The Morphlet Transform : A Multiscale Representation for Diffeomorphisms

We describe a multiscale representation for diffeomorphisms. Our representation allows synthesis – e.g. generate random diffeomorphisms – and analysis – e.g. identify the scales and locations where the diffeomorphism has behavior that would be unpredictable based on its coarse-scale behavior. Our representation has a forward transform with coefficients that are organized dyadically, in a way that is familiar from wavelet analysis, and an inverse transform that is nonlinear, and generates true diffeomorphisms when the underlying object satisfies a certain sampling condition. Although both the forward and inverse transforms are nonlinear, it is possible to operate on the coefficients in the same way that one operates on wavelet coefficients; they can be shrunk towards zero, quantized, and can be randomized; such procedures are useful for denoising, compressing, and stochastic simulation. Observations include: (a) if a template image with edges is morphed by a complex but known transform, compressing the morphism is far more effective than compressing the morphed image. (b) One can create random morphisms with and desired self-similarity exponents by inverse transforming scaled Gaussian noise. (c) Denoising morpishms in a sense smooths the underlying level sets of the object.