Ten lectures on wavelets
Nonlinear total variation based noise removal algorithms
Proceedings of the eleventh annual international conference of the Center for Nonlinear Studies on Experimental mathematics : computational issues in nonlinear science: computational issues in nonlinear science
On the convergence of the coordinate descent method for convex differentiable minimization
Journal of Optimization Theory and Applications
Mathematical Programming: Series A and B
Proceedings of the 27th annual conference on Computer graphics and interactive techniques
Geometric partial differential equations and image analysis
Geometric partial differential equations and image analysis
Oscillating Patterns in Image Processing and Nonlinear Evolution Equations: The Fifteenth Dean Jacqueline B. Lewis Memorial Lectures
Convergence of a block coordinate descent method for nondifferentiable minimization
Journal of Optimization Theory and Applications
Wavelet Algorithms for High-Resolution Image Reconstruction
SIAM Journal on Scientific Computing
Convergence of alternating optimization
Neural, Parallel & Scientific Computations
Image Processing And Analysis: Variational, Pde, Wavelet, And Stochastic Methods
Image Processing And Analysis: Variational, Pde, Wavelet, And Stochastic Methods
Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations (Applied Mathematical Sciences)
A coordinate gradient descent method for nonsmooth separable minimization
Mathematical Programming: Series A and B
Split Bregman Algorithm, Douglas-Rachford Splitting and Frame Shrinkage
SSVM '09 Proceedings of the Second International Conference on Scale Space and Variational Methods in Computer Vision
Augmented Lagrangian Method, Dual Methods and Split Bregman Iteration for ROF Model
SSVM '09 Proceedings of the Second International Conference on Scale Space and Variational Methods in Computer Vision
Linearized Bregman Iterations for Frame-Based Image Deblurring
SIAM Journal on Imaging Sciences
The Split Bregman Method for L1-Regularized Problems
SIAM Journal on Imaging Sciences
IEEE Transactions on Image Processing
Bregmanized Nonlocal Regularization for Deconvolution and Sparse Reconstruction
SIAM Journal on Imaging Sciences
Wavelet frame based surface reconstruction from unorganized points
Journal of Computational Physics
De-noising by soft-thresholding
IEEE Transactions on Information Theory
Image decomposition via the combination of sparse representations and a variational approach
IEEE Transactions on Image Processing
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X-ray computed tomography (CT) has been playing an important role in diagnostic of cancer and radiotherapy. However, high imaging dose added to healthy organs during CT scans is a serious clinical concern. Imaging dose in CT scans can be reduced by reducing the number of X-ray projections. In this paper, we consider 2D CT reconstructions using very small number of projections. Some regularization based reconstruction methods have already been proposed in the literature for such task, like the total variation (TV) based reconstruction (Sidky and Pan in Phys. Med. Biol. 53:4777, 2008; Sidky et al. in J. X-Ray Sci. Technol. 14(2):119---139, 2006; Jia et al. in Med. Phys. 37:1757, 2010; Choi et al. in Med. Phys. 37:5113, 2010) and balanced approach with wavelet frame based regularization (Jia et al. in Phys. Med. Biol. 56:3787---3807, 2011). For most of the existing methods, at least 40 projections is usually needed to get a satisfactory reconstruction. In order to keep radiation dose as minimal as possible, while increase the quality of the reconstructed images, one needs to enhance the resolution of the projected image in the Radon domain without increasing the total number of projections. The goal of this paper is to propose a CT reconstruction model with wavelet frame based regularization and Radon domain inpainting. The proposed model simultaneously reconstructs a high quality image and its corresponding high resolution measurements in Radon domain. In addition, we discovered that using the isotropic wavelet frame regularization proposed in Cai et al. (Image restorations: total variation, wavelet frames and beyond, 2011, preprint) is superior than using its anisotropic counterpart. Our proposed model, as well as other models presented in this paper, is solved rather efficiently by split Bregman algorithm (Goldstein and Osher in SIAM J. Imaging Sci. 2(2):323---343, 2009; Cai et al. in Multiscale Model. Simul. 8(2):337---369, 2009). Numerical simulations and comparisons will be presented at the end.