Optimal pixel-level self-repairing
authentication method for grayscale
images under a minimax criterion of
distortion reduction
Che-Wei Lee
Wen-Hsiang Tsai
Optimal pixel-level self-repairing authentication method
for grayscale images under a minimax criterion
of distortion reduction
Che-Wei LeeNational Chiao Tung University Department of Computer Science and
Information Engineering Hsinchu, Taiwan 30010 Wen-Hsiang Tsai
National Chiao Tung University Department of Computer Science and
Information Engineering Hsinchu, Taiwan 30010
and
Asia University
Department of Information Communication Taichung, Taiwan 41354
E-mail:[email protected]
Abstract. A new blind pixel-level self-repairing grayscale image authen-tication method, which is optimal under a minimax criterion of image-distortion reduction, is proposed. By dividing the grayscale range into bins, a three-bit bin code which provides the double functions of tampering localization and data repairing is generated as the authentication signal for each pixel in the cover image. The optimality in choosing three bits of a pixel as the authentication signal under a minimax criterion of minimizing the total maximum distortion resulting from authentication signal embed-ding and tampered pixel repairing is proved. Experimental results show the effectiveness of the proposed method. © 2012 Society of Photo-Optical Instrumentation Engineers (SPIE). [DOI:10.1117/1.OE.51.5.057006]
Subject terms: fragile watermarking; grayscale image authentication; image tamper-ing; tampering localization; tampered data repairing.
Paper 111480 received Nov. 27, 2011; revised manuscript received Feb. 25, 2012; accepted for publication Mar. 26, 2012; published online May 22, 2012.
1 Introduction
With the era of cloud computing coming, data stored origin-ally in personal computers mostly will eventuorigin-ally be moved to and processed in powerful servers at far ends. However, how can one be sure that personal data accessed from cloud servers are intact? Undoubtedly, this problem of data security has become a significant issue in the age of cloud computing. This study explores the security issue of keeping digital-image data. The use of an digital-image authentication technique provides a solution to this issue. A new grayscale image authentication method is proposed in this paper. By embed-ding fragile authentication signals into a cover image to be protected to create a stego-image, illicit modifications made to the stego-image may be localized to pixel-level precision by the proposed method such that the integrity and fidelity of the original image content can be checked.
An example of scenarios of applying the method goes like this. A lawyer, say Bob, always saves the critical grayscale document images of one of his clients, say Alice, into a cloud server through the internet. Each of such images is a stego-image yielded by the proposed method with almost no difference in appearance from the original cover image. One day, Bob retrieves from the far-end server the stego-image of a document required for a court session scheduled for Alice the next day. For the purpose of checking the authenticity of the stego-image, Bob fetches a private key that he keeps per-sonally and authenticates the image by the proposed method with the key as an input. Unfortunately, a portion of the stego-image is found to be unidentical to its original content and marked out by the method as an alert to Bob, indicating that the image has been tampered with illegally at the server site. Instead of abandoning the attacked stego-image, Bob may try to repair the tampered image portion by using the
proposed method and may keep the resulting image for further uses in court or in other later activities.
Several fragile watermarking techniques for image authentication have been proposed in the past, and they
may be categorized into two approaches: block-wise1–7and
pixel-wise.8–11 Methods of the former approach embed
fragile watermarks as authentication signals into nonover-lapping blocks of the cover image and identify possible tam-pered image parts in the unit of block. One weakness of such block-level authentication methods is that the detail of the
tampered image part cannot be located precisely.9 On the
other hand, methods of the second approach8–11authenticate
images at the pixel level such that tampered image parts can be identified pixel by pixel, yielding a detailed
tampering-localization result. Liu et al.8 generated a binary image
that is mapped from the difference image computed from the cover image and its so-called chaotic pattern. And the least-significant-bit (LSB) plane was used to accommodate the binary image as the fragile watermark for use in later image authentication. Because of the binary nature of the embedded fragile watermark, the LSB of a tampered pixel value may coincide with the watermark bit, yielding a high erroneous pixel-authentication rate up to 50%. To deal with this phenomenon, a statistical fragile watermarking method which utilizes probability distributions computed from the original pixels and the tampered ones to locate the tampered
pixels was proposed in Zhang and Wang.9 However, the
method only works in the case that the tampering ratio is
smaller than 1.1% (Ref. 10). As an improvement, Zhang
and Wang10 proposed later a fragile watermarking method
for authenticating grayscale images using a hierarchical mechanism, which embeds watermark data derived from the pixels and blocks of the cover image into the LSBs of all the pixels. In the authentication process, tampered blocks are identified first, and tampered pixels within the identified blocks are located subsequently.
