IEICE TRANS. FUNDAMENTALS, VOL.E97–A, NO.1 JANUARY 2014

384

PAPER

Adaptive Reversible Data Hiding via Integer-to-Integer Subband
Transform and Adaptive Generalized Diﬀerence Expansion Method
Taichi YOSHIDA†a) , Student Member, Taizo SUZUKI††b) , and Masaaki IKEHARA†c) , Members

SUMMARY
We propose an adaptive reversible data hiding methodwith superior visual quality and capacity in which an adaptive generalized
diﬀerence expansion (AGDE) method is applied to an integer-to-integer
subband transform (I2I-ST). I2I-ST performs the reversible subband transform and the AGDE method is a state-of-the-art method of reversible datahiding. The results of experiments we performed objectively and perceptually show that the proposed method has better visual quality than conventional methods at the same embedding rate due to low variance in thefrequency domain.
key words: adaptive reversible data hiding, adaptive generalized diﬀerence expansion, frequency domain, integer-to-integer subband transform

1.

Introduction

With the ongoing development of computer and communication networks, many data hiding methods, which are techniques to embed information in signals, have been proposedfor various purposes such as copyright protection, authentication, covert communication, and forensic tracking [1]–[19]. In typical data hiding for images, ‘cover-image’, ‘messages’, and ‘stego-image’ refer to ‘original image’, ‘embedded information’, and ‘image with embedded information’,respectively. Stego-images are evaluated in four aspects: 1)
robustness, which is the resistance to modifications such as
malicious attacks and image processing, 2) capacity, whichis the maximum volume of messages under a given visualquality, 3) security, which indicates whether undesirable receivers illegally extract messages or not, and 4) visual quality, which is an objective, perceptual measure of the degradation of the stego-image compared with the cover-image
and the invisibility of embedding.
Generally, data hiding methods are categorized as either watermarking or steganography. The former pursues
a high robustness while sacrificing the capacity while thelatter pursues a high capacity while sacrificing the robustness, i.e., they are in a trade-oﬀ relationship. Hence, watermarking is mainly utilized for copyright protection, authentication, and so on [1], [2], and steganography applicaManuscript received April 1, 2013.
Manuscript revised August 19, 2013.
†
The authors are with the Department of Electronics andElectrical Engineering, Keio University, Yokohama-shi, 223-8522
Japan.
††
The author is with the Faculty of Engineering, Information and Systems, University of Tsukuba, Tsukuba-shi, 305-8573
Japan.
a) E-mail: yoshida@tkhm.elec.keio.ac.jpb) E-mail: taizo@cs.tsukuba.ac.jpc) E-mail: ikehara@tkhm.elec.keio.ac.jp
DOI: 10.1587/transfun.E97.A.384

tions are covert communication, forensic tracking, and so on[3]. Steganography is evaluated by its capacity, bit-per-pixel(bpp), at the same visual quality, or vice versa.
Reversible methods can completely restore not only
messages but also the cover-image if the stego-image has
not been modified. The reversibility of the cover-image isconsiderably important for military, medical, and artistic images. Some reversible data hiding methods have been proposed based on the histogram shifting (HS) technique [4]–[8]. These methods trade a quite limited embedding capacity for high visual quality. Recent works [7], [8] haveused HS with image interpolation. HS embedding is applied
for diﬀerences between original and interpolated pixels, andthere are typically few distortions. These methods deliver
excellent visual quality at a low capacity. However, if a highcapacity is required, the visual quality is drastically damaged due to iterative embedding.
Some reversible data hiding methods in the frequency
domain have also been proposed [9]–[11]. In [9], HS is applied for the quantized discrete cosine transform (DCT). In[10], least significant bits (LSBs) are adaptively replacedin the frequency domain obtained via an integer-to-integer
wavelet transform. For the joint photographic experts group(JPEG) [11], [20] has proposed embedding messages in thequantized DCT coeﬃcients according to the allocation andnumber of zero/non-zero coeﬃcients. However, they cannot
achieve a high capacity because they cannot be iteratively
applied.
For the realization of high capacity reversible data hiding, the diﬀerence expansion (DE) method has been proposed [12]. This method divides images into two pixel pairs
and then one pixel value is predicted from the other. A 1-bit
message is embedded in the diﬀerence between the original
and predicted pixels, called a stego-diﬀerence. The stegopixel is produced by adding the predicted pixel and stegodiﬀerence. This method embeds 1 bit into 2 pixels, i.e., its
capacity is 0.5 bpp in one embedding process. In addition, it
can be iteratively applied while the embedded pixel values
have no-over/underflow.
This DE has been expanded into prediction error expansion (PEE) methods [13]–[15] and a generalized DE[16]–[19]. The PEE methods improve prediction and embedding algorithms while the generalized DE embeds arbitrary bits into arbitrary pixels. In [18], n log2 k (k = 2m ,m ∈ N) bits are embedded in n + 1 (n ∈ N) pixels, where kdetermines the capacity of stego-images. To improve the visual quality, [19] has proposed an adaptive generalized DE

