Robustness relative to geometrical distortions is one of the greatest challenges in the watermarking of images and videos.
The attack known as the “print & scan” attack is one of the most devastating attacks known.
In one set of problems related to copyright protection for authors, when an image is watermarked, a watermark presenting copyright information is inserted into the image. This image, thus protected, risks being subjected to modifications (intentional or unintentional). These modifications are called attacks on the watermarked document. Any attack can lead to a partial erasure of the watermark. The more virulent the attack, the greater is the deterioration of the quality of the watermarked image and the greater are the chances that the detector will lose the watermark. There is therefore a need to propose a technique for watermarking images (including text-type images) that is robust against “print & scan” attacks).
Printing and scanning a document (an image) leads to a multitude of attacks:                Adding noise, due to the scanner and the printer;        <<Scaling>> due to the scanner and the printer;        <<Shearing>> due to the scanner and the printer;        conversion of color to gray scale, due to the scanner and the printer;        conversion of depth (for example from 255 bpp to 2, 8, 16 . . . bpp), due to the scanner and the printer;        analog-to-digital conversion;        rotation, due to the scanner and the printer;        shifting due to the scanner and the printer;        change of image format (tif/pdf/jpg/ . . . ), due to the scanner;        cropping due to the scanner;        compression, typically JPG or J2K, due to the scanner;        Etc.        
First of all, one of the pioneering articles on the use of multiplicative watermarking (and its comparison with additive watermarking) was presented in the following article: I. J. Cox, J. Kilian, T. Leighton, and T. Shamoon. Secure spread spectrum watermarking for multimedia. IEEE Transactions on Image Processing, 6(12):1673-1687, 1997.
The authors of this article present chiefly three watermark insertion equations:v′i=vi+αxi v′i=vi(1+α)xi v′i=vi(eαxi)
Here v′i is a watermarked image (or an image coefficient, vi is the original image (or an image coefficient), xi is a watermark (or a watermark coefficient) inserted into the image (or the image coefficient), and α is a weighting factor (often a simple numerical value, sometimes dependent on the image: the term used then is watermark weighting coefficient). The greater the strength of the watermark α the greater is the risk that the watermarked image will be degraded, and the greater the robustness of the watermark.
The measurement of similarity used in this article (and very often taken up in other work on watermarking) is the following:
      sim    ⁡          (              X        ,                  X          *                    )        =                    X        *            ·      X                                X          *                ·                  X          *                    
X* representing an extracted mark and X the original mark.
These studies have introduced the concept of spread-spectrum watermarking. This concept comes from the theory of communications and is expressed in watermarking by a spreading of the watermark throughout the image (or through all its transformed coefficients). This generally leads to a watermark of very low amplitude occupying all the coefficients of the image.
The following article is also known: V. Solachidis and I. Pitas. “Circularly symmetric watermark embedding in the 2D DFT domain”. IEEE Transactions on Image Processing, 10(11):1741-1753, 2001. In this article:                the watermark is inserted additively or multiplicatively in the Fourier domain and the watermark bits are distributed on concentric circle segments forming a ring in the medium frequencies. A weighting is planned for the watermark (to increase the insertion strength “α”);        the detection of the watermark uses a computation of 2D normalized cross-correlation between the potentially marked coefficients and the watermark. The detection is called <<blind>> detection as the detector needs only the watermarked image and the original watermark.        
The method of correlation used by Solachidis & Pitas is the following:
  C  =            ¥                        k          ⁢                                          ⁢          1                -        1            N        ⁢                  ¥        N                              k          ⁢                                          ⁢          2                -        1              ⁢          W      ⁡              (                              k            ⁢                                                  ⁢            1                    ,                      k            ⁢                                                  ⁢            2                          )              ⁢                  M        ′            ⁡              (                              k            ⁢                                                  ⁢            1                    ,                      k            ⁢                                                  ⁢            1                          )            
where W represents the watermark and M′ represents the watermarked coefficients.
This computation of correlation provides only one value (the scalar C) and this value will be used to determine whether the watermark is present in the image as a function of the detection threshold. This (scalar) value does not enable a decision to be taken optimally, as compared with the solution proposed by the invention (which provides a 2D correlation map.
Besides, the position and the shape of the watermark in the Fourier domain are two crucial aspects.
With regard to the position of the watermark, the scientific community working on watermarks has been habitually trying to modify (watermark) the mean and low frequency coefficients of the transform domains (DCT, wavelets, Fourier). The modification of the high frequency coefficients is reputed to give rise to an invisible watermark but such a modification (of the high frequencies) is supposed to offer low robustness against distortions (and especially relative to the compression algorithms). Conversely, a watermarking of the low-frequency coefficients (close to the center of the Fourier domain) is supposed to be highly robust against every type of distortion but is also very rapidly visible. To put it briefly, if a watermark is placed in the high frequencies, its strength of insertion (α into the equations 1 and 2) could be very great without in any way degrading the quality of the watermarked image but will offer low robustness whereas a watermark placed in the low frequencies will be far more robust but will have to be modulated by very low strength, or else it will be visible. For these reasons, and as in most watermarking methods in the literature, Solachidis and Pitas in the technique presented in their article have opted for watermarking in the medium frequencies.
With regard to the shape of the watermark, Solachidis and Pitas propose a ring-shaped watermark in the Fourier domain. One drawback is that the correlation between the original watermark and the extracted watermark (watermark to be tested) is skewed by the ring shape. In short, the correlation indicates that the two noises resemble each other but this resemblance is due to the shape of the ring. In practice, the detection threshold should be raised to take account of this skew. Indeed, if the detection threshold is not accurately set, there is a risk of obtaining false negatives or false positives (which are detection errors). It may be recalled that the detection threshold makes it possible to determine if the image is watermarked or not: if the correlation is above the threshold, the image is considered to be watermarked. If the correlation is below the threshold, the detector will declare the image to be not watermarked.
Several scenarios can be envisaged:                a watermarked image gives rise to a correlation above the threshold: the term used here is true positive (the mark has been correctly detected);        a watermarked image gives rise to a correlation below the threshold: the term used here is “false negative” (a watermark has been “missed”);        a non-watermarked image gives rise to a correlation below the threshold: the term used here is “true negative” (the mark has not been detected because it is not present);        a non-watermarked image gives rise to a correlation above the threshold: the term used here is “false positive” (a mark that is not present has been detected).        