Test and Improvement of Robustness for Digital Watermarking
X. WANG, K. YEN, A. CABALLERO
Department of Electrical & Computer Engineering
Department of Construction Management
Florida International University
10555 W. Flagler Street
Miami Florida 33174, USA
Abstract: The use of digital multimedia for profit is pressing for the development of a robust and indelible method of protecting the rights of intellectual property of the content creators. For the most part the common denominator of many current watermarking techniques has been the use of pseudo-noise (PN) sequences in various forms to embed (or scramble) the message. The first half of this paper is an overview of robustness testing of the current state of digital watermarking against JPEG compression. And the second half presents an improvement of robustness for digital watermarking in frequency domains.
Key-words: Robustness, watermark, data hiding, human visual system (HVS), bit error rate.
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1 Introduction
As the Internet becomes populous, people concern more about the issue of copyright protection for digital data such as images and audio. Digital watermarking technique hides data in images or audio to indicate the data owner or recipient. One of the important issues to be considered regarding copyright protection by digital watermarking is robustness. In this paper, we first tested the robustness for different digital watermarking algorithms. Bit error rate is used to show the resistance of an algorithm when a watermark is attacked by JPEG compression. In this way, the comparison of robustness between different watermarking algorithms is possible. After all the tests done, watermarking in DCT domain has been found to have the highest robustness performance among algorithms selected. Then, after a careful exploration of statistics in frequency domains, such as DCT domain, Wavelet-Transform domain, an improvement for watermarking technique in frequency domains is proposed, which normalizes the data before checking the correlation coefficient. The simulation results demonstrated that this new approach could decrease the bit error rate efficiently. In some of the simulations, bit error rate has been decreased to 0.17% with the new approach comparing with 0.81% by using the other methods.
2 Robustness Testing
We begin with the explanation of embedding and detection procedure for digital watermarking. During embedding procedure, we assume that the original image is a discrete random process. Here the original image could be the data in the space domain or frequency domain.
(1)
Here N is the size of the selected block of host data.
Assume that the watermark is another discrete random process described by
(2)
where Wi = 1 or –1, k is the gain factor, and 1≤ i ≤N.
Since the watermark information is bit information, it is composed of 1 and 0. If the value 1 is to be embedded into the original image, the watermarked image is
= (3)
For the value of 0 to be embedded, we have
= (4)
Here is the transformed back of the original image, then will be compressed using JPEG standard. So we have
= + (5)
where is a noise, which is added to during the compression. In the detection process, the image will be decompressed first, and then be transformed to a corresponding domain.
Using the same key to generate watermark , the correlation coefficient between * and can be computed. Now if 1 is the value embedded, we have
(6)
and the expectation value of R, E(R), is k. On the other hand, if 0 is embedded,
(7)
then E(R) = -k. As a result, a convenient way to judge the embedded bit is to check whether the correlation coefficient is positive or negative. If it is positive, bit 1 is found; otherwise, 0.
Since bits 1 and 0 are supposed to be found in any host image, two normally distributed curves are expected for the correlation coefficient, as shown in Figure 1. The left one indicates the distribution for R if 0 is embedded, and the right one is the distribution for R if 1 is embedded. We can see that Region A and Region B cause the bit errors. For region A, actually bit 1 is embedded into the image but because the value of R is less than 0, it is taken as bit 0. It is the same for region B. If k is lager enough, the two curves will be more separated from each other. So region A and region B will be smaller, and thus the bit error rate can be improved. If a correlation coefficient is located in region A, false negative errors are produced possibly; if in region B, false positive errors are possible.
Figure 1. The curves of the correlation coefficient if multi-bits are embedded
The possibility of these two types of errors are derived based on a first-order autoregressive image model [1] and given below.
(11)
and
(12)
where
Here, Pfp represents the probability of false positive and Pfn the false negative, represents the variance of the watermark pixels, and represents the variance of the image pixels. If the watermark pattern W(x,y) only consists of the integers {-1,1} and the number of -1s is equal to the number of 1s, the variance of the watermark equals to k2. The errors represented by Pfp and Pfn can be minimized by increasing the gain factor “k”. Using larger k, however, will decrease the visual quality of the watermarked image.
3 Watermarking Algorithms Review
3.1 Algorithm in Space Domain
IW (x,y) = I(x,y) + k×W(x,y) (13) where IW (x, y) represents a pixel at position (x, y) of the watermarked image; I(x, y) is a pixel of the original image and W(x, y) watermark data at the position (x, y).
The method can be improved by pre-filtering the image before applying the watermark. If we can reduce the correlation between the host image and the PN (Pseudo-random Noise) sequence, we can increase the immunity of the watermark to additional noise. By applying the FIR edge enhancement filter shown in Equation 14, the robustness of the watermark can be improved with no loss of capacity and very little reduction of image quality [2].
(14)
3.2 Algorithm of Watermarking in DCT Domain
(15)
One such technique utilizes the comparison of middle-band DCT coefficients to encode a single bit into a DCT block. To begin with, we define the middle-band frequencies (FM) of an 8×8 DCT block as shown in Figure 2.