0091-3286/2012/$25.00 © 2012 SPIE
In this study, a method for pixel-level grayscale image authentication using fragile authentication signals with an additional capability for repairing attacked image parts auto-matically is proposed. The method is based on the concept of compressing a number of the most significant bits (MSBs) of
a pixel’s gray value into a shorter “bin code” for use both as
an authentication signal for the pixel and as an index for gen-erating the data for repairing the pixel when it is found to have been tampered with. The bin code is generated from
a bin-mapping scheme which transforms each pixel’s gray
value into one of eight “bins,” coded by three bits. It is
proved that the choice of using three bits out of eight ones in a pixel as the bin code is optimal under a minimax
criterion of reducing the total maximum pixel-level gray-value distortion resulting both from authentication signal embedding and from tampered pixel repairing.
The proposed method has at least four merits. (1) First,
different from other methods12,13which generate the
authen-tication signal and the repairing data as two separate items, the proposed method uses the above-mentioned single bin code to function as the two items simultaneously, leading to use of less storage for embedding these data in the image. (2) The use of less storage leads further to the possibility of conducting more precise pixel-level authentication because it becomes now possible to allow every pixel to include the pixel authentication signal (saved as the three LSBs) in addi-tion to the original pixel content (kept in the five MSBs). Note that most related methods with data-repairing
capabil-ities authenticate images at the block level,14–16 yielding
coarser tampering localization and data repairing results. (3) Furthermore, a secret key is used in the proposed method for randomly choosing pixels for embedding the generated authentication signals, thus increasing the security of the stego-image yielded by the proposed method. (4) Finally, because of the first merit of using less storage for authentica-tion and repairing data menauthentica-tioned previously, the proposed
method is blind17in nature—no information other than the
image itself is needed for conducting the data-repairing
pro-cess. Note that the methods of Refs.2,3,6,8and 12need
to know the prior information of the hidden digital signatures
or watermarks17used in the authentication process. Besides,
extra information like codebooks or other overhead data is required in some existing methods with data-repairing capabilities.6,12,16
The remainder of this paper is organized as follows. In
Sec. 2, the details of the proposed method are described.
Table 1 Bins, bin numbers, bin codes, and representative values of bins used in this study.
Bin (interval) Bin number (integer) Bin code (binary number) Representative value of bin [0, 3] 0 000 2 [4, 7] 1 001 6 [8, 11] 2 010 10 [12, 15] 3 011 14 [16, 19] 4 100 18 [20, 23] 5 101 22 [24, 27] 6 110 26 [28, 31] 7 111 30
Fig. 1 Illustration of bin code (authentication signal) generation and embedding. (a) Mapping five-bit MSBs to a three-bit bin code. (b) Bin codes embedded into pixels randomly selected by a secret key K .
In Sec.3, the previously mentioned optimality of the choice of three bits as the bin code for use as the authentication sig-nal is proved. Some experimental results showing the
feasi-bility of the proposed method are presented in Sec.4. And
finally conclusions are made in Sec. 5.
2 Proposed Method for Pixel-Level Grayscale Image Authentication and Self-Repairing
2.1 Authentication Signal Generation and Embedding In the proposed method for grayscale image authentication and self-repairing, the eight-bit gray value g of each pixel in
an input image is divided into two parts—the five MSBs of
g and the remaining three LSBs. The former are used to generate an authentication signal for the pixel itself, with the signal also working as an index for generating the data for
repairing the pixel’s gray value when the pixel is found to
have been tampered with. The five MSBs ideally are expected to be embedded directly in a randomly selected pixel elsewhere and can be retrieved later for use in the two previously mentioned purposes of authentication-signal and repairing-data generations. However, due to the limited data-hiding capacity in the image, it is difficult to embed the large-volume data consisting of such MSBs of all the pixels into the input image; and even if they could be embedded, noticeable distortion would be created. Consequently, we propose in this study to use a bin-mapping scheme for the purpose of compressing these MSB data before embedding them. Specifically, we map the gray-value range specified by the five MSBs into eight equal-length intervals called bins, with each bin being indexed by an integer called a bin number, or equivalently, by a three-bit binary number, called a bin code. The eight bins and their corresponding
bin numbers and bin codes are shown in Table 1. The bin
code of each pixel is then taken as the authentication signal of the pixel and embedded into the three remaining LSBs
(the previously mentioned second part) of another pixel ran-domly chosen by a preselected secret key. An illustration of these ideas of authentication signal (bin code) generation and
embedding is given in Fig.1, and the detail is described as an
algorithm in the following.