c 2014 The Institute of Electronics, Information and Communication Engineers
Copyright YOSHIDA et al.: ADAPTIVE REVERSIBLE DATA HIDING VIA INTEGER-TO-INTEGER SUBBAND TRANSFORM

385

Fig. 1

Proposed data hiding algorithm. Notations are defined in Sect. 3.1 and channel M is 4.

(AGDE) method that adaptively decides k according to thevariance of each cover-block. If the variance is high and k
is large, the embedding raises the massive distortions. The
AGDE method eﬃciently distributes the distortions for better visual quality than the non-adaptive method [18], thusachieving both an excellent visual quality and a high capacity.
In this paper, we propose an adaptive reversible datahiding method with superior visual quality and capacity by
applying the AGDE method [19] to the integer-to-integer
subband transform (I2I-ST), which is briefly outlined inFig. 1. Since natural images usually have a lower varianceof each block of subbands in the frequency domain than one
in the space domain, the distortions in the frequency domain
are objectively lower at the same capacity. I2I-ST and its inverse transform are used for reversibility and to distributethe distortions to neighborhoods after embedding. Thus,
the proposed method improves the visual quality. In thispaper, we utilize the integer discrete cosine transform (IntDCT) [21]–[24], which is one of the most popular I2I-STs[25]–[27]. Experimental results objectively and perceptually show better visual quality than the conventional methods at the same embedding rate due to the low variance inthe frequency domain.
The rest of this paper is organized as follows: Sect. 2
summarizes the DE based data hiding methods and the IntDCT. We then propose our adaptive reversible data hiding
method in the frequency domain obtained via the I2I-ST and
the AGDE in Sect. 3 and introduce its implementing structure in Sect. 4. Section 5 shows the results of several datahiding methods. We conclude in Sect. 6.
2.

Review

2.1 Generalized DE Method

w, respectively, and w j should be represented by log2 k [bit](k = 2m , m ∈ N). k is a capacity parameter. an,k (x) is defined
as

⎥
⎢⎢⎢
⎢⎢⎢ 2(k − 1) nj=0 x j + n(k − 1) ⎥⎥⎥⎥⎥⎥⎦ ,an,k (x) = ⎢⎣(2)2(n + 1)
where · is the floor function. (1) means to embed n log2 k
bits in n + 1 signals, and k controls the embedding rate.
In the extraction, from (1), x and w are reversibly restored from y, which is defined as⎢⎢⎢⎥⎥⎥⎢⎢⎢⎢ k − 1 n yi − y0n(k − 1) ⎥⎥⎥⎥+
x j = y0 + ⎢⎢⎣⎥⎥n + 1 i=1k2(n + 1) ⎦y j − y0,+kw j−1 = y j − kx j + an,k (x) ( j  0).
(3)

2.2 AGDE Method
The generalized method in Sect. 2.1 was recently expanded
into the AGDE method [19], which improves the visualquality by adaptively deciding on the k according to the variance of each cover-block.
As outlined in Sect. 2.1, this method also embeds w inx to derive y, from (1) with k. In general, grayscale images
are presented in 8-bit, i.e., 0 ≤ x j ≤ 255. y j should also be
within [0, 255]. To represent this range condition, sets A and
Ak are introduced, as follows:
A = { x = [x0 , x1 , · · · , xn ] | 0 ≤ x j ≤ 255},
Ak = { x ∈ A | 0 ≤ kx0 − an,k (x) ≤ 255,
0 ≤ kx j − an,k (x) ≤ 256 − k ( j = 1, 2, · · · , n)}.
(4)

