we take a look at how complementary theDCTR features are in comparison to the other rich models.This experiment was run only for J-UNIWARD at 0.4 bpnzAC.The DCTR seems to well complement PSRMQ3 as this20,870-dimensional merger achieves so far the best detectionof J-UNIWARD, decreasing EOOB by more than 3% w.r.t.the PSRMQ3 alone. Next, we report on the computationalcomplexity when extracting the feature vector using a Matlabcode. The extraction of the DCTR feature vector for oneBOSSbase image is twice as fast as JRM, ten times faster

226 IEEE TRANSACTIONS ON INFORMATION FORENSICS AND SECURITY, VOL. 10, NO. 2, FEBRUARY 2015

Fig. 8. Detection error EOOB for the side-informed NPQ for quality factors75 and 95 when steganalyzed with the proposed DCTR and other rich featuresets.

TABLE VI

DETECTION OF J-UNIWARD AT PAYLOAD 0.4 BPNZAC WHEN MERGING

VARIOUS FEATURE SETS. THE TABLE ALSO SHOWS THE FEATURE

DIMENSIONALITY AND TIME REQUIRED TO EXTRACT A SINGLE

FEATURE FOR ONE BOSSBASE IMAGE ON AN INTEL I5

2.4 GHZ COMPUTER PLATFORM

than SRMQ1, and almost 200 times faster than the PSRMQ3.Furthermore, a C++ (Matlab MEX) implementation takesonly between 0.5–1 sec.

V. CONCLUSION

This paper introduces a novel feature set for steganalysisof JPEG images. Its name is DCTR because the features arecomputed from noise residuals obtained using the 64 DCTbases. Its main advantage over previous art is its relatively lowdimensionality (8,000) and a significantly lower computationalcomplexity while achieving a competitive detection acrossmany JPEG algorithms. These qualities make DCTR a goodcandidate for building practical steganography detectors andin steganalysis applications where the detection accuracy andthe feature extraction time are critical.

The DCTR feature set utilizes the so-called undecimatedDCT. This transform has already found applications insteganalysis in the past. In particular, the reference featuresused in calibration are essentially computed from the undec-imated DCT subsampled on an 8 × 8 grid shifted w.r.t. theJPEG grid. The main point of this paper is the discovery thatthe undecimated DCT contains much more information that isquite useful for steganalysis.

In the spatial domain, the proposed feature set can beinterpreted as a family of one-dimensional co-occurrences(histograms) of noise residuals obtained using kernels formedby DCT bases. Furthermore, the feature set can also be viewedin the JPEG domain as a projection-type model with orthonor-mal projection vectors. Curiously, we were unable to improvethe detection performance by forming two-dimensional co-occurrences instead of first-order statistics. This is likelybecause the neighboring elements in the undecimated DCT arequalitatively different projections of DCT coefficients, makingthe neighboring elements essentially independent.

We contrast the detection accuracy and computationalcomplexity of DCTR with four other rich modelswhen used for detection of five JPEG steganographicmethods, including two side-informed schemes. Thecode for the DCTR feature vector is available fromhttp://dde.binghamton.edu/download/feature_extractors/ (notefor the reviewers: the code will be posted upon acceptance ofthis manuscript).

Finally, we would like to mention that it is possible thatthe DCTR feature set will be useful for forensic applications,such as [24], since many feature sets originally designed forsteganalysis found applications in forensics. We consider thisas a possible future research direction.

APPENDIX

ORTHONORMALITY OF PROJECTION VECTORS IN

UNDECIMATED DCT

Here, we provide the proof of orthonormality (6) of vectorsp(k,l)

a,b defined in (5). It will be useful to follow Figure 9 foreasier understanding. For each a, b, 0 ≤ a, b ≤ 7, the (i, j)thDCT basis pattern B(i, j ) positioned so that its upper left cornerhas relative index (a, b) is split into four 8 × 8 subpatterns:κ stands for cirκ le, μ stands for diaμond, τ for τ riangle, andσ for σ tar:

κ(i, j )mn =

⎧⎪⎨

⎪⎩B(i, j )

m−a,n−ba ≤ m ≤ 7

b ≤ n ≤ 7

0 otherwise

HOLUB AND FRIDRICH: LOW-COMPLEXITY FEATURES FOR JPEG STEGANALYSIS 227

Fig. 9. Diagram showing the auxiliary patterns κ (cirκle), μ (diaμond),τ (τ riangle), and σ (σ tar). The black square outlines the position of the DCTbasis pattern B(i, j).