2.2 Image Authentication, Tampering Detection, and Data Repairing
During the image authentication process, an authentication signal is computed from the five MSBs of every pixel p. Also, the authentication signal embedded in the three LSBs
of the pixel p0 corresponding to p, which was randomly
selected previously in Algorithm 1, is retrieved. The two
authentication signals then are compared with each other. If mismatching occurs, pixel p is regarded as having been tampered with. In this case, we use again the three LSBs
of pixel p0, which is also the bin code of p, as an index
to generate a data item for use in repairing the tampered gray values of p. The generated data item is taken to be the middle value of the bin indexed by the bin code, which is called the representative value of the bin and denoted by
M. Specifically, M is computed as M¼ dða þ bÞ∕2e for a
bin with range [a, b] whered·e specifies the integer ceiling
operation. The representative value M for each bin used in
this study is shown in the rightmost column of Table 1,
though it may be computed analytically directly (for the
detail, see Sec.3later). Finally, after padding three trailing
0’s to M, the result is used to repair the tampered pixel. A
diagram illustrating the above idea of authentication signal
matching and tampered pixel detection is shown in Fig.2.
And another diagram illustrating the idea of
tampered-pixel repairing is shown in Fig.3. Detailed algorithms
imple-menting these ideas are described subsequently.
In Algorithm1, Steps 2.2 through 2.4 are used to show
how the concept of bin mapping of our method is applied.
Algorithm 1 Authentication-signal generation and embedding.
Input: a grayscale cover image I, a random number generator f , and a secret key K Output: a stego-image Iswith authentication signals embedded.
Steps.
Step 1. (Beginning of looping) In a raster-scan order, select a pixel p from the image I.
Step 2. Authentication-signal generation and embedding) Perform the following steps to generate an authentication signal for p and embed it into another randomly selected pixel.
2.1 Transform the gray value of p into eight bits, b7; b6; : : : ; b0.
2.2 Transform the five MSBs, b7; b6; : : : ; b3, of p into an integer d.
2.3 Map the integer d into a bin indexed by a bin number B computed by the function B ¼ bd∕4c where b:c specifies the integer floor function. 2.4 Transform B into a three-bit bin code s ¼ c2c1c0for use as the authentication signal for p.
2.5 Select randomly a pixel p0in I other than p using the input random number generator f with the input key K as the seed, and regard pixel p0
as corresponding to p.
2.6 Embed the three-bit authentication signal s ¼ c2c1c0of p into p0 by replacing the three LSBs of p0with s.
Step 3. (End of looping) If there remain unprocessed pixels in I, then go to Step 1; otherwise, take the final I as the desired stego-image Is.
In practice, these steps may be reduced to be simply as follows for use in real applications:
2.2 Take the three MSBs b7, b6, b5of p to yield a bin code
denoted as s¼ c2c1c0.
2.3 An Illustrative Example
An example is given here to illustrate the above algorithms. Given a pixel p in the input image with gray value 133, or equivalently, with gray value 10000101 in binary, the five
MSBs and the three LSBs of p are m¼ 10000 and
l¼ 101, respectively. The integer value of m is 16 and
the bin mapping of it results in the bin numberb16∕4c ¼ 4,
so we get to know that the bin into which m falls is indexed by four. The binary form 100 of this bin number four, i.e., the bin code is 100, which is then taken as the authentication
signal s for p. Also, assume that another pixel p0, say, of
gray value 231 is selected randomly to be corresponding to p using a certain random number generator f with a pre-selected secret key K as the seed. The binary form of 231 is 11100111. So, the three LSBs of this binary number are
replaced by the authentication signal s¼ 100 of p, resulting
in a binary value of 11100100, or an integer of 228, which
is then taken to be the new gray value of pixel p0 as the
authentication signal generation and embedding result
con-ducted by Algorithm1.
Now, suppose that the original gray value 133 of pixel p becomes 99, or 01100011 in binary, due to illicit tampering. Then, in the authentication process conducted
by Algorithm 2, the five MSBs of this tampered binary
gray value, namely, 01100, or 12 in decimal, is used to obtain
the computed authentication signal s¼ 011 by the bin
map-pingb12∕4c ¼ 310 ¼ 0112. On the other hand, applying the
random number generator f with the secret key K used
before as the seed, we select again the pixel p0corresponding
to p with gray value 228 as mentioned previously. The
authentication signal for pixel p presumably is embedded
at p0. To extract it, we transform again the gray value 288
of p0 into the binary form 11100100. We then take the
three LSBs s0¼ 100 as the extracted authentication signal.
