Why are edges in spatial images represented as edges in their Fourier transform image? Here is a well-known image and its Fourier Transform (magnitude).


If I understand correctly the theory behind the FFT, each pixel in the FFT image represents a certain 2D sine wave with frequency depending on the distance from the center of the image, and orientation depending on angle with the horizontal. Intensity of these pixels indicates the coefficients with which each sine wave is added, which, combined with (hidden here) phase information, gives the original image back if we do an inverse Fourier transform.  

While I fail to grasp is that when treating real images such as the ones presented, why are edges in the images visible as edges in the frequential domain ?
In this example, there is a diagonal line in the FFT image (let's forget about the vertical and horizontal line which I think are artefacts based on the way FFT is computed, needs a periodical image, etc). This diagonal line is probably caused the girl's hat. But as I understand it, the line in FFT domain means a sum of sines oriented in the same way but with different frequencies. How does that result in an edge when we convert back ? Since edges are high frequency information, wouldn't an edge be represented by one very bright point in the FFT ?
Does it have to do with adding different sines so that they cancel each other out on some portions of the image ? Does the phase image has anything to do with it ?

A more compelling example might be the following image set :
An edge can be described as a sum of sinusoids of different frequencies with phase all equal to $\pi/2$ (or $-\pi/2$) at the location of the edge.

You might find this paper interesting, it is about how different combinations of sinusoids make up different types of features.

Svetha Venkatesh, Robyn Owens On the classification of image features