Adaptive Reversible Data Hiding in Frequency
Domain via Integer-to-Integer Transform
Taichi Yoshida∗ , Yusuke Okamura∗ , Taizo Suzuki† , and Masaaki Ikehara∗∗ EEE Dept., Keio Univ., Yokohama, Kanagawa 223-8522, Japan

E-mail: { yoshida, yokamura, ikehara }@tkhm.elec.keio.ac.jp† EEE Dept., College of Eng., Nihon Univ., Koriyama, Fukushima 963-8642, Japan

E-mail: taizo@ee.ce.nihon-u.ac.jp

Abstract—This paper proposes an adaptive reversible datahiding algorithm for images, which embeds significant information in frequency domain based on integer-to-integer transform.
The embedding method is realized by modifying state-of-the-art
one according to integer-to-integer transformed coefficients. Itovercomes a problem where the conventional method often generates particular artifacts. The simulation results show that ourproposed algorithm outperforms the conventional one about thevisual quality of embedded images, objectively and perceptually,
while keeping embedding capacity.

I. I NTRODUCTION
Image Data hiding is a technique to embed useful information into images for various purposes, such as contentauthentication, protect privacy, forensic tracking and so on [1].
Especially, it is one of the most significant technique to protect
the copyright information of contents.
Recently, several reversible data hiding algorithms havebeen proposed [1]–[5]. Different from irreversible one, it isable to restore original images after data extraction. Thereversibility is essential when the original image should be
losslessly restored, such as medical, military, and art images.
The method in [3] has large embedding capacity. It multiplies
original signals by 2m (m ∈ N) and embeds significant
information in the lower bits. To avoid the overflow, it subtracts an uniform value depending on an average of original
signals from embedded signals. Moreover, to improve a visualquality of embedded images, an adaptive method [5] has been
proposed. It divides an image into some blocks, and switches
a budget of embedding information according to a varianceof each block, adaptively. If the variance is low, a large
budget is embedded, because a distortion between original
and embedded signals is small, and vice versa. Thus, theadaptive method achieves both excellent visual quality andlarge embedding capacity. However, particular artifacts often
appear at the high embedding capacity. It violates a policy ofdate hiding that hiding information has not to be recognized.
In this paper, we solve the problem by using an adaptive reversible data hiding algorithm in frequency domain
based on integer-to-integer transform, while keeping the highembedding performance. In natural images, a variance ofeach block in frequency subbands is lower than one inspatial domain, generally. Moreover, distortions generated byembedding significant information spread to neighborhood

pixels via the inverse transform, and are not perceptually
distinct by human visual system in spatial domain. For a
rapid calculation, the proposed algorithm is based on nonlapped transforms. Because, in a case of lapped transforms, averification that embedded signals do not overflow/underflow
is a complicate process. We utilize the integer discrete cosine
transform (IntDCT) [6]–[8] as one of the simplest and the mostpopular integer-to-integer transform techniques. In simulation
results, our proposed method outperforms the conventional
ones, objectively and perceptually.
II. R EVIEW
A. Reversible Data Hiding
In [3], a high capacity reversible data hiding algorithm has
been proposed. Let x ∈ Zn+1 , y ∈ Zn+1 and w ∈ Zn be
original signals, embedded signals and embedded information,respectively. The algorithm embeds w in x to derive y withan arbitrary capacity parameter k, where k = 2m (m ∈ N).
Elements of w are restricted to be represented by log2 k bits.
Hence, the algorithm can embed n log2 k bits into n+1 signals,
and the capacity parameter controls the embedding rate. It is
formulated as{
kxt − an,k (x)for t = 0,(1)yt =kxt − an,k (x) + wt−1 for t ̸= 0,
where xt , yt and wt are elements of x, y and w, respectively,t ∈ Z, 0 ≤ t ≤ n,∑n⌊⌋2(k − 1) t=0 xt + n(k − 1)an,k (x) =,2(n + 1)
and ⌊·⌋ is a floor function. Since (1) is an integer transform,
x and w are losslessly restored from y. The inverse transformis defined as⌋ ⌊⌊⌋⌋n ⌊n(k − 1)
yt − y0k − 1 ∑ yt − y0++,xt = y0 +n + 1 t=1k2(n + 1)k(2)where wt (1 ≤ t ≤ n) is extracted as
wt = yt − kxt + an,k (x).