Comparing this signal s0¼ 100 with the computed
authen-tication signal s¼ 011 bit by bit, we decide that the pixel p
has been tampered with because the bits in each of three corresponding bit pairs are different.
We now have to repair p using the information of the
extracted authentication signal s0¼ 1002. Since 1002¼
410, s0specifies a bin indexed by four. Because the interval
of bin four is [16, 19], we get the representative value M of
this bin to be M¼ 1710¼ 100012 according to Table 1.
After padding three trailing zeros to M in the binary form,
we get a gray value of 100010002¼ 13610, which is finally
taken to be the new gray value of p as the tampering repairing result. As a comparison, note that p originally
has the gray value of 13310, which shows that the repairing
result is close to the original value.
3 Proof of Optimality of Proposed Method for Image Distortion Reduction
In the proposed method presented above, the eight bits of
each pixel’s gray value is separated into two parts, five
MSBs and three LSBs, with the former used for keeping the pixel content and the latter used for embedding the authentication signal. It seems that we may generalize this
specific choice of pixel-bit division, ðm; lÞ ¼ ð5; 3Þ, where
m denotes the number of MSBs and l the number of
LSBs with mþ l ¼ 8. For example, we may choose
alterna-tively to use two LSBs in a pixel for embedding the authen-tication signal and the remaining six bits for keeping the
pixel content, so thatðm; lÞ ¼ ð6; 2Þ. Or, by a reverse
con-sideration, we may choose to adopt ðm; lÞ ¼ ð4; 4Þ as
well. Is there a criterion to decide which choice is better? The answer proposed in this study is to consider the resulting image distortion.
Fig. 3 Diagram of tampered pixel repairing (detail to be described in Algorithm2).
It will be proved in this section that the choice of
m¼ 5 and l ¼ 3 as done in this study is optimal in the
sense of minimizing the resulting total image distortion incurred both by authentication signal embedding and by tampered pixel repairing. The proof is conducted in a step-by-step reasoning manner as described in the following. 3.1 Proof of the Optimal Choice of the Number of
Bits for Use as the Authentication Signal
Stage 1—optimization criterion consideration in terms of
resulting image distortion.
1. First, we consider simultaneously at the pixel level the
maximum distortion D1 resulting from the process of
embedding authentication signals as well as the
max-imum distortion D2 resulting from the process of
repairing tampered pixels, and take their sum
D¼ D1þ D2 as the criterion function for
optimiza-tion in choosing the values ofðm; lÞ, i.e., for dividing
the eight bits of a pixel’s gray value into two parts for
the purposes described previously. The goal is to obtain a choice of (m, l) which minimizes the value of D, or equivalently, the maximum distortion coming
Algorithm 2 Image authentication, tampering detection, and data repairing.
Input: a stego-image Isgenerated by Algorithm1presumably, an originally white authentication image Ia, and the random number generator f
and the secret key K used in Algorithm1.
Output: an image Ir with tampered pixels, if any, being repaired.
Step 1. (Beginning of looping for pixel authentication) Take in a raster-scan order a pixel p from Is, and perform the following steps.
Stage 1—computation of authentication signals.
1.1 Transform the gray value of p into eight bits b7; b6; : : : ; b0.
1.2 Transform the five MSBs b7; b6; : : : ; b3of p into an integer d.
1.3 Map the integer d into a bin indexed by a bin number B computed by B ¼
d∕4
.
1.4 Transform B into a three-bit bin code s ¼ c2c1c0which is also regarded as an authentication signal, called the computed authentication
signal.
Stage 2—extraction of the hidden authentication signal
1.5 Use the random number generator f and the input key K as the seed to select from Israndomly a pixel p0corresponding to p, where a previously
embedded authentication signal for p is located presumably. 1.6 Transform the gray value of p0into eight bits b0
7; b60; : : : ; b00, extract the three LSBs to form a string s0¼ b20b10b10, called the extracted
authentication signal.
Stage 3—authentication signal matching and tampered pixel marking
1.7 Match the computed authentication signal s ¼ c2c1c0and the extracted one s0¼ b20b10b00 bit by bit; and if mismatching occurs, regard p as
having been tampered with and mark its corresponding pixel on the authentication image Iaas a black point.
1.8 (End of looping) If there remain unprocessed pixels in Is, then go to step 1; otherwise, take the final Iaas a new authentication image Ia0for use in
the next stage of the algorithm for image repairing. Stage 4—tampered pixel repairing
Step 2 (Beginning of looping for tampered pixel repairing) For each black point pain Ia0 selected in the raster-scan order, perform the following
steps.