A high capacity reversible data hiding method has previously been proposed by Wang et al. [18]. Initially, themethod divides a cover-image into several blocks, called
cover-blocks, which contain n + 1 pixels. Let x ∈ Zn+1 be
the cover-block and an element of x be its pixel. The method
embeds w ∈ Zn in x to derive y ∈ Zn+1 , where w and y aremessages and the stego-block, respectively. This embedding
is defined as⎧⎪⎪
for j = 0
⎨kx j − an,k (x)yj = ⎪,(1)⎪⎩kx j − an,k (x) + w j−1 for j  0Consequently, if x satisfies x ∈ Ak , y ∈ A because of thedefinition of y in (1), and y has no-over/underflow.
In this method, k of each cover-block is determined according to a given threshold parameter T and the varianceof the cover-block V(x). k is experimentally restricted as{1, 2, 4, 8} and determined as follows:

where x j , y j , and w j ( j = 0, 1, ..., n) are elements of x, y, andThis determination is shown in Fig. 2. Let N and kq be

1. If V(x) ≤ T/49 and x ∈ A8 , escape algorithm withk = 8.
2. If V(x) ≤ T/9 and x ∈ A4 , escape algorithm with k = 4.
3. If V(x) ≤ T and x ∈ A2 , escape algorithm with k = 2.
4. Else k = 1.

IEICE TRANS. FUNDAMENTALS, VOL.E97–A, NO.1 JANUARY 2014386

Fig. 3 Lifting structures of a 2 × 2 matrix, where r0 r3 − r1 r2 = 1 andr1  0 (R means a rounding operator).

Fig. 2

Capacity parameter determination of the AGDE algorithm.

the cover-block number and k of q-th cover-block (q =0, 1 · · · , N − 1), respectively. The AGDE method emN−1
log2 kq bits in (n + 1)N signals. Compared
beds n q=0
with the generalized method, the AGDE method achieves
a higher peak signal-to-noise ratio (PSNR) between coverand stego-images, at the same capacity, because it adopts
the optimal kq without over/underflow to minimize the overall distortions with a given capacity. Capacity parameters
for each cover-block are regarded as side information required at extraction. We show how to embed them in thestego-image in Sect. 4.1.

Fig. 4⎡⎤⎡⎤0 0 0⎥⎥⎢⎢⎢0 0 1 0⎥⎥⎥ ⎢⎢⎢ 1⎥⎢⎢⎢0 1 0 0⎥⎥⎥ ⎢⎢⎢ 0
1 0 0⎥⎥⎥⎥⎥⎥⎥ ⎢⎢⎢⎥
T = ⎢⎢⎢⎢⎢⎢⎣1 0 0 0⎥⎥⎥⎦ ⎢⎢⎢⎣ 0 −1 1 0⎥⎥⎥⎥⎦
0 0 0 1 −1 0 0 1⎡⎤⎡⎢⎢⎢1 0 1/2 1/2 ⎥⎥⎥ ⎢⎢⎢−1 0⎢⎢⎢0 1 1/2 −1/2⎥⎥⎥ ⎢⎢⎢ 0 −1⎥⎥ ⎢⎢× ⎢⎢⎢⎢
0 ⎥⎥⎥⎥ ⎢⎢⎢⎢ 1
0⎢⎢⎣0 0 1
⎦⎣
0 0 0
1
0 −1

2.3 IntDCT
Discrete cosine transform (DCT) is one of the most usefulsubband transform techniques and is applied for many signal processing applications, especially image/video coding
standards. In the type-II DCT matrix C with a channel number M, its (u, v) element cu,v (u, v = 0, 1, · · · , M − 1) is defined asu(v + 1/2)π2cu,v =
βu cos,(5)MM√
where βu is 1/ 2 (u = 0) or 1 (u  0).
IntDCT is an I2I-ST realized by the lifting factorizationof DCT [21]–[24]. A 2 × 2 matrix can be factorized into thelifting structures as



 

0
1
0 1 r1 1
r0 r1
,
(6)
= r3 −1
r2 r3
1 0 1 r0r−11
r11
where r0 r3 − r1 r2 = 1, and r1  0 [28]. Since the lifting
structures with the rounding operations as shown in Fig. 3(a)achieve I2I-ST, original signals are reversibly restored via its
inverse operations as shown in Fig. 3(b). In this paper, forsimplicity, we utilize the 8-channel IntDCT in [21], which
has least rounding operators and shows eﬃcient image coding performance. The IntDCT is shown in Fig. 4, where
Cqp = cos(pπ/q), S qp = sin(pπ/q), and the 4-channel lifting
based Walsh-Hadamard transform is defined as

8-channel IntDCT in [21].

