Similarity between two geometric shapes

I have two shapes in a 2D space, not necessarily convex, and I'd like to compare how similar they are. How can I define a robust distance metric to measure their similarity, and how can I compute it?

I am looking for a method which provides a short distance in case of:

• scaling;

• rotation;

• perhaps local scaling or rotation.

I see two possible solutions:

• transform the shapes into pixel-based matrices (bitmap) and compute a Levenshtein distance (but without enough robustness in the distance, in case of rotation for instance);

• transform the shapes into graphs and try to define a distance between them.

One approach would be to use SIFT (or SURF, or other similar methods) to align one object to the other to account for scaling and rotation, and then compute a pixelwise distance based on the aligned images.

The right algorithm to use for alignment will depend upon the nature of the 2D images. If they are natural-color photographs taken using a camera, SIFT, SURF, etc. might work well. If they are black-and-white geometric shapes like shown in the question, you might do better to use image moments (e.g., Hu moments) or some other approach.

StackExchange Computer Science Q#22781, answer score: 2