2.1 For the pixel p0in I
scorresponding to pa, use the input random number generator f with the input key K as the seed to select randomly a pixel
p0 0where a previously embedded authentication signal for p0is located presumably.
2.2 Transform the gray value of p0 0into eight bits b0 0
7; b60 0; : : : ; b00 0, extract the three LSBs b20 0b10 0b00 0, and transform b20 0b10 0b00 0into an integer B0 0
which specifies the index of the bin into which the gray value of p0falls.
2.3 Repair the tampered pixel p0by the following steps.
2.3.1 Derive the representative value M of the bin indexed by B0 0.
2.3.2 Transform M into a five-bit binary string r7r6r5r4r3.
2.3.3 Pad three trailing 0’s to r7r6r5r4r3to get an eight-bit string T ¼ r7r6r5r4r3000.
2.3.4 Transform T into an integer d0and replace the gray value of p0 with d0as the repairing result.
Step 3. (End of looping) If there remain unprocessed black pixels in Ia, then go to step 2; otherwise, take the final Isas the desired output image Ir.
from authentication signal embedding and tampered pixel repairing for each pixel.
2. Since mþ l ¼ 8, we just have to choose an optimal
value for l under the above-mentioned minimax
criter-ion, and take the value of m to be m¼ 8− l.
Stage 2—derivation of distortion incurred by authentication
signal embedding.
3. As mentioned, l LSBs of a pixel p are used to compose a bin code which is then taken to be the authentication
signal s of p and embedded in another pixel p0 (see
step 2 of Algorithm1). And this will incur a maximum
gray-value change of 2l− 1 coming from either of the
two cases of bit changes from l 0’s to l 1’s and from
l 1’s to l 0’s.
4. Therefore, the maximum gray-value distortion occur-ring at each pixel resulting from authentication-signal
embedding is D¼ 2l− 1.
Stage 3—derivation of distortion resulting from tampered
pixel repairing.
5. The width of the total range of gray values specified by
the m MSBs of a pixel is 2mwhich is divided into 2l
bins (see step 2 of Algorithm1), so the width Wbin of
each bin is
Wbin ¼ 2m∕2l¼ ð28−lÞ∕2l¼ 28−2l
because mþ l ¼ 8, as explained before.
6. Accordingly, if the range of the x’th bin Bxis denoted
by [L, R], then it is easy to figure out that
L¼ ðx− 1Þ × 28−2 land R¼ x× 28−2 land, where x¼
1; 2; : : : ; 2l (see Table 1 for numerical examples
of½L; R).
7. Then, the representative value M of Bx (computed in
step 2 of Algorithm 2), which is the middle value
between L and R, is just
M¼ ðL þ RÞ∕2 ¼ f½ðx−1Þ×28−2l þ ðx×28−2l−1Þg∕2
¼ x×28−2l−27−2l−2−1:
8. With M as the representative value for all the gray
values in bin Bx used in repairing a tampered pixel
p, the maximum gray-value difference D0 between
the repaired m MSBs of pixel p and the original m
ones is M− L (or R − M) which may be computed
to be
D0¼ M− L
¼ ðx× 28−2 l− 27−2 l− 2−1Þ− ðx − 1Þ × 28−2 l ¼ 27−2 l− 2−1:
9. Since we pad l trailing zeros to the m MSBs of the
representative value M (see step 2 in Algorithm 2)
to compose an eight-bit number to repair the tampered pixel p, the maximum gray-value distortion after repairing p is
D2¼ D0× 2lþ ð2l− 1Þ;
where the term 2l− 1 specifies the partial distortion
coming from the extreme case that the original last l bits of p are all 1’s.
10. By using the result of D0derived previously in Eq. 8,
D2 may be derived in more detail to be
D2¼ ð27−2 l− 2−1Þ× 2lþ ð2l− 1Þ
¼ 27−lþ 2l−1− 1:
Stage 4—minimization of the overall distortion.
11. The maximum gray-value distortion D considered for a pixel as mentioned previously in Eq. 1 now can be computed from the results of Eqs. 4 and 10 above to be
D¼ D1þ D2¼ ð2l− 1Þ þ ð27−lþ 2l−1− 1Þ
¼ 27−lþ 3× 2l−1− 2:
12. Taking the derivative of D with respect to l, we get
dD∕dl ¼ 27−l× ln 2 × ½dð7 − lÞ∕dl
þ 3× 2l−1× ln 2 × ½dðl − 1Þ∕dl
¼ 27−l× ln 2 × ð−1Þ þ 3 × 2l−1× ln 2 × ðþ1Þ
¼ ln 2× ð3 × 2l−1− 27−lÞ;
where ln 2 is the natural logarithm value of 2.