0
0
1
0

⎤
0⎥⎥
⎥
0⎥⎥⎥⎥
⎥.
0⎥⎥⎥⎥
⎦
1

(7)

The rotation matrices in Fig. 4 can be lifting-factorized as(6). Note that the frequency response of IntDCT is notstrictly the same as the one of DCT due to the rounding errors.
3.

Adaptive Reversible Data Hiding via I2I-ST and
AGDE Method

3.1 Formulation
In this paper, we propose an adaptive reversible data hiding method by applying the AGDE method in Sect. 2.2 tothe frequency domain obtained via the IntDCT in Sect. 2.3
as the I2I-ST method. This proposed method can easily beapplied for any I2I-ST method with minor modifications.
Referring to Fig. 1, initially, we divide the cover-image
into some cover-blocks with size M×(n+1). Experimentally,
we set M = 8 in this paper, but the number of M can beeasily adapted to an arbitrary number [23]. Let X ∈ Z M×(n+1)
be the cover-block, where Z M×(n+1) denotes a set of integer
matrices with size M × (n + 1).
Next, X is transformed by the IntDCT as y j = Φ(x j ),where x j and y j are j-th column vectors of X and the transformed block Y ∈ Z M×(n+1) , respectively, and Φ means thefunction of the IntDCT operation in Fig. 4. In the proposed
method, let W ∈ Z M×n and k ∈ Z M be messages and capacity parameters of one cover-block. We embed W with k in

YOSHIDA et al.: ADAPTIVE REVERSIBLE DATA HIDING VIA INTEGER-TO-INTEGER SUBBAND TRANSFORM

387

Y, and the produced stego-block in the frequency domain isdenoted as Z ∈ Z M×(n+1) . This embedding is defined as⎧⎪⎪
for j = 0
⎨ki yi, j − an,ki (ỹi )zi, j = ⎪,(8)⎪
⎩ki yi, j − an,ki (ỹi ) + wi, j−1 for j  0
where i = 0, 1, · · · , M − 1, and j = 0, 1, · · · , n. yi, j , zi, j , andwi, j are (i, j) elements of Y, Z, and W, ỹi is i-th row vectorof Y, and ki is i-th element of k. Needless to say, the rangeof wi, j is restricted by ki . The subband transform is applied
for each column vector of the cover-block. The embedding
is applied for each row vector, i.e., each frequency subband
in the frequency domain, because the variance of a region
across some subbands is usually higher than one in only one
subband. In particular, the variance of a region consistingof the direct current and other subbands critically increases.
Therefore, we define the embedding of messages into eachsubband of Y as (8).
Finally, the stego-block in the space domain S ∈from the inverse IntDCT of Z. This proZ M×(n+1) is derived M−1
log2 ki bits into M(n + 1) pixels. It iscess embeds n i=0
the same embedding rate as the AGDE. At the receiver, thecover-block and messages are reversibly restored from thestego-block via the inverse process.
3.2 Determination of Capacity Parameters
In this section, we show the determination of the capacity parameters k at each cover-block. Initially, to avoid
over/underflow, similar to sets A and Ak , we define new sets
B and Bk as
B = X ∈ Z M×(n+1) | 0 ≤ xi, j ≤ 255 ,(9)c−p,i (k p − 1) + α ≤ êi, j
Bk = {X ∈ B | 0 − M−1
 M−1 p=0
≤ 255 − p=0 c+p,i (k p − 1) − α},from the inverse DCT of Ẑ as Ê = CT Ẑ. Its (i, j) element
êi, j is then defined as (11). Similarly, let E be E = CT Z. The
relation of E and Ê is represented as⎧⎪⎪
for j = 0êi, j
⎪
⎪
⎪
⎪
⎨
M−1
ei, j = ⎪
,
(13)
⎪
⎪
êi, j +c p,i w p, j−1 for j  0
⎪
⎪
⎪
⎩
p=0
where ei, j is (i, j) element of E. From the definitionof W, wi, j is restricted in [0, ki − 1]. Hence, the intervalterm of (13) for j  0 is derived as of the second 
M−1 −
M−1 +
c
(k
−
1),
Therefore, if 0 −p=0 p,i p
p=0 c p,i (k p − 1) .
 M−1−
p=0 c p,i (k p − 1) ≤ êi, j in Bk , ei, j satisfies 0 ≤ ei, j with arbitrary messages, i.e., the pixel avoids an underflow. The overflow verification is defined in the same way. Finally, similar
to the case of A and Ak , if X ∈ Bk , there is no-over/underflow
in the corresponding S and E. However, for S, the condition
does not exactly guarantee no-over/underflow, as discussed
later.
By using the subset Bk and a given threshold parameter T , the determination algorithm of k shown in Fig. 5 isdefined as follows:
1. The initial k is determined as⎧⎪
1 for V(ỹi ) > T
⎪
⎪
⎪
⎪
⎪
⎪
⎨2 for T/9 < V(ỹi ) ≤ T
ki = ⎪,⎪⎪
4 for T/49 < V(ỹi ) ≤ T/9
⎪
⎪
⎪
⎪
⎩
8 for V(ỹi ) ≤ T/49
where V(ỹi ) is a variance of ỹi .
2. If X ∈ Bk , escape algorithm with k.

