Edges are not the best defined features in images.
However, they can be associated, locally, at a certain scale, with relative variations in intensity along a first direction, combined with a relative smoothness in a complementary (for instance orthogonal) second direction. When looking along the first direction, the 1D intensity profile exhibit variations, faster than in the other direction. Several typical edges profiles exist, as depicted below.
Those are typically enhanced by derivative-like operators, which are weakly sensitive to flat, linear or generally regular intensity variations. In other words, they behave as band-pass or high-pass operators.
In the other direction, gradient detectors offer more smoothing (more low-pass).
Another vision is to smooth an image, and subtract this result from the original image. By removing low-frequencies, higher ones are relatively amplified. But to avoid noise amplification, some smoothing is often included in the process.
Hence, linear or not, edge operators tend to emphasize mid- to high frequencies relatively to the lower ones. However, all this may happen at different scales, depending on the edgy objects of the picture.
Now, Is this new information reliable?
Not really, if you rely on mere filtering or enhancement. It is easy to produce spurious edges. But within a restoration or inverse problem context, with models and an objective function to minimize, more reliable information can be obtained, as long as the models mimic well the physics of your acquisition scheme.