13. Setting dD∕dl ¼ 0, we can get the following
equation
ln 2 × ð3 × 2l−1− 27−lÞ ¼ 0;
which may be solved to get 27−l¼ 3× 2l−1or
equiva-lentlyð27−lÞ∕ð2l−1Þ ¼ 28−2l¼ 3.
14. Taking the base-two logarithm values of the two sides of the above equality and simplifying the result, we get finally the solution of l as:
l¼ 4− ½log23∕2;
which may be evaluated explicitly to be approximately equal to 3.2075.
15. Accordingly, since l is the number of LSBs which should be an integer, it is taken to be the integers three and four for which the corresponding values
of the gray-value distortion D are Dð3Þ ¼ 27−3þ 3×
23−1− 2 ¼ 28 and Dð4Þ ¼ 27−4þ 3× 24−1− 2 ¼ 32,
respectively. Therefore, the optimal l is finally decided to be three, which is exactly the number of bits we use to compose an authentication signal as described previously. This completes the proof.
4 Experimental Results
Many experiments have been conducted to test the proposed
method, and one result is shown in Fig.4, where Fig.4(a)is
an input surveillance image with the size of 480× 360. The
result of applying Algorithm 1 to generate and embed
with a PSNR value of 37.51. Actually, a general lower bound may be computed for this PSNR value, as done by the following reasoning.
1. With l being the number of bits in a pixel used for embedding the authentication signal, the largest mean square error value MSE of a stego-image with
respect to the cover image is ð2l− 1Þ2 because at
each pixel, the largest gray-value difference is
2l− 1 after an l-bit authentication signal is embedded
there, as described previously.
2. Accordingly, the peak-signal-to-noise-ratio value
PSNR by definition is just PSNR ¼ 10 × log10ð2552∕MSEÞ ¼ 10× log10½2552∕ð2l− 1Þ2 ¼ 20× log10½255∕ð2l− 1Þ ¼ 20× log10ð255∕7Þ ≍31.23;
where 255 is the maximum gray value of an eight-bit pixel and l is three for our case here.
3. That is, the lowest bound for the PSNR value is approximately 31.23, which means that the quality of the stego-image is good enough for general applications.
Back to the presentation of the first case in our
experimen-tal results, Fig. 5(a) shows a tampering result with a
tampering ratio of 0.74% in which two numbers “3” and
“7” on the car plate shown in Fig. 4(b) were replaced
with fake numbers “7” and “5,” respectively. Figure 5(b)
shows the obtained authentication image after applying
stages 1 through 3 of Algorithm 2 to Fig. 5(a). As can be
seen, the tampered pixels covered by the fake numbers have been detected correctly. However, some noise points
can be seen to appear in Fig.5(b). These noise points indicate
that the pixels in the original image corresponding to these noise points are also erroneously identified as having been tampered with. The reason for this noise phenomenon is explained in the following.
If a pixel A is identified as having been tampered with, it means that the authentication signal of a pixel B, which is embedded at pixel A, is also damaged. This in turn means that B will also be found to have been tampered with,
even when B is in fact not so. This effect of mutual affection leads to erroneous marking of single points in the authenti-cation image as tampered pixels, creating a pepper-and-salt
noise phenomenon like that seen in Fig.5(b). To remove this
effect, we applied the median filtering operation to eliminate such noise points before performing the pixel-repairing
operations described in stage 4 of Algorithm 2. The final
authentication image resulting from doing so to Fig. 5(b)
is shown in Fig.5(c), in which, as can be seen, most
pep-per-and-salt points have been eliminated, but 90 false accep-tance pixels and one false rejection pixel are left. To deal further with this authentication image, image repairing
was conducted and the result is shown in Fig. 5(d), in
which we see that the original numbers“3” and “7” have
been repaired successfully at their original positions. Also, with the tampered pixel repaired, the image has a PSNR value of 45.6 with respect to the stego-image shown in Fig.4(b).