μ(i, j )mn =

⎧⎪⎨

⎪⎩B(i, j )

m−a,8+n−ba ≤ m ≤ 7

0 ≤ n < b

0 otherwise

τ(i, j )mn =

⎧⎪⎨

⎪⎩B(i, j )

8+m−a,n−b0 ≤ m < a

b ≤ n ≤ 7

0 otherwise.

σ(i, j )mn =

⎧⎪⎨

⎪⎩B(i, j )

8+m−a,8+n−b0 ≤ m < a

0 ≤ n < b

0 otherwise

In Figure 9 top, the four patterns are shown using fourdifferent markers. The light-color markers correspond to zeros.The first 64 elements of p(i, j )

a,b are simply projections of

κ(i, j )mn onto the 64 patterns forming the DCT basis. The next

64 elements are projections of μ(i, j )mn onto the DCT basis, the

next 64 are projections of τ(i, j )mn , and the last 64 are projections

of σ(i, j )mn . We will denote these projections with the same Greek

letters but with a single index instead: (κ(i, j )1 , . . . , κ

(i, j )64 ),

(μ(i, j )1 , . . . , μ

(i, j )64 ), (τ

(i, j )1 , . . . , τ

(i, j )64 ), and (σ

(i, j )1 , . . . , σ

(i, j )64 ).

In terms of the introduced notation,

p(i, j )Ta,b · p(k,l)

a,b =64∑

r=1

κ(i, j )r κ(k,l)

r +64∑

r=1

μ(i, j )r μ(k,l)

r

+64∑

r=1

τ(i, j )r τ (k,l)

r +64∑

r=1

σ(i, j )r σ (k,l)

r . (11)

Note that the sum κ(i, j ) +μ(i, j ) +τ (i, j ) +σ (i, j ) is the entireDCT mode (i, j) split into four pieces and rearranged backtogether to form an 8 × 8 block (Figure 9 botom). For fixeda, b, due to the orthonormality of DCT modes (i, j) and (k, l),κ(i, j ) +μ(i, j ) +τ (i, j ) +σ (i, j ) and κ(k,l) +μ(k,l) +τ (k,l) +σ (k,l)

are thus also orthonormal and so are their projections onto theDCT basis (because the DCT transform is orthonormal):

64∑

r=1

(κ(i, j )r + μ

(i, j )r + τ

(i, j )r + σ

(i, j )r )

×(κ(k,l)r + μ(k,l)

r + τ (k,l)r + σ (k,l)

r ) = δ(i, j ),(k,l). (12)

The orthonormality now follows from the fact that the LHSof (12) and the RHS of (11) have the exact same valuebecause the sum of every mixed term in (12) is zero (e.g.,∑64

r=1 κ(i, j )r τ

(k,l)r = 0, etc.). This is because the subpatterns

κ(i, j ) and τ (k,l) have disjoint supports (their dot product in thespatial domain is 0 and thus the product in the DCT domainis also 0 because DCT is orthonormal).

ACKNOWLEDGMENT

The U.S. Government is authorized to reproduce and dis-tribute reprints for Governmental purposes notwithstandingany copyright notation there on. The views and conclusionscontained herein are those of the authors and should not beinterpreted as necessarily representing the official policies,either expressed or implied of AFOSR or the U.S. Govern-ment.

REFERENCES

[1] I. Avcibas, N. D. Memon, and B. Sankur, “Steganalysis of watermark-ing techniques using image quality metrics,” Proc. SPIE, vol. 4314,pp. 523–531, Jan. 2001.

[2] P. Bas, T. Filler, and T. Pevný, “‘Break our steganographic system’: Theins and outs of organizing BOSS,” in Proc. 13th Int. Conf. Inf. Hiding,Prague, Czech Republic, May 2011, pp. 59–70.