Figure 2. Three types of the frequency components
Three frequency components, low, middle, and high, exist. FL is used to denote the lowest frequency components of the block, and FH denotes the higher frequency components. FM, the middle frequency components are usually chosen to embed a watermark pattern to provide additional resistance to the lossy compression, while avoiding significant modification of the cover image.
The watermarking procedure can be made somewhat more adaptive by slightly altering the embedding process to the method shown below in Equation 16. It is called as Adaptive DCT Watermarking [3] and is presented below.
(16)
3.3 Algorithm in DWT Domain
Another possible domain for watermark embedding is the wavelet domain. DWT (Discrete Wavelet Transform) separates an image into a lower resolution approximation image (LL) as well as horizontal (HL), vertical (LH), and diagonal (HH) detail components. The process can then be repeated to compute multiple “scale” wavelet decomposition, as the one-scale wavelet transform shown in Figure 3.
Figure 3. One-scale 2-dimensional Discrete Wavelet Transform
One of the many advantages of the wavelet transform is that it is believed to have more accurately model aspects of the human visual system (HVS) as compared to the FFT or DCT. This allows us to use higher energy watermarks in regions that the HVS is known to be less sensitive to, such as the high-resolution detail bands {LH and HL}. Embedding watermarks in these regions allow to increase the robustness of a watermark, at little to no additional impact on image quality [2][4]. One of the most straightforward techniques is to use a similar embedding technique that is used in the DCT domain, with the embedding of a CDMA sequence in the detail bands according to Equation 17.
(17)
4 Improvement
Equations 11 and 12 tell us that the variance of a host data affects the error rate. The larger the variance is, the higher possibility of error will be. Unfortunately, usually data in the frequency domain has a large variance. Data in the middle frequency band of DCT domain, from a particular 8×8 block that caused bit error, is checked (See Table 1). The range of data varies greatly. There are even negative data in the block. The data referred to Table 1 is indicated as Matrix A. The watermark pattern for bit 1 is also checked and is indicated as Matrix B.
The correlation coefficient for A and B is
(18)
So the corresponding bit is mistaken as value 0, although it should be 1. By carefully checking the two matrixes, we notice that
(19)
This value contributes too much to the correlation coefficient and makes R less than 0. By checking other data in the blocks, most of the blocks that cause the bit error share the same feature, which is, some of the data are too big or too small compared with others. In other words, the data are not evenly distributed in the block. The data has to be more evenly distributed. Equation 20 gives a solution to this problem. Before the correlation, all the data are normalized as
(20)
Figure 4. The visualized relationship between x and y in Equation 20
By this way of the normalization, matrix A has different component values as shown in Table 3. The new correlation coefficient after normalization is
So the detected bit for this block is 1. It is correct.
In the following we have two results from simulation without and with a process of normalization. Both cases try to extract the origin information from the same watermark images as shown in Figure 5. We can clearly see less error dots in the right image.
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Figure 5. Comparison of two images of extracted watermark information.
Left image is the result without a process of normalization; right image is a result with a process of normalization. To see a complete comparison, we implemented an experiment with different JPEG quality rate (Q). The result is shown in Figure 6. Any Q between 100 and 30, normalization process helps improve the robustness. Experiment shows that normalization process also works for DWT algorithm. This result is shown in Figure 7.
5 Conclusions
Among all the algorithms, the one in DCT domain has the highest robustness. Probably one reason is that the compression theory for JPEG file is based on DCT domain. Robustness is very important for an algorithm to be used in the practical world. Any attempt to improve robustness is valuable. Like in the space domain, where the robustness is improved by using the edge-enhancement filter, in frequency domains (also called as transformed domains), the robustness could be improved by using the normalization of data before the correlation checking. The normalization of data before checking the correlation coefficient is an efficient way to improve the robustness in frequency domains. In some cases, the data in the frequency domain is distributed widely. While calculating the correlation coefficient, we find that some of the data will contribute strongly, leaving the others useless. But the hypothesis behind the correlation based watermarking algorithms is that the distribution of the data should be even. Every one data should have an equal chance to affect the calculation of correlation coefficient. Normalization process helps the data in frequency domains to maintain this required pattern. The simulation demonstrated that this new approach decreases the bit error rate efficiently, so robustness is improved. They show that when the Q rate is bigger than 30, the normalization of data do improve the robustness, both in DCT domain and DWT domain. From a mathematical view, normalization decreases the standard deviation of the host image data. Equations 11 and 12 pointed out that the probability of false negative or false positive is related to. Less means less error probability.
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Table 1 Matrix A, data in the middle frequency band in a 8×8 block
Table 2 Matrix B, the watermark pattern for bit 1
Table 3 New matrix A, normalized from the original matrix A in Table 1
Figure 6. Bit error rate between DCT and improved DCT
Figure 7. Bit error rate between DWT and improved DWT
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