Another experimental result of replacing the entire car
plate with a fake one is shown in Fig. 6. Compared with
the previous experimental result with the tampering ratio being 0.74%, the tampering ratio in this case was raised
to be 2.25%. It can be seen in Fig.6(b)that the phenomenon
of noise points caused by the effect of mutual affection becomes more conspicuous than that in the previous case because of the higher tampering ratio. After noise
elimina-tion was performed on Fig. 6(b), the final authentication
image of Fig.6(c)was obtained, which includes 551 false
acceptance pixels (due to the reason that the five MSBs of each of them coincide with those of the original image) and 16 false rejection pixels (due to the reason that their authentication signals embedded in the tampered area were destroyed). Finally, the repaired image in which the original car plate reappeared clearly with a PSNR
value of 36.38 is shown in Fig.6(d). Some relevant statistics
of the two cases mentioned above are given in Table2.
Fig. 4 Generation of stego-image from an input surveillance image. (a) Input image taken by a monitor. (b) Stego-image with PSNR value 37.51.
Fig. 5 Authentication result of a surveillance image taken by a moni-tor with tampered area. (a) Image with modification of two car-plate numbers. (b) Authentication image with noise. (c) Final authentication image. (d) Final repairing result with PSNR 45.60 with respect to stego-image.
To show the relation of the performance of tampering localization and repairing to the degree of tampering as well as the use of median filtering, the statistics of the false judgments (including false acceptance pixels and false rejection pixels) and the PSNR values of a series of repaired images listed in the order of increasing tampering ratios are
given in Table 2. In addition, an illustration of the statistics
is shown in Fig.7. Note that the total numbers of false
accep-tance pixels plus false rejection pixels comprises the ordinate
of the number of falsely judged pixels in Fig. 7.
In a subsequent experiment, we used another test image,
Lena, of size 512× 512 as shown in Fig.8(a), and the
stego-image yielded by the proposed method is shown in Fig.8(b)
whose PSNR value is 39.34.
In this experiment, we selected the area of Lena’s hair and
modified it by adding a rose-flower shape of 2084 pixels to
it. The modification result is shown in Fig.9(a). Figure9(b)
shows the authentication result without noise elimination,
and the final authentication image is shown in Fig. 9(c),
in which 2041 tampered pixels of the flower were detected and most isolated points were removed after median filtering. Finally, we repaired each of those detected pixels by refer-encing the bin code as the authentication signal embedded in
a certain pixel whose position in Fig.9(a)was located by a
key. The repairing result in this case is shown in Fig.9(d),
and the PSNR value with respect to the stego-image is 47.00.
Some other statistics about this case are given in Table3.
As done in the previous experiments using a surveillance image, we also gradually extended the tampered area in the Lena image to test the effectiveness of the proposed method.
Table3 lists the statistics of our experiments conducted in
this way. Furthermore, an illustration corresponding to the
statistics of Table3 is shown in Fig.10.
According to the results and statistics of all the conducted experiments, the proposed method is seen to be effective enough till the tampering ratio reaches about 10%. This over-all result is better than that of the method described in Ref.9, which works effectively when the tampering ratio is smaller than 1.1%.
Table4 lists a comparison of the proposed method with
other pixel-level image authentication methods8,9in terms of
capabilities of self-recovery and tampered pixel detection. We conducted an experiment that was also conducted in
Ref.9 with 2084 tampered pixels. The experimental result
Fig. 6 Authentication result of a surveillance image taken by a monitor with tampered area. (a) Image with modification of entire car plate. (b) Authentication image with noise. (c) Final authentication image. (d) Final repairing result with PSNR 36.38 with respect to stego-image.
Table 2 Statistics of experiments using a surveillance image of Fig.4(a).
Surveillance image (480 × 360)
Total # of tampered pixels (tampering ratio)
PSNR of recovered image with respect to stego-image
Total # of false acceptance pixels
Total # of false rejection pixels
Case 1 shown in Fig.5 784 (0.45%) 45.60 90 1
Case 2 shown in Fig.6 3895 (2.25%) 36.38 551 16
Case 3 (not shown) 8640 (5%) 29.33 2026 59
Case 4 (not shown) 17280 (10%) 23.94 4201 265
Case 5 (not shown) 34560 (20%) 17.87 8009 2411
Case 6 (not shown) 51840 (30%) 13.59 15079 9775
Case 7 (not shown) 69120 (40%) 10.42 30211 21997
Case 8 (not shown) 86400 (50%) 8.16 53353 34174
Fig. 7 Relations of performances among tampering ratios, false tam-pering detection, and tamtam-pering repairing using surveillance image of Fig.4(a).
is exactly that of Fig.9given above. From Table4, it can be seen that the proposed method provides better performance in the aspects of tampered pixel detection and tampering-ratio limitation, and has the additional self-recovery capabil-ity. In addition, due to the characteristic of pixel-level authentication, we can recover the tampered area by the unit of pixel and so can recognize the detailed part existing in the original image after the recovery work.