[3] S. Bayram, A. E. Dirik, H. T. Sencar, and N. Memon, “An ensembleof classifiers approach to steganalysis,” in Proc. 20th Int. Conf. PatternRecognit. (ICPR), Istanbul, Turkey, Aug. 2010, pp. 4376–4379.

[4] R. Böhme, “Weighted stego-image steganalysis for JPEG covers,” inProc. 10th Int. Workshop Inf. Hiding, vol. 5284, pp. 178–194, Jun. 2007.

[5] L. Breiman, “Bagging predictors,” Mach. Learn., vol. 24, no. 2,pp. 123–140, Aug. 1996.

[6] C. Chen and Y. Q. Shi, “JPEG image steganalysis utilizing bothintrablock and interblock correlations,” in Proc. IEEE Int. Symp. CircuitsSyst. (ISCAS), Seattle, WA, USA, May 2008, pp. 3029–3032.

[7] R. Cogranne and F. Retraint, “Application of hypothesis testing theoryfor optimal detection of LSB matching data hiding,” Signal Process.,vol. 93, no. 7, pp. 1724–1737, Jul. 2013.

[8] H. Farid and L. Siwei, “Detecting hidden messages using higher-orderstatistics and support vector machines,” in Proc. 5th Int. Workshop Inf.Hiding, Oct. 2002, pp. 340–354.

[9] J. Fridrich, M. Goljan, and D. Hogea, “Steganalysis of JPEG images:Breaking the F5 algorithm,” in Proc. 5th Int. Workshop Inf. Hiding,Oct. 2002, pp. 310–323.

228 IEEE TRANSACTIONS ON INFORMATION FORENSICS AND SECURITY, VOL. 10, NO. 2, FEBRUARY 2015

[10] J. Fridrich and J. Kodovský, “Rich models for steganalysis of dig-ital images,” IEEE Trans. Inf. Forensics Security, vol. 7, no. 3,pp. 868–882, Jun. 2011.

[11] J. Fridrich, T. Pevný, and J. Kodovský, “Statistically undetectable JPEGsteganography: Dead ends challenges, and opportunities,” in Proc. 9thACM Multimedia Security Workshop, Sep. 2007, pp. 3–14.

[12] L. Guo, J. Ni, and Y.-Q. Shi, “An efficient JPEG steganographicscheme using uniform embedding,” in Proc. 4th IEEE Int. WorkshopInf. Forensics Security, Tenerife, Spain, Dec. 2012, pp. 169–174.

[13] V. Holub and J. Fridrich, “Random projections of residuals for digitalimage steganalysis,” IEEE Trans. Inf. Forensics Security, vol. 8, no. 12,pp. 1996–2006, Dec. 2013.

[14] V. Holub and J. Fridrich, “Universal distortion design for steganographyin an arbitrary domain,” EURASIP J. Inf. Security, vol. 2014, no. 1,pp. 1–13, 2014.

[15] F. Huang, J. Huang, and Y.-Q. Shi, “New channel selection rule forJPEG steganography,” IEEE Trans. Inf. Forensics Security, vol. 7, no. 4,pp. 1181–1191, Aug. 2012.

[16] A. D. Ker, “Implementing the projected spatial rich features on a GPU,”Proc. SPIE, vol. 9028, pp. 1801–1810, Feb. 2014.

[17] A. D. Ker and T. Pevný, “Identifying a steganographer inrealistic and heterogeneous data sets,” Proc. SPIE, vol. 8303,pp. 83030N-1–83030N-13, Jan. 2012.

[18] A. D. Ker and T. Pevný, “Calibration revisited,” in Proc. 11th ACMMultimedia Security Workshop, Sep. 2009, pp. 63–74.

[19] J. Kodovský and J. Fridrich, “Quantitative structural steganalysis ofJsteg,” IEEE Trans. Inf. Forensics Security, vol. 5, no. 4, pp. 681–693,Dec. 2010.

[20] J. Kodovský and J. Fridrich, “Steganalysis of JPEG images using richmodels,” Proc. SPIE, vol. 8303, pp. 83030A-1–83030A-13, Jan. 2012.

[21] J. Kodovský, J. Fridrich, and V. Holub, “Ensemble classifiers forsteganalysis of digital media,” IEEE Trans. Inf. Forensics Security, vol. 7,no. 2, pp. 432–444, Apr. 2012.

