How to do edge detection using Curvelet transform?
If you decide to stick to curvelets, you can find a precise description of an edge dectection algorithm is  Edge detection in microscopy images using curvelets, Tobias Gebäck and Petros Koumoutsakos, BMC Bioinformatics, 2009. The pdf is there. And you are very lucky: you can get, for free, an  implementation, CurveletUtils. It provides a Matlab source code for a GUI implementing the edge detection method. I hope you can get started with that.

If you care of other multiscale transforms providing different oriented  edges at different scales, you can take a look at a review paper  on 2D directional wavelets (including curvelets, shearlets, contourlets, etc.). It includes morphological and non-linear wavelets.

By the way, I do not understand why you call Canny and Sobel "morphological edge detectors".