B. Adaptive Reversible Data Hiding
The adaptive reversible data hiding in spatial domain has
been proposed [5]. For images, it divides an image into someblocks which contain n + 1 signals. At each block, it switches
the capacity parameter k according to a variance of signals,
adaptively. k of one block is determined depending on thegiven threshold parameter T and the variance of the block
V (x). From experiences, k is restricted as {1, 2, 4, 8}, and thedetermination is defined as follows:
1) If V (x) ≤ T /49 and x ∈ A8 , k = 8.
2) If V (x) ≤ T /9 and x ∈ A4 , k = 4.
3) If V (x) ≤ T and x ∈ A2 , k = 2.
4) Otherwise k = 1.
In general, grayscale images are represented by 8-bit, i.e.,0 ≤ xt ≤ 255. If x ∈ Ak , the corresponding y derivedby (1) with k is verified in [0, 255], i.e., y does not have
overflow/underflow signals. The set Ak is defined as follows:

Fig. 1: A lifting structures of a 2 × 2 matrix D whose
determinant is 1 (R means a rounding operator).

Fig. 2: The 8-channel IntDCT in [6].

A = { x = [x0 , x1 , · · · , xn ] | 0 ≤ xt ≤ 255}
Ak = { x ∈ A | 0 ≤ kx0 − an,k (x) ≤ 255,0 ≤ kxt − an,k (x) ≤ 256 − k (1 ≤ t ≤ n)}(3)

C. IntDCT
Discrete cosine transform (DCT) is one of the most usefulfrequency transform techniques, applied for many signal processing applications. The (i, j) element of type-II DCT matrix
C with M channel (i, j = 0, 1, · · · , M − 1) is represented as√()2i(j + 1/2)πci,j =αi cos,(4)MM√where αi is 1/ 2 (i = 0) or 1 (i ̸= 0).
The IntDCT is realized by lifting factorization of DCT. Let
D be a 2 × 2 matrix whose determinant is 1. D is factorized
into the lifting structures as[] [][][]
10 1 b10a b
D== d−1,(5)c d1
1 0 1 a−1bb
where b ̸= 0 [9] as shown in Fig. 1. The lifting structure with rounding operators transforms integer signals into
integer ones. Hence, original signals are losslessly restoredfrom transformed ones via its inverse operation. Based on
the lifting factorization, many IntDCTs have been proposed[6]–[8]. In this paper, we utilize an IntDCT which has theleast rounding operators and shows efficient image coding
performance [6]. The 8-channel IntDCT is shown in Fig. 2,where Cqp = cos(pπ/q), Sqp = sin(pπ/q), and

0 0 1 010 0 0
0 1 0 0  01 0 0


T =
1 0 0 0  0 −1 1 0
−1 0 0 1
0 0 0 1
(6)
1 0 1/2 1/2−1 0 0 0
0 1 1/2 −1/2  0 −1 0 0

.
×
0 0
10  10 1 0
0 00
10 −1 0 1

III. A DAPTIVE R EVERSIBLE DATA H IDING IN F REQUENCYD OMAIN VIA I NT DCT
A. Algorithm
We propose a new adaptive reversible data hiding algorithm.
It is realized by applying the adaptive reversible data hiding
method in Sec. II-B to the frequency domain.
Initially, we divide an image into some blocks with size
M × (n + 1). M corresponds to the channel of IntDCT and an
arbitrary number M can be easily obtained by [8]. This paper,
however, utilizes an 8-channel IntDCT in [6], i.e., M = 8.
Next, a block is transformed by the IntDCT as yj = Φ(xj )(j ∈ Z, 0 ≤ j ≤ n), where xj and yj are j-th column vectorof the input block X ∈ ZM ×(n+1) and the transformed block
Y ∈ ZM ×(n+1) , respectively, and Φ means a function of theIntDCT operation in Fig. 2. Similarly to (1), the embedded
block Z ∈ ZM ×(n+1) is derived from Y with an embedded
information W ∈ ZM ×n and capacity parameters k ∈ ZM .
We apply the embed algorithm to each subband of Y, i.e.,applying (1) to each row vector of Y. The proposed embedalgorithm is defined as{
ki yi,j − an,ki (ỹi )for j = 0,zi,j =(7)ki yi,j − an,ki (ỹi ) + wi,j−1 for j ̸= 0,where i ∈ Z, 0 ≤ i ≤ M − 1, yi,j , zi,j and wi,j are (i, j)elements of Y, Z and W, ỹi is i-th row vector of Y, and ki
is i-th element of k, respectively. Needless to say, a range ofwi,j is restricted by ki .
Finally, the embedded block in the spatial domain is derived−1
from the inverse IntDCT∑M of Z, described as Φ (Z). Thisalgorithm embeds n i=1 log2 ki bits into M (n + 1) pixels.
It is the same capacity rate as in Sec. II-B. Through an inverse
process of the proposed algorithm, both of the original image
and the embedded information are losslessly restored.