where α is an arbitrary number as a margin and c±u,v is (u, v)
element of C± defined as
⎧
⎧
⎪
⎪
⎪
⎪
if cu,v ≥ 0⎨cu,v if cu,v ≥ 0⎨0+−cu,v = ⎪, cu,v = ⎪,⎪⎪⎩0⎩cu,v if cu,v < 0
if cu,v < 0(10)
and cu,v is (u, v) element of the DCT matrix C in (5). êi, j isdefined with yi, j and ỹi as
M−1êi, j =
c p,i k p y p, j − an,k p (ỹ p ) .

(11)

p=0

Let Ẑ ∈ Z M×(n+1) be the stego-block in the frequency domain
with W = 0 driven by (8), where 0 is a null matrix, i.e., its(i, j) element ẑi, j is defined asẑi, j = ki yi, j − an,ki (ỹi ),(12)and zi, j and ẑi, j are related as zi, j = ẑi, j + wi, j−1 ( j  0). LetÊ ∈ R M×(n+1) be the stego-block in the space domain derived

Fig. 5

Proposed determination of capacity parameters.

IEICE TRANS. FUNDAMENTALS, VOL.E97–A, NO.1 JANUARY 2014388

3. Let k̂ and P be a maximum of k and a set {i | ki = k̂},respectively.
4. If k̂ = 1, escape algorithm with k.
5. k p = k̂/2, where p (p ∈ P) is an index whose V(ỹ p ) is
the maximum of V(ỹi ). Go to Step 2.
As mentioned above, the condition X ∈ Bk does
not exactly guarantee that the corresponding S has noover/underflow pixels. The rounding errors in each lifting
step cannot be exactly predicted due to depending on input signals. If the optimal k is required, over/underflow ischecked after S is calculated with arbitrary k, iteratively.
However, in this case, the determination should wait until
the previous blocks are finished, i.e., in a non-parallel process. This is not useful because it requires a high computational cost and takes a long time to process. Therefore,
the complexity is reduced by introducing the margin α asdefined in (9).
4.

Implementation StructureThis section briefly presents an implementation algorithm[19].

6. The encoded LSBs and the residual messages are embedded into X2 and the stego-blocks are set to S 2 .
7. The stego-image is constructed from S 1 and S 2 .
The details of the classification in Step 3 are shown
here. Let N and L be the block number and length of theencoded LM, respectively L is stored as the header of the encoded LM. Since L is up to 2MN, a length of the header Lh
is log2 (2MN −1) , where ∗ is the ceiling function. Hence,
to store the header and the encoded LM via the LSB replacement, X1 should be a set of the former (L + Lh )/M(n + 1)
blocks. Of course, X2 is a set of the residual latter blocks.
For the simulation discussed later in Sect. 5, the conditions are as follows: We used a raster scan as the predetermined order. The LM is encoded by Huﬀman coding andarithmetic coding and the replaced LSBs are encoded byarithmetic coding [29], if necessary. In the extraction, we
can divide the stego-image into S 1 and S 2 via reading thefirst Lh LSBs of the stego-image along the predeterminedorder.
If the stego-image should be locked, the above embedding procedure is not needed, and the stego-image is simply
derived from the proposed method. In this case, the LM isrecognized as the key.

