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Random tree Besov priors – Towards fractal imaging

  • *Corresponding author: Hanne Kekkonen

    *Corresponding author: Hanne Kekkonen 
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  • 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 variables. 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 $ \tau $ 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.

    Mathematics Subject Classification: Primary: 60F17, 65C20, 42C40; Secondary: 28A80.


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  • Figure 1.  (a) Photograph featuring fractal-like structures such as boundaries of clouds. (b) The three-dimensional structure of human lungs follows a self-similar rule leading to a fractal dimension. Also shown is an X-ray tomography slice of lungs. Images courtesy of Wikimedia Commons

    Figure 2.  On the left: all the nodes where $ t_{j, k}<\beta $, with $ t_{j, k}\sim\mathcal{U}[0, 1] $ and $ \beta\in[0, 1] $. On the right: the proper subtree where only nodes with direct connection to the root node are included

    Figure 3.  From left to right; Weierstrass function with Hausdorff dimension 1.5, sixth iteration of the space filling Hilbert curve with Hausdorff dimension 2, and Menger sponge with Hausdorff dimension $ \log_3(20)\approx2.7 $

    Figure 4.  Original blocks signal and the observed noisy signal with signal to noise ratio 3

    Figure 5.  The denoised signal using pruning algorithm on top left and the denoised signal that was attained using hard thresholding on top right. Below them are the wavelet trees corresponding to the estimators

    Figure 6.  Prior draws from the 1-dimensional semi-Gaussian random tree Besov prior with Haar wavelets. The wavelet densities are $ \beta = 0.6 $ on the left, $ \beta = 0.75 $ on the middle and $ \beta = 0.9 $ on the right

    Figure 7.  Measured accelerometer data on left and the denoised signal on right

    Figure 8.  Original sharp image and image with Gaussian noise with variance $ 0.01 $

    Figure 9.  From left to right; Denoised images and close-ups attained using pruning, tree enforced soft thresholding and soft thresholding algorithms. From the close-ups we can clearly see that the tree based methods denoise smooth areas better than soft thresholding

    Figure 10.  Prior draws from the 2-dimensional semi-Gaussian random tree Besov prior with Daubechies 2 wavelets. The wavelet densities are $ \beta = 0.3 $ on the left, $ \beta = 0.6 $ on the middle and $ \beta = 0.9 $ on the right

    Table Algorithm 1.  Pseudocode for finding the minimising $ t $ and $ g $ recursively. In practical calculations we found that replacing $ g_{jk} = m_{jk}/2 $ by $ g_{jk} = m_{jk} $ gives better results

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