Random tree besov priors – towards fractal imaging
H.N. Kekkonen (TU Delft - Statistics)
Matti Lassas (University of Helsinki)
Eero Saksman (University of Helsinki)
Samuli Siltanen (University of Helsinki)
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Abstract
We propose alternatives to Bayesian prior distributions that are frequently used in the study of inverse problems. Our aim is to construct priors that have similar good edge-preserving properties as total variation or Mumford-Shah priors but correspond to well-defined infinite-dimensional random variables, and can be approximated by finite-dimensional random vari-ables. We introduce a new wavelet-based model, where the non-zero coefficients are chosen in a systematic way so that prior draws have certain fractal behaviour. We show that realisations of this new prior take values in Besov spaces and have singularities only on a small set τ with a certain Hausdorff dimension. We also introduce an efficient algorithm for calculating the MAP estimator, arising from the the new prior, in the denoising problem.