Watermarking with Side Information

1. Watermarking with Side Information Multimedia Security

2. The basic model for a digital watermarking system • Both the embedder and the detector ignore the available information about the cover work. Cover work interference Co N Received message Message coding Scaling Detection m Source message Embedder

3. Informed watermarking (Private Watermarking) • The detector is provided with information about the original, unwatermarked cover work. • One can subtract this information from the received work, eliminating any interference between the cover work and the watermark. • With an informed detector, the detectability of the watermark is affected only by the distortions to which the watermarked work is subjected after embedding; it is not affected by the cover work itself.

4. Partially Informed Embedding (E-Fixed-LC/D-LC) • The embedder examines the cover work during the modification of the message mark in preparation for embedding; the message mark is scaled so that one can obtain a fixed linear correlation detection value. Cover work interference Co N Received message Message coding Modification Detection m Source message Embedder

5. Embedding as an Optimization Problem There are two main application dependent scenarios : • Find the added mark that maximize robustness while keeping within a prescribed limit on perceptual distortion. • Find the added mark that minimizes perceptual distortion while maintaining a prescribed level of robustness.

6. A more complicated and challenging problem For example, one may specify a maximum level of distortion and a minimum level of robustness. If the watermark cannot be embedded within both constraints, one or the other should be automatically relaxed. Thus, the embedder might try to maximize robustness while staying within the maximum distortion. However, if the highest robustness attainable is lower than a specified minimum, the perceptual constraint is relaxed. To balance the tradeoff between fidelity and robustness!! Key Point : we need methods for measuring perceptual distortion and robustness!!

7. A. Optimizing with respect to a Detection Statistic • The simplest way to measure robustness is to assume that a watermark with a high detection value is more robust than one with a low detection value, so that the Detection Statistic itself becomes the measure of robustness. • Thus, when embedding a watermark with fixed robustness, the user specifies a desired detection value with the minimum possible distortion. • For example, in the E-Fixed-LC embedder, we specify a desired linear correlation as the sum of the threshold, , and a “strength” parameter, .

8. The assumption that high detection values indicate high robustness is generally true for linear correlation. A high detection value means the watermarked vector is far from the detection threshold, and more distortion of any type can be applied before the watermark is undetectable. • However, this assumption is not true for many other detection statistics, including normalized correlation.

9. Why the E-Fixed-LC algorithm is inappropriate for normalized correlation? Embedding region Detection region : Projected Vectors before embedding Wr : Projected Vectors after embedding

10. What should we do? • We need to fix the normalized correlation at some desired value, , rather than fixing the linear correlation! • For each work, represented by ‘ ’, the embedder finds the closest point in marking space, represented by ‘ ’, that has the desired normalized correlation with the watermark.

11. Y Embedding region Detection region : Projected Vectors before embedding (Vo) Wr X : Projected Vectors after embedding (Vw)

12. How to implement the fixed normalized correlation embedder? • The embedding region, for normalized correlation, is the surface of a cone, centered on the reference vector, . Therefore, we are trying to find the closest point on a cone to a given point, . Clearly, this point will lie in the plane that contains both and .

13. So the problem can be reduced to two dimensions by using Gram-Schmidt orthonormalization approach to find two orthogonal coordinate axes for this plane:

14. Each point on the plane can be expressed with an x and y coordinate as . The X-axis is aligned with the reference vector, . The coordinateds of are: Note that is guaranteed to be positive.

15. The embedding region intersects with the XY plane along two lines. Because is positive, the closest point will be on the upper line. This line is described by the vectors

17. E_BLK_Fixed_CC/D_BLK_CC: 1.Use the extraction function Ex(.) to extract a vector, , from the unwatermarked image, . 2. Compute a new vector in marking space, , that is acceptably close to but lies within the detection region for the desired watermark, 3.Perform the inverse of the extraction operation on to obtain the watermarked image, .

18. As in the E_BLK_BLIND embedding algorithm, we encode a 1-bit message, m , by determining the ‘Sign’ of a given 8x8 reference mark, :

19. Fragility of a Fixed Normalized Correlation Strategy for Information Embedding Does a larger value of implying higher robustness? Unlike with linear correlation, higher normalized correlation do not necessarily lead to higher robustness. Linear correlation measures the magnitude of the signal ,normalized correlation essentially measures the signal-to-noise ratio.

20. The signal to noise ratio can be high even if the signal is very weak, as long as the noise is proportionally weak. So, if we add a fixed amount of noise to a weak signal, the resulting signal-to-noise ratio plummets (筆直落下).

21. With a high value of , the E-BLK-Fixed-CC embedder yields a vector closer to the cone. Wr Perturbing this vector in almost any direction is likely to send it outside the detection region. A low value of yields a vector deeper within the cone, And thus more resilience to noise.

22. B. Optimizing with respect to an Estimate of Robustness: The problem with the E_BLK_Fixed_CC algorithm is that its “strength” parameter, , is based on the wrong measure. Normalized correlation does not measure robustness. A correct measure should estimate the amount of , white, Gaussian noise that can be added to the embedded vector, , before it is expected to fall outside the detection region.

23. The reason for basing this measure on Gaussian noise, rather than on some more realistic form of degradation, is that correlation coefficients are not inherently robust against it. If we ensure that our watermarks are robust against this type of noise, it helps to ensure that they are robust against other types of degradations, for which the correlation coefficient may be better suited.

26. Detection region Wr

27. We now wish to make an embedder that holds constant, rather than holding constant. Contours of constant for a given threshold, as shown in the above figure, are hyperbolas. Note that each hyperbola has a positive and a negative part. We wish to ensure that the correlation between the watermark and the selected vector, , is positive. Thus, we restrict the embedder to only the positive portion of the hyperbola.

28. In other words, the embedder will not only hold constant, but also ensure that is positive. Thus, the embedding region in N-dimensional marking space is not a cone but one half of an N-D, two –sheet hyperboloid. Note: A hyperboloid is obtained by rotating a hyperbola about one of its axes. With hyperbola oriented like those in the above figure, a rotation about the y-axis will result in a single surface called a one-sheet hyperboloid. A rotation about the x-axis will result in two surfaces (one for the positive part and one for the negative part), or a two-sheet hyperboloid. We restrict the embedding region to the positive sheet of the two-sheet hyperboloid.

29. E_BLK_Fixed_R/D_BLK_CC: The E_BLK_Fixed_R embedder is essentially the same as the E_BLK_Fixed_CC embedder, except that it maintains a constant instead of normailzed correlation. The target value of is provided by the user as a “strength” parameter. Rather than finding analytically we perform a simple search in the same XY plane as used for fixed normalized correlation embedding.