Noise verification

Given a tested image JJ and a verification key K=(ki)1≤i≤nK=(ki​)1≤i≤n​, the goal of noise verification is to recover and check the trails of noises in JJ at all regions kiki​. For each kiki​, we pick an atomic enveloping region vivi​ determined by:

vi≜(xiul−δix,yiul−δiy,xilr+δix,yilr+δiy)vi​≜(xiul​−δix​,yiul​−δiy​,xilr​+δix​,yilr​+δiy​)

where δixδix​ and δiyδiy​ are the width and the height of kiki​:

δix=xilr−xiul+1δiy=yilr−yiul+1δix​=xilr​−xiul​+1δiy​=yilr​−yiul​+1

Since any enveloping region is so small that spectral analysis cannot give reliable results, hence to filter the distortions of noises (i.e. the trails of high energy) we compare gradients of the region and the contained distortion region; one way to do that is using the Laplacian filter. Let abla2abla2 denote the Laplacian operator, calculate the mean of each enveloping region vivi​:

v‾i=1∣vi∣∑(x,y)∈vi(∇2vi)(x,y)vi​=∣vi​∣1​(x,y)∈vi​∑​(∇2vi​)(x,y)

and the mean of corresponding distortion region:

k‾i=1∣ki∣∑(x,y)∈ki(∇2vi)(x,y)ki​=∣ki​∣1​(x,y)∈ki​∑​(∇2vi​)(x,y)

where ∣vi∣∣vi​∣ and ∣ki∣∣ki​∣ are respectively the area of vivi​ and of kiki​. Then compare the deviation (c.f.~\cref{equ:noise_recovery,equ:noise_difference}):

ei≜∣v‾i−k‾i∣ei​≜∣vi​−ki​∣

with some energy threshold. Using the noise tuning, we experimentally accept the existence of the atomic watermarked wiwi​ when ei≥5ei​≥5.

If there is a distortion region where the deviation eiei​ is lower than the threshold then the image JJ is immediately rejected, otherwise JJ is accepted.

Remark. From the construction of enveloping regions from distortion regions, the areas can be simply calculated by ∣ki∣=δix×δix∣ki​∣=δix​×δix​ and ∣vi∣=9×∣ki∣∣vi​∣=9×∣ki​∣.

Figure 6:

The figure on the left shows an enveloping region of size 9 × 9 9×9, its distortion region is of size 3 × 3 3×3 located at the center, numbers at each pixel are the RGB color values. The right one shows the enveloping region after applying the Laplacian convolution.