B. Determination of Capacity Parameters
In this section, we show the determination of capacity
parameters. As mentioned above, embedded signals should bein [0, 255], and then the new verification is proposed.
Similar to Sec. II-B, we define a new set Bk as{}B = X ∈ ZM ×(n+1) | 0 ≤ xi,j ≤ 255 ,∑M −1(8)Bk = {X ∈ B | 0 − p=0 c−p,i (kp − 1) + α ≤ ei,j∑M −1 +
≤ 255 − p=0 cp,i (kp − 1) − α}.
where notations are same as the previous section, ci,j is (i, j)element of the DCT matrix C in (4), α is a arbitrary number±
as a margin, and c±i,j is (i, j) element of C defined as{{ci,j if ci,j ≥ 00if ci,j ≥ 0c+, c−. (9)i,j =i,j =0if ci,j < 0ci,j if ci,j < 0ei,j is defined with yi,j and ỹi asei,j =M−1∑{}
cp,i kp yp,j − an,kp (ỹp ) .

(10)

p=0∑nei,j is (i, j) element of Z with W = 0. p=0 c±p,i (kp − 1)
are the maximum predicted negative and positive value ofΦ−1 (W), which means distortions due to embedding with k.
Similar to Sec. II-B, if X ∈ Bk , corresponding Φ−1 (Z) withk and W is guaranteed to avoid the overflow/underflow.
Hence, the determination algorithm of k is defined with thegiven threshold parameter T as follows:1) The initial k is determined as
8 if V (ỹi ) ≤ T /494 if V (ỹ ) ≤ T /9iki =2ifV(ỹi) ≤ T
1 otherwise
where V (ỹi ) is a variance of ỹi .
2) Let k̂ be a maximum of k, and P = {i | ki = k̂}.
3) If X ∈ Bk or k̂ = 1, escape algorithm with k.
4) kp = kp /2, where (p ∈ P ) is an index whose V (ỹp ) ismaximum of V (ỹi ). Go to Step 2.
However, if X ∈ Bk , it is not exactly guaranteed thatΦ−1 (Z) does not overflow/underflow, because the rounding
error in each lifting step cannot be exactly predicted and itdepends on input signals. If the optimal k is required, Y isembedded with arbitrary k and W, and the overflow/underflow
is checked, iteratively. However, the iteration requires the highcomputational cost, and takes a long time to process dueto the non-parallel process, i.e., a current block has to wait
until the previous blocks are finished. Therefore, to reduce thecomplexity, we define as (8).
IV. S IMULATIONAs a simulation, the proposed method is compared withthe conventional ones. Six images (Lena, Airplane, Barbara,Boat, Goldhill, and Baboon) are used as test images (8bit grayscale), shown in Fig. 3. As conventional methods,(a) Lena(b) Airplane(c) Barbara(d) Boat(e) Goldhill(f) Baboon

