Naive approach for inserting a signature in a digital photograph

    In this example of the use of the Fourier transforms one seeks to insert a signature in the photograph of the Chinese Musician. Of course, this mark should serve for the identification of the image without damaging too much the aesthetic of it.
Photograph to be signed & its Fourier transform

    In order to realize that, on can insert the signature not directly on the image but in the Fourier space, without modifying too much the image: it is then evident to modify rather the high resolution Fourier coefficients than the low resolution ones because these latter are associated with sine waves of low frequencies and of strong amplitudes.
It should be noted, however, that in order to obtain a real value signed image one has to keep satisfied the symmetry relationship F(-h,-k)=F*(h,k): the signature should then be inserted symmetrically around F(0,0).
The signature text is a rectangular area of 50*20 pixels, all zero excepted the text 'Hello' and is then inserted in the FT simply by replacing the corresponding Fourier coefficients of the original image:

Fourier transform with the inserted signature keeping the symmetry around F(0,0)

    The signed photograph is then obtained by the inverse Fourier transform of this modified FT, and is aesthetically equivalent to the original image:

Sign photograph

    When signed, the photograph can be identified by simply calculating its Fourier transform:

The hidden signature blows up when calculating the Fourier transform of the image

    This approach is very simple but is not robust for the different processes that can be applied to the signed image, such as format changes, compressions, image size modifications…