  We propose alternatives to Bayesian a priori 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 variables. We
introduce a new wavelet-based model, where the non zero coefficient 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 some Besov spaces and have
singularities only on a small set $\tau$ that has a certain Hausdorff
dimension. We also introduce an efficient algorithm for calculating the MAP
estimator, arising from the the new prior, in denoising problem.
