Imagine that a line divides an image in two regions which (slightly) differ in terms of texture and color. It is not a perfect, artificial line but rather a thin transition zone. I want to build a neural network which is able to infer geometrical information on this line (orientation and offset). The image may also contain other elements which are not relevant for the task. Now, would a classical CNN be suitable for this task? How complex should it be in terms of number of convolutions (and number of layers, in general)?
A long shot: these guys have worked on a problem that might be relevant. They define "semantic" lines as lines delimiting significant regions or objects in an image. To detect such lines, they use the conv layers from a pre-trained VGG16 net and then add their own layers on top. The cool thing about their approach is that they run both classification and regression in parallel on the same network.
You might be able to adopt a similar technique to determine where the line is, and then run some simple analysis on the extracted line to determine the offset and the orientation.