They didn't really morph, these were approaches to deal with a fundamental issue in signal representation in higher dimensions that comes up when you use tensor

They didn't really morph, these were approaches to deal with a fundamental issue in signal representation in higher dimensions that comes up when you use tensor product representations. Simply put, wavelets are not good at capturing edges in signals of dimension > 1 (although they are great at it for dimension 1) The fundamental (functional analysis) research in this area split in a few directions. One as you mention, mostly driven by imaging research, but also frame theory and non tensor-product basis, etc.