So if we can isolate the edges of an image, and we know that the apparant sharpness of an image is dependant on the strength of the edges, then we should be able to increase the sharpness of an image by boosting the power in the edges, which means boosting the power in the high frequencies. This is easily done through a linear combination of the original image with the image resulting from a highpass filter. And since all these operations are linear, we can perform this addition prior to multiplying by the filter rather than afterwards, and thereby simply incorporate it into the filter.

If we call the transform of the original image A and a fully attenuating highpass filter H, then the transform of the highpassed image B(u,v) = A(u,v)*H(u,v). Therefore we can create any linear combination C = aA + bB = aA + b(A*H) = A(a + bH) and therefore we can create our sharpening filter H'(u,v) = (a + bH(u,v)). By selecting a good ratio of a to b as well as choosing the right cutoff frequency for the filter, we can therefore create natural looking sharpening of the photo.