by Triet M. Le and Luke Rogers

In image decompositions, one is interested in decomposing $$f$$ into $$u+v$$ where $$u$$ and $$v$$ have different features. In a variational approach, such a decomposition is achieved by solving the following variational problem
$$ \inf_{(u,v)∈X1×X2} \{ tF_1(u) + F_2(v) : f = u + v\}$$,
where $$F_1$$ and $$F_2$$ are positive functionals defined on some function spaces $$X1$$ and $$X2$$ respectively. Often the scale parameter $$t$$ is fixed a prior. In this paper, we address the problem of selecting the scale parameter $$t$$ in a multiscale fashion, and introduce the notion of interpolating scales that are stable with respect to the functional or energy being minimized. The motivation comes from Peetre’s K-functional in interpolation theory.

UCLA Computational and Applied Math Technical report 10-63

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