High-order total variation regularization approach for axially symmetric object tomography from a single radiograph
Raymond H. Chan Haixia Liang Suhua Wei Mila Nikolova Xue-Cheng Tai
Inverse Problems & Imaging 2015, 9(1): 55-77 doi: 10.3934/ipi.2015.9.55
In this paper, we consider tomographic reconstruction for axially symmetric objects from a single radiograph formed by fan-beam X-rays. All contemporary methods are based on the assumption that the density is piecewise constant or linear. From a practical viewpoint, this is quite a restrictive approximation. The method we propose is based on high-order total variation regularization. Its main advantage is to reduce the staircase effect while keeping sharp edges and enable the recovery of smoothly varying regions. The optimization problem is solved using the augmented Lagrangian method which has been recently applied in image processing. Furthermore, we use a one-dimensional (1D) technique for fan-beam X-rays to approximate 2D tomographic reconstruction for cone-beam X-rays. For the 2D problem, we treat the cone beam as fan beam located at parallel planes perpendicular to the symmetric axis. Then the density of the whole object is recovered layer by layer. Numerical results in 1D show that the proposed method has improved the preservation of edge location and the accuracy of the density level when compared with several other contemporary methods. The 2D numerical tests show that cylindrical symmetric objects can be recovered rather accurately by our high-order regularization model.
keywords: high-order total variation augmented Lagrangian method. radiograph Tomography Abel inversion
Model distortions in Bayesian MAP reconstruction
Mila Nikolova
Inverse Problems & Imaging 2007, 1(2): 399-422 doi: 10.3934/ipi.2007.1.399
The Bayesian approach and especially the maximum a posteriori (MAP) estimator is most widely used to solve various problems in signal and image processing, such as denoising and deblurring, zooming, and reconstruction. The reason is that it provides a coherent statistical framework to combine observed (noisy) data with prior information on the unknown signal or image which is optimal in a precise statistical sense. This paper presents an objective critical analysis of the MAP approach. It shows that the MAP solutions substantially deviate from both the data-acquisition model and the prior model that underly the MAP estimator. This is explained theoretically using several analytical properties of the MAP solutions and is illustrated using examples and experiments. It follows that the MAP approach is not relevant in the applications where the data-observation and the prior models are accurate. The construction of solutions (estimators) that respect simultaneously two such models remains an open question.
keywords: regularization MAP estimation modeling. restoration
Analytical bounds on the minimizers of (nonconvex) regularized least-squares
Mila Nikolova
Inverse Problems & Imaging 2008, 2(1): 133-149 doi: 10.3934/ipi.2008.2.133
This is a theoretical study on the minimizers of cost-functions composed of an l 2 data-fidelity term and a possibly nonsmooth or nonconvex regularization term acting on the differences or the discrete gradients of the image or the signal to restore. More precisely, we derive general nonasymptotic analytical bounds characterizing the local and the global minimizers of these cost-functions. We first derive bounds that compare the restored data with the noisy data. For edge-preserving regularization, we exhibit a tight data-independent bound on the l norm of the residual (the estimate of the noise), even if its l 2 norm is being minimized. Then we focus on the smoothing incurred by the (local) minimizers in terms of the differences or the discrete gradient of the restored image (or signal).
keywords: Regularization Non-convex analysis Inverse problems Variational methods Edge restoration Image restoration Signal restoration Non-smooth analysis.
Two-phase approach for deblurring images corrupted by impulse plus gaussian noise
Jian-Feng Cai Raymond H. Chan Mila Nikolova
Inverse Problems & Imaging 2008, 2(2): 187-204 doi: 10.3934/ipi.2008.2.187
The restoration of blurred images corrupted with impulse noise is a difficult problem which has been considered in a series of recent papers. These papers tackle the problem by using variational methods involving an L1-shaped data-fidelity term. Because of this term, the relevant methods exhibit systematic errors at the corrupted pixel locations and require a cumbersome optimization stage. In this work we propose and justify a much simpler alternative approach which overcomes the above-mentioned systematic errors and leads to much better results. Following a theoretical derivation based on a simple model, we decouple the problem into two phases. First, we identify the outlier candidates---the pixels that are likely to be corrupted by the impulse noise, and we remove them from our data set. In a second phase, the image is deblurred and denoised simultaneously using essentially the outlier-free data. The resultant optimization stage is much simpler in comparison with the current full variational methods and the outlier contamination is more accurately corrected. The experiments show that we obtain a 2 to 6 dB improvement in PSNR. We emphasize that our method can be adapted to deblur images corrupted with mixed impulse plus Gaussian noise, and hence it can address a much wider class of practical problems.
keywords: two-phase approach Mumford-Shah functional. impulse noise Image deblurring

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