7,891 research outputs found
The curvelet transform for image denoising
We describe approximate digital implementations of two new mathematical transforms, namely, the ridgelet transform and the curvelet transform. Our implementations offer exact reconstruction, stability against perturbations, ease of implementation, and low computational complexity. A central tool is Fourier-domain computation of an approximate digital Radon transform. We introduce a very simple interpolation in the Fourier space which takes Cartesian samples and yields samples on a rectopolar grid, which is a pseudo-polar sampling set based on a concentric squares geometry. Despite the crudeness of our interpolation, the visual performance is surprisingly good. Our ridgelet transform applies to the Radon transform a special overcomplete wavelet pyramid whose wavelets have compact support in the frequency domain. Our curvelet transform uses our ridgelet transform as a component step, and implements curvelet subbands using a filter bank of a` trous wavelet filters. Our philosophy throughout is that transforms should be overcomplete, rather than critically sampled. We apply these digital transforms to the denoising of some standard images embedded in white noise. In the tests reported here, simple thresholding of the curvelet coefficients is very competitive with "state of the art" techniques based on wavelets, including thresholding of decimated or undecimated wavelet transforms and also including tree-based Bayesian posterior mean methods. Moreover, the curvelet reconstructions exhibit higher perceptual quality than wavelet-based reconstructions, offering visually sharper images and, in particular, higher quality recovery of edges and of faint linear and curvilinear features. Existing theory for curvelet and ridgelet transforms suggests that these new approaches can outperform wavelet methods in certain image reconstruction problems. The empirical results reported here are in encouraging agreement
Unsupervised bayesian convex deconvolution based on a field with an explicit partition function
This paper proposes a non-Gaussian Markov field with a special feature: an
explicit partition function. To the best of our knowledge, this is an original
contribution. Moreover, the explicit expression of the partition function
enables the development of an unsupervised edge-preserving convex deconvolution
method. The method is fully Bayesian, and produces an estimate in the sense of
the posterior mean, numerically calculated by means of a Monte-Carlo Markov
Chain technique. The approach is particularly effective and the computational
practicability of the method is shown on a simple simulated example
MR image reconstruction using deep density priors
Algorithms for Magnetic Resonance (MR) image reconstruction from undersampled
measurements exploit prior information to compensate for missing k-space data.
Deep learning (DL) provides a powerful framework for extracting such
information from existing image datasets, through learning, and then using it
for reconstruction. Leveraging this, recent methods employed DL to learn
mappings from undersampled to fully sampled images using paired datasets,
including undersampled and corresponding fully sampled images, integrating
prior knowledge implicitly. In this article, we propose an alternative approach
that learns the probability distribution of fully sampled MR images using
unsupervised DL, specifically Variational Autoencoders (VAE), and use this as
an explicit prior term in reconstruction, completely decoupling the encoding
operation from the prior. The resulting reconstruction algorithm enjoys a
powerful image prior to compensate for missing k-space data without requiring
paired datasets for training nor being prone to associated sensitivities, such
as deviations in undersampling patterns used in training and test time or coil
settings. We evaluated the proposed method with T1 weighted images from a
publicly available dataset, multi-coil complex images acquired from healthy
volunteers (N=8) and images with white matter lesions. The proposed algorithm,
using the VAE prior, produced visually high quality reconstructions and
achieved low RMSE values, outperforming most of the alternative methods on the
same dataset. On multi-coil complex data, the algorithm yielded accurate
magnitude and phase reconstruction results. In the experiments on images with
white matter lesions, the method faithfully reconstructed the lesions.
Keywords: Reconstruction, MRI, prior probability, machine learning, deep
learning, unsupervised learning, density estimationComment: Published in IEEE TMI. Main text and supplementary material, 19 pages
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