Fig. 3: Test images

the Luo’s and Peng’s methods [4], [5], which are the stateof-the-art methods of data hiding, are used. According to[5], the block size is 4 × 4, and then n = 15. A set ofcapacity parameters is called the location map (LM). The
implementation algorithm, which embeds the LM and theembedded information and losslessly extracts them, is similar
to [5]. Details are shown in [5].
Table I shows the objective visual quality of embedded
images, measured by the peak signal-to-noise ratio (PSNR).
The rate is represented in bits per pixel (bpp). At 1.2 [bpp]in Baboon, all methods are not able to embed. The proposed
method shows the better results than the conventional ones,especially for images with rich high frequency components
such as Barbara. The capacity-distortion curves are shown inFig. 4. The proposed method outperforms at the high capacity
rate, and a lowest rate, in which the proposed method shows
the best result, depends on features of the image. If images
have rich high frequency components, the lowest rate is lower,and vice versa. However, the maximum capacity rate of theproposed method is less than one of the Peng’s method,
because of the determination of capacity parameters.
Fig. 5 shows a particular area of the embedded images in
Lena at 1.0 bpp. Although the particular distortions in Peng’s
method are perceived, the distortions in the proposed images
is similar to the ordinary random noise. On the other hand, due
to the histogram shifting algorithm processing pixel by pixel,
an excessive contrast enhancement and a particular asperous
artifact occurs in textures of Luo’s images. It is the policyof data hiding that the embedding information has not to beunderstood. In that sense, the proposed method has better
perceptual results because the particular distortion has a higher
risk of recognizing the hiding than a random noise.
V. C ONCLUSION
In this paper, we proposed an adaptive reversible data hiding
in frequency domain via IntDCT. The proposed method shows
better performance than the conventional ones, especially, forthe high capacity rate and images with rich high frequency
components,.
ACKNOWLEDGMENT
This work in supported by Grant-in-Aid for JSPS Fellows.

TABLE I: PSNR of embedded images in [dB]
Image
Lena

Airplane

BarbaraBoatGoldhill

Baboon

Rate
0.8
1.0
1.2
0.8
1.0
1.2
0.8
1.0
1.2
0.8
1.0
1.2
0.8
1.0
1.2
0.8
1.0
1.2

Luo’s [4]
35.63
32.63
29.47
39.08
35.81
32.81
32.37
28.67
24.43
30.95
27.51
24.17
32.11
28.75
25.32
23.62
19.56
-Peng’s [5]
35.18
32.67
30.44
38.17
35.41
32.99
32.48
29.52
26.77
31.53
28.94
26.60
31.58
28.96
26.61
24.68
22.26
-Prop.
36.24
33.70
31.60
38.07
35.58
33.31
35.84
33.08
30.46
33.06
30.47
28.21
32.83
30.26
27.88
35.05
22.46
-

(a) Lena

Fig. 5: A particular area of embedded images of Lena at 1.0[bpp]: (1st row) Original (2nd row) Luo’s [4], (3rd row) Peng’s[5], (4th row) Prop.
(b) BarbaraFig. 4: The capacity-distortion curve.

R EFERENCES[1] Q. Cheng and T. S. Huang, “An additive approach to transform-domain
information hiding and optimum detection structure,” IEEE Trans. Multimedia, vol. 3, no. 3, pp. 273–284, Sep. 2001.
[2] B. Yang, M. Schmucker, W. Funk, C. Busch, and S. Sun, “Integer DCTbased reversible watermarking for images using companding technique,”in SPIE, Security, Steganography, and Watermarking of Multimedia
Contents, Jan. 2004, pp. 405–415.
[3] C. Wang, X. Li, and B. Yang, “High capacity reversible image watermarking based on integer transform,” in In Proc. Int. Conf. Image Process.,
Sep. 2010, pp. 217–220.
[4] L. Luo, Z. Chen, M. Chen, X. Zeng, and Z. Xiong, “Reversible image

watermarking using interpolation technique,” IEEE Trans. Inf. Forensics
Security, vol. 5, no. 1, pp. 187–193, Mar. 2010.
[5] F. Peng, X. Li, and B. Yang, “Adaptive reversible data hiding scheme
based on integer transform,” Signal Process., vol. 92, no. 1, pp. 54–62,Jan. 2012.
[6] S. Fukuma, K. Ohyama, M. Iwahashi, and N. Kambayashi, “Lossless 8point fast discrete cosine transform using lossless hadamard transform,”in Tech. Rep. IEICE, Oct. 1999.
[7] S. Chokchaitam, M. Iwahashi, and S. Jitapunkul, “A new unified lossless/lossy image compression based on a new integer DCT,” IEICE Trans.
Inf. Syst., vol. E88-D, no. 2, pp. 403–413, Feb. 2005.
[8] T. Suzuki and M. Ikehara, “Design of block lifting-based discrete cosine
transform type-II and IV,” in IEEE Digital Signal Process. Signal Process.
Educ. Workshop, Jan. 2009, pp. 480–484.
[9] F. A. M. L. Bruekers and A. W. M. van den Enden, “New networks
for perfect inversion and perfect reconstruction,” IEEE J. Sel. AreasCommun., vol. 10, no. 1, pp. 129–137, Jan. 1992.