4.1 Embedding Algorithm
4.2 Extraction Algorithm
We focus on how to treat the capacity parameters of eachblock. A set of the capacity parameters is called a locationmap (LM). The embedding algorithm shown in Fig. 6 is implemented as follows:
1. The cover-image is divided into some cover-blocks thatare indexed along the predetermined order.
2. k of each cover-block is determined and the LM is constructed as a sequence of k.
3. The cover-blocks are classified into two categories X1
and X2 according to the length of the encoded LM.
4. A part of the messages is embedded into X1 according
to the LM, and the stego-blocks in the space domain
are set to S 1 .
5. The LSBs of S 1 are replaced with the encoded LM andthen encoded.

Fig. 6The extraction procedure corresponding to the algorithm in
Sect. 4.1 is as follows:
1. By reading the LSBs, the stego-image is divided into
S 1 and S 2 and the encoded LM is extracted.
2. From S 2 , X2 is restored, and the replaced LSBs andresidual messages are extracted based on the LM.
3. The LSBs of S 1 are restored from the replaced LSBs,
X1 is restored, and the messages are extracted.
4. The cover-image is restored from X1 and X2 .
5.

Simulation

We performed simulations to compare the proposed method

Flowchart of data embedding.

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389

Fig. 7

Fig. 8

Test images.

Embedding RD curves.

IEICE TRANS. FUNDAMENTALS, VOL.E97–A, NO.1 JANUARY 2014390
Table 1

Embedding rates [bpp], when PSNR is 30 dB.

Image
Lena
Airplane
BarbaraBoat
Goldhill
BaboonLuo’s [8]
1.16
1.39
0.90
0.84
0.91
0.50

Table 2
Image

Peng’s [19]
1.23
1.46
0.96
0.91
0.92
0.47

Prop.
1.34
1.50
1.23
1.05
1.03
0.50

PSNRs of stego-images [dB].

Rate
0.8
1.0
1.2

Luo’s [8]
35.63
32.63
29.47

Peng’s [19]
35.18
32.67
30.44

Prop.
36.24
33.70
31.60

Airplane

0.8
1.0
1.2

39.08
35.81
32.81

38.17
35.41
32.99

38.07
35.58
33.31

Barbara

0.8
1.0
1.2

32.37
28.67
24.43

32.48
29.52
26.77

35.84
33.08
30.46

Boat

0.8
1.0
1.2

30.95
27.51
24.17

31.53
28.94
26.60

33.06
30.47
28.21

Goldhill

0.8
1.0
1.2

32.11
28.75
25.32

31.58
28.96
26.61

32.83
30.26
27.88

Baboon

0.8
1.0
1.2

23.62
19.56
-

24.68
22.26
-

25.05
22.46
-

Lena

with Luo’s and Peng’s methods [8], [19], which are state-ofthe-art methods of reversible data hiding, using the MATLAB programming language. According to [19], the block
size of Peng’s method is 4 × 4, so n = 15 in both Peng’s andthe proposed methods. We set α = 0 in (9) and experimentally determined the threshold parameter T so that it was alarge enough integer number to achieve the desired capacity.
The three methods were applied to six standard test images(512 × 512, 8-bit grayscale), shown in Fig. 7. A random bit
string was used as the message.
Figure 8 shows the embedding rate-distortion (RD)curves. The visual qualities of the stego-images are objectively measured using PSNR. The embedding rate is presented in bpp. Table 1 shows the embedding rates when
the PSNRs are 30 dB. From these figures and the table, it isclear that the proposed method achieves the highest embedding capacity. Moreover, the proposed method has a better visual quality at high embedding rates. Unfortunately,at low embedding rates, Luo’s HS method delivers a better visual quality because the proposed method embeds not
only messages but also the LM. In fact, the encoded LM seriously aﬀects the embedding rates, e.g., the encoded LM
with 15 × 103 bits at 0.2 bpp extends to 23% in overall capacity.
Table 2 objectively shows the visual qualities at highembedding rates. At 1.2 bpp in Baboon, none of the methods are able to embed. The proposed method shows better results than the conventional ones, especially for images

