General Regularization Term Research Articles

Generalized Arnoldi--Tikhonov Method for Sparse Reconstruction

This paper introduces two new algorithms, belonging to the class of Arnoldi--Tikhonov regularization methods, which are particularly appropriate for sparse reconstruction. The main idea is to consider suitable adaptively defined regularization matrices that allow the usual 2-norm regularization term to approximate a more general regularization term expressed in the $p$-norm, $p\geq 1$. The regularization matrix can be updated both at each step and after some iterations have been performed, leading to two different approaches: the first one is based on the idea of the iteratively reweighted least squares method and can be obtained considering flexible Krylov subspaces; the second one is based on restarting the Arnoldi algorithm. Numerical examples are given in order to show the effectiveness of these new methods, and comparisons with some other already existing algorithms are made.

A General Framework for Sparsity-Based Denoising and Inversion

Estimating a reliable and stable solution to many problems in signal processing and imaging is based on sparse regularizations, where the true solution is known to have a sparse representation in a given basis. Using different approaches, a large variety of regularization terms have been proposed in literature. While it seems that all of them have so much in common, a general potential function which fits most of them is still missing. In this paper, in order to propose an efficient reconstruction method based on a variational approach and involving a general regularization term (including most of the known potential functions, convex and nonconvex), we deal with i) the definition of such a general potential function, ii) the properties of the associated “proximity operator” (such as the existence of a discontinuity), and iii) the design of an approximate solution of the general “proximity operator” in a simple closed form. We also demonstrate that a special case of the resulting “proximity operator” is a set of shrinkage functions which continuously interpolate between the soft-thresholding and hard-thresholding. Computational experiments show that the proposed general regularization term performs better than <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ℓ</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> -penalties for sparse approximation problems. Some numerical experiments are included to illustrate the effectiveness of the presented new potential function.

An adaptive regularization parameter choice strategy for multispectral bioluminescence tomography

Bioluminescence tomography (BLT) provides an effective tool for monitoring physiological and pathological activities in vivo. However, the measured data in bioluminescence imaging are corrupted by noise. Therefore, regularization methods are commonly used to find a regularized solution. Nevertheless, for the quality of the reconstructed bioluminescent source obtained by regularization methods, the choice of the regularization parameters is crucial. To date, the selection of regularization parameters remains challenging. With regards to the above problems, the authors proposed a BLT reconstruction algorithm with an adaptive parameter choice rule. The proposed reconstruction algorithm uses a diffusion equation for modeling the bioluminescent photon transport. The diffusion equation is solved with a finite element method. Computed tomography (CT) images provide anatomical information regarding the geometry of the small animal and its internal organs. To reduce the ill-posedness of BLT, spectral information and the optimal permissible source region are employed. Then, the relationship between the unknown source distribution and multiview and multispectral boundary measurements is established based on the finite element method and the optimal permissible source region. Since the measured data are noisy, the BLT reconstruction is formulated as l(2) data fidelity and a general regularization term. When choosing the regularization parameters for BLT, an efficient model function approach is proposed, which does not require knowledge of the noise level. This approach only requests the computation of the residual and regularized solution norm. With this knowledge, we construct the model function to approximate the objective function, and the regularization parameter is updated iteratively. First, the micro-CT based mouse phantom was used for simulation verification. Simulation experiments were used to illustrate why multispectral data were used rather than monochromatic data. Furthermore, the study conducted using an adaptive regularization parameter demonstrated our ability to accurately localize the bioluminescent source. With the adaptively estimated regularization parameter, the reconstructed center position of the source was (20.37, 31.05, 12.95) mm, and the distance to the real source was 0.63 mm. The results of the dual-source experiments further showed that our algorithm could localize the bioluminescent sources accurately. The authors then presented experimental evidence that the proposed algorithm exhibited its calculated efficiency over the heuristic method. The effectiveness of the new algorithm was also confirmed by comparing it with the L-curve method. Furthermore, various initial speculations regarding the regularization parameter were used to illustrate the convergence of our algorithm. Finally, in vivo mouse experiment further illustrates the effectiveness of the proposed algorithm. Utilizing numerical, physical phantom and in vivo examples, we demonstrated that the bioluminescent sources could be reconstructed accurately with automatic regularization parameters. The proposed algorithm exhibited superior performance than both the heuristic regularization parameter choice method and L-curve method based on the computational speed and localization error.

Generalized regularization term for non-parametric multimodal image registration

In the field of non-rigid medical image registration, many regularizers based on first- or second-order derivatives have been studied. In this paper, a new regularization term based on fractional order derivatives is proposed for the registration of multimodal (e.g., medical) images. It can be seen as a generalization of the diffusion and curvature smoothing terms, but with this approach it is possible to obtain better registration results in terms of both similarity of the images and smoothness of the transformation. This registration scheme is tested on two realistic medical imaging scenarios, comparing the obtained results with the optimally registered diffusion and curvature cases.

Stability of the Minimizers of Least Squares with a Non-Convex Regularization. Part II: Global Behavior

Many estimation problems amount to minimizing a piecewise Cm objective function, with m ≥ 2, composed of a quadratic data-fidelity term and a general regularization term. It is widely accepted that the minimizers obtained using non-convex and possibly non-smooth regularization terms are frequently good estimates. However, few facts are known on the ways to control properties of these minimizers. This work is dedicated to the stability of the minimizers of such objective functions with respect to variations of the data. It consists of two parts: first we consider all local minimizers, whereas in a second part we derive results on global minimizers. In this part we focus on data points such that every local minimizer is isolated and results from a Cm-1 local minimizer function, defined on some neighborhood. We demonstrate that all data points for which this fails form a set whose closure is negligible.

Stability of the Minimizers of Least Squares with a Non-Convex Regularization. Part I: Local Behavior

Many estimation problems amount to minimizing a piecewise Cm objective function, with m ≥ 2, composed of a quadratic data-fidelity term and a general regularization term. It is widely accepted that the minimizers obtained using non-convex and possibly non-smooth regularization terms are frequently good estimates. However, few facts are known on the ways to control properties of these minimizers. This work is dedicated to the stability of the minimizers of such objective functions with respect to variations of the data. It consists of two parts: first we consider all local minimizers, whereas in a second part we derive results on global minimizers. In this part we focus on data points such that every local minimizer is isolated and results from a Cm-1 local minimizer function, defined on some neighborhood. We demonstrate that all data points for which this fails form a set whose closure is negligible.