Considering a minimization of the Mumford and Shah energy functional for purposes of image segmentation, I understand why we'd want to enforce the first two terms of the energy. I don't understand the benefit of minimizing the final term, which represents border length of an image segment:
If the idea is to find an "ideal" boundary to segment the image, why should the boundary be constrained by length? What if the image is naturally segmented into a very large partition? Wouldn't minimizing this term then have an adverse affect on the convergence to the "ideal" partition?
To summarize into a single question:
What is the intuition behind minimizing boundary length for segmentation?
Energy minimization methods usually have 2 terms. the "data term", which gets smaller the more the proposed solution fits the data exactly, and the "regularization term" or "smoothing term" whose purpose is to prevent overfitting. Since the data is usually noisy or lacking, fitting exactly to noisy data usually produces very jagged solutions that are not easily extrapolated. That is, if you would have just kept the data term in your problem and added new data, the solution would change drastically, whereas the regularization term keeps the solution (boundary in your case) in a more or less physical shape.
In short: the regularization term encodes prior knowledge of the problem domain, that the boundaries are relatively smooth, etc.