Fig. 9 Particular areas of stego-images of Lena at 1.0 bpp: (1st row)
Cover-image (2nd row) Luo’s [8], (3rd row) Peng’s [19], (4th row) Prop.

with rich high frequency components such as Barbara. On
the other hand, in the case of very smooth images such as
Airplane, the proposed method shows results comparablewith Luo’s. The rate range, in which the proposed method
has the best result, is wide if the image has rich high frequency components, and vice versa. However, the DE methods are ineﬀective for extremely textured images such as
Baboon. Figure 9 shows specific areas of the stego-imagesof Lena at 1.0 bpp. Although the particular distortions inPeng’s method are perceived, the distortions in the proposed
images are not easily noticeable because they are similar toordinary random noise. This is because the distortions produced by embedding in the frequency domain are randomly
spread to the neighborhood pixels via the inverse frequency

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transform with oscillated atoms of high frequency subbands.
Also, the proposed method does not produce an excessive
contrast enhancement, as Luo’s method does. According
to the visual policy of data hiding that the modification ofembedding should not be perceptually recognized, the proposed method has better perceptual results because the particular distortion has a higher risk of recognizing the hiding
than a random noise. Consequently, our method objectively
and perceptually shows better visual quality than the conventional methods at the same embedding rate.
6.

Conclusion

In this paper, we proposed an adaptive reversible data hiding method in the frequency domain obtained via integer-tointeger subband transform (I2I-ST) and the adaptive generalized diﬀerence expansion (AGDE) method. Natural images in the frequency domain usually have a lower variance. In addition, the distortions generated by embedding
are distributed to neighborhoods via the inverse subband
transform. Therefore, the AGDE method in the frequency
domain obtained by an I2I-FT is more suitable for reversible
data hiding than one in the space domain. As a result, theproposed method objectively and perceptually showed an
excellent performance. For even better visual quality, we
should apply other I2I-FTs and adaptive threshold parameters for each subband.
Acknowledgments
This work was supported by Grant-in-Aid for JSPS Fellows,
and JSPS KAKENHI Grant Number 25820152.
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Taichi Yoshidareceived the B.E., M.E.,
and Ph.D. degrees in electrical engineering from
Keio University, Yokohama, Japan, in 2006,
2008, and 2013, respectively. He is currently
a Ph.D. at Keio University, Yokohama, Japan,
under the supervision of Prof. Masaaki Ikehara,
and a Research Fellow of the JSPS. His researchinterests are in the field of filter bank design andits image coding application.

Taizo Suzuki
received the B.E., M.E.
and Ph.D. degrees in electrical engineering from
Keio University, Yokohama, Japan, in 2004,
2006 and 2010, respectively. He joined Toppan
Printing Co., Ltd., Tokyo, Japan, from 2006 to
2008. From 2008 to 2011, he was a Research
Associate of Global COE (C-12). From 2010 to2011, a Research Fellow of the JSPS and a Visiting Scholar at Video Processing Group, the University of California, San Diego, La Jolla, CA,
respectively. From 2011 to 2012, he was an Assistant Professor in the Department of Electrical and Electronic Engineering, College of Engineering, Nihon University, Koriyama, Japan. Since
2012, he is an Assistant Professor in the Faculty of Engineering, Information and Systems, University of Tsukuba, Tsukuba, Japan. His current
research interests are filter bank theory/design and its application for image/video.

Masaaki Ikehara
received the B.E., M.E.
and Dr.Eng.degrees in electrical engineeringfrom Keio University, Yokohama, Japan, in
1984, 1986, and 1989, respectively. He was Appointed Lecturer at Nagasaki University, Nagasaki, Japan, from 1989 to 1992. In 1992, hejoined the Faculty of Engineering, Keio University. From 1996 to 1998, he was a visiting researcher at the University of Wisconsin, Madison, and Boston University, Boston, MA. He iscurrently a Full Professor with the Department
of Electronics and Electrical Engineering, Keio University. His researchinterests are in the areas of multirate signal processing, wavelet image coding, and filter design problems.