I´m currently reading the paper Edge analysis and identification using the continuous shearlet transform (by K. Guo, D. Labate and W. Lim). In this paper it is written that not only the detection of edges is important, it is also important to know more about the geometrical structure of the boundary curves (edges). (This is why shearlets are more usefull than wavelets in this particular task.) My question is: Why is it important to also know the geometrical structure of these edges? Does anyone have an example? In which application is it needed?
Thank you in aticipation.
Wavelets (speaking intuitively) provide you some understanding of the frequencies in the image and where those frequencies are located. If you look at the Haar wavelet transform, for example, it breaks the image into four smaller images: LL, LH, HL, and HH. (Here, L means low frequencies and H means high frequencies; the first/second letter represents the first/second dimension, respectively). By eye, we can look at the wavelet transform of an image and see approximately where in the image the "low" frequencies are and the "high" frequencies. You can lookup "Haar Wavelet of image" and see many examples.
If you isolate one pixel in the Wavelet transform (setting all other values to 0) and invert transform, you'll get a basis function of what that pixel represents. And you'll see that you can't tell much about the original image. All you see is that there's an edge in the original image, but you can't tell more than that.
Although the Wavelet transform is invertible (meaning no information is lost) when looking at the transform it's difficult for our brains to extract some of the information out. What is the higher order structure of the image? Perhaps, if we had some representation of the figure that was invertible (to ensure that no information was lost) and told us what the structure of the data was locally then we could use this information to our advantage. And the Shearlet is an example of such a concept. If you look at the basis function of a Shearlet, then you see a curved structure. So you see that you not only see that there's an edge present, but you get some understanding of what that edge looks like locally.
This can sometimes lead to the image being more compressible. And this can be useful in compressed sensing, for example.
(Note: this was all broad strokes and not meant to be a rigorous discussion.)