[22] L. Li, H. T. Sencar, and N. Memon, “A cost-effective deci-sion tree based approach to steganalysis,” Proc. SPIE, vol. 8665,pp. 86650P-1–86650P-7, Feb. 2013.

[23] Q. Liu, “Steganalysis of DCT-embedding based adaptive steganogra-phy and YASS,” in Proc. 13th ACM Multimedia Security Workshop,Sep. 2011, pp. 77–86.

[24] Q. Liu and Z. Chen, “Improved approaches to steganalysis and seam-carved forgery detection in JPEG images,” ACM Trans. Intell. Syst. Tech.Syst., vol. 5, no. 4, pp. 39:1–39:30, 2014.

[25] I. Lubenko and A. D. Ker, “Going from small to large data in steganaly-sis,” Proc. SPIE, vol. 8303, pp. 83030M-1–83030M-10, Jan. 2012.

[26] S. Lyu and H. Farid, “Steganalysis using higher-order image statis-tics,” IEEE Trans. Inf. Forensics Security, vol. 1, no. 1, pp. 111–119,Mar. 2006.

[27] T. Pevný and J. Fridrich, “Merging Markov and DCT fea-tures for multi-class JPEG steganalysis,” Proc. SPIE, vol. 6505,pp. 650503-1–650503-14, Feb. 2007.

[28] T. Pevný and J. Fridrich, “Novelty detection in blind steganaly-sis,” in Proc. 10th ACM Multimedia Security Workshop, Sep. 2008,pp. 167–176.

[29] Y. Q. Shi, C. Chen, and W. Chen, “A Markov process based approachto effective attacking JPEG steganography,” in Proc. 8th Int. WorkshopInf. Hiding, Jul. 2006, pp. 249–264.

[30] T. H. Thai, R. Cogranne, and F. Retraint, “Statistical model of quantizedDCT coefficients: Application in the steganalysis of Jsteg algorithm,”IEEE Trans. Image Process., vol. 23, no. 5, pp. 1980–1993, May 2014.

[31] A. Westfeld, “Generic adoption of spatial steganalysis to trans-formed domain,” in Proc. 10th Int. Workshop Inf. Hiding, Jun. 2007,pp. 161–177.

[32] A. Westfeld and A. Pfitzmann, “Attacks on steganographic systems,” inProc. 3rd Int. Workshop Inf. Hiding, Sep./Oct. 1999, pp. 61–75.

[33] T. Zhang and X. Ping, “A fast and effective steganalytic techniqueagainst JSteg-like algorithms,” in Proc. ACM Symp. Appl. Comput.,Melbourne, FL, USA, Mar. 2003, pp. 307–311.

[34] C. Zitzmann, R. Cogranne, L. Fillatre, I. Nikiforov, F. Retraint,and P. Cornu, “Hidden information detection based on quantizedLaplacian distribution,” in Proc. IEEE ICASSP, Kyoto, Japan, Mar. 2012,pp. 1793–1796.

Vojtech Holub is currently a Research and Develop-ment Engineer with Digimarc Corporation, Beaver-ton, OR, USA. He received the Ph.D. degree fromthe Department of Electrical and Computer Engi-neering, Binghamton University, Binghamton, NY,USA, in 2014. The main focus of his disserta-tion was on steganalysis and steganography. Hereceived the M.S. degree in software engineeringfrom Czech Technical University in Prague, Prague,Czech Republic, in 2010.

Jessica Fridrich (M’05) is currently a Professor ofElectrical and Computer Engineering with Bingham-ton University, Binghamton, NY, USA. She receivedthe Ph.D. degree in systems science from Bing-hamton University, in 1995, and the M.S. degreein applied mathematics from Czech Technical Uni-versity, Prague, Czech Republic, in 1987. Her maininterests are in steganography, steganalysis, digi-tal watermarking, and digital image forensics. Herresearch work has been generously supported by theU.S. Air Force and the Air Force Office of Scientific

Research. Since 1995, she has received 19 research grants totaling over $9million for projects on data embedding and steganalysis that lead to over160 papers and seven U.S. patents. She is a member of the Association forComputing Machinery.