To reveal further the characteristics of the proposed
method, an image-authentication method12 based on the
similar concept of using compressed codes was used as a comparison. As can be observed in Table 5, the proposed method can recover tampered areas at the pixel level, instead
of at the block level as done in Ref.12. And it is also noted
that in Ref.12an auxiliary data item, a code book, is needed
for image repairing. This leads to inconvenience and non-blindness in the image-recovery process because extra storage space is required for the auxiliary data and the image-repairing work cannot be done without referring to the auxiliary data. On the contrary, the proposed method is characterized as blindness.
Some issues deserve further investigation in the future, for example, noise attacks. Though this kind of attacks can be detected with the aid of human vision in the proposed method, a feasible criterion which can be used to distinguish noise points in the authentication image caused by mutual affection from those resulting from noise attack is desired.
Fig. 8 Generation of stego-image from another image. (a) Input image Lena. (b) Stego-image with PSNR 39.34.
Fig. 9 Authentication result of a grayscale image with an added flower shape composed of 2084 pixels. (a) Image with modification of a hair portion. (b) Authentication image with noise. (c) Final authen-tication image. (d) Final repairing result with PSNR 47.00 with respect to stego-image of (b).
Table 3 Statistics of experiments using image Lena of Fig.8(a).
Lena (512 × 512)
Total # of tampered pixels (tampering ratio)
PSNR of recovered image with respect to stego-image
Total # of false acceptance pixels
Total # of false rejection pixels
Case 1 shown in Fig.9 2084 (0.79%) 47.00 43 12
Case 2 (not shown) 13100 (5%) 32.58 4 92
Case 3 (not shown) 26200 (10%) 25.93 87 430
Case 4 (not shown) 52400 (20%) 19.19 1169 4617
Case 5 (not shown) 78720 (30%) 14.12 8114 18866
Case 6 (not shown) 104640 (40%) 10.85 27678 42277
Case 7 (not shown) 131072 (50%) 8.52 65399 65477
Fig. 10 Relations of performances among tampering ratios, tamper-ing detection, and tampertamper-ing repairtamper-ing ustamper-ing image Lena of Fig.8(a). Lee and Tsai: Optimal pixel-level self-repairing authentication method for grayscale images: : :
5 Conclusions
A grayscale image-authentication method with a capability of localizing tampered image regions and repairing them at the pixel level has been proposed. Based on a bin-mapping scheme of dividing the five-bit grayscale into eight bins, a three-bit bin code is generated for use as an authentication signal for each input image pixel. The authentication signals are embedded into other pixels selected randomly by a secret key. The signals are utilized not only for detecting and loca-lizing tampered pixels but also for generating representative values for repairing the tampered pixels. This double-function merit of the authentication signal leads to the pos-sibility of pixel-level tampering detection and the blindness characteristic of the proposed method. Also shown is a proof of the optimality of the proposed method in choosing three bits out of the eight ones of a pixel as an authentication signal under a minimax criterion of minimizing the maximum total gray-value distortion incurred by authentication signal embedding and tampered pixel repairing. Experimental results have shown the effectiveness of the proposed method for authenticating and repairing tampered real images. Future works may be directed to extending the method to deal with color images.
Acknowledgments
This work is supported financially by the National Science Council, Taiwan, ROC under Project No. 99-2631-H-009-001.
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Che-Wei Lee received his BS degree in civil engineering and MS in electrical engineering from National Cheng Kung University, Tai-nan, Taiwan, in 2002 and 2005, respectively. He has been a PhD student in the depart-ment of computer science at National Chiao Tung University since 2005. His research interests include digital watermarking, image processing and video technologies.
Wen-Hsiang Tsai received his BS degree in electrical engineering from National Taiwan University, Taiwan, in 1973, MS in electrical engineering from Brown University in 1977, and PhD in electrical engineering from Purdue University in 1979. Since 1979, he has been with National Chiao Tung Univer-sity (NCTU), Taiwan, where he is now a chair professor of computer science. His cur-rent research interests include computer vision, information security, video surveil-lance, and autonomous vehicle applications.
Table 4 Comparison of performance of proposed method with those of Refs.8and9.
Authentication methods Pixel-level Recoverable
# of correctly detected pixels
out of 2084 tampered pixels Limitation of tampering ratio
Method in Ref.8 Yes No Around 1042 Unrestricted
Method in Ref.9 Yes No 1996 Smaller than 1.1%
