Probabilistic regularization in inverse optical imaging (original) (raw)
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A Semiblind Regularization Algorithm for Inverse Problems with Application to Image Deblurring
SIAM Journal on Scientific Computing
In many inverse problems the operator to be inverted depends on parameters which are not known precisely. In this article we propose a functional that involves as variables both the solution of the problem and the parameters on which the operator depends. We first prove that the functional, even if it is non-convex, admits a global minimum and that its minimization naturally leads to a regularization method. Then, using the popular Alternating Direction Multiplier Method (ADMM), we describe an algorithm to identify a stationary point of the functional. The introduction of the ADMM algorithm lets us easily introduce some constraints on the reconstructions like non-negativity and flux conservation. Since the functional is non-convex a proof of convergence of the method is given. Numerical examples prove the validity of the proposed approach.
Regularization theory in image restoration-the stabilizing functional approach
IEEE Transactions on Acoustics, Speech, and Signal Processing
This paper presents several aspects of the application of regularization theory in image restoration. This is accomplished by extending the applicability of the stabilizing functional approach to 2-D ill-posed inverse problems. Image restoration is formulated as the constrained minimization of a stabilizing functional. The choice of a particular quadratic functional to be minimized is related to the a priori knowledge regarding the original object through a formulation of image restoration as a maximum a posteriori estimation problem. This formulation is based on image representation by certain stochastic partial differential equation image models. The analytical study and computational treatment of the resulting optimization problem are subsequently presented. As a result, a variety of regularizing filters and iterative regularizing algorithms are proposed. A relationship between the regularized solutions proposed and optimal Wiener estimation is also identified. The filters and algorithms proposed are evaluated through several experimental results.
DEVELOPMENT OF DETERMINISTIC AND STOCHASTIC ALGORITHMS FOR INVERSE PROBLEMS OF OPTICAL TOMOGRAPHY
A semi log plot of number of nodes (unknown parameters) versus reconstruction time per frame using methods (given in the legend) (Tab. (3.1)). Polynomial fits corresponding to each method (Appendix AII) are also plotted in this figure. 3.7 (a) An example of reconstructed dynamic data set using SVD method. Corresponding time points are shown on top of the each figure. (b) Recovered a contrast (maximum value in the target region) versus time. 3.8 Reconstructed a distributions using (a) nonlinear (b) linear iterative (c) SVD (d) linear efficient, and (e) SVD efficient methods with 1% noise in the data for two inhomogeneities. Flowchart: Flowchart of implementation steps. 4.1 The cross-sectional plots through the center of the inclusion of the images List of Figures xii shown in Fig. (4.2) 4.2 The reference and recovered gray-level a images (a) Reference, (b), (c), and (d) are respectively obtained from BS, DBS-B and DBS filters 67 4.3 Pseudo-time evolution of the parameters from BS, DBS-B and DBS filters for an absorbing object with one inclusion. (a) h, (b) r and (c) of the recovered inclusion. 4.4a (a) The recovered gray-level a images for the two inhomogeneity case: (a) Reference, (b) from the BS and (c) from the DBS filter. 68 4.4b Pseudo-time evolution of the parameters, in the two inclusions, estimated through the BS and DBS filters: (a) h, (b) r and (c) 69 4.5 Time history of RMSE of parameters: (a) h, (b) r and (c) 70 4.6 Time history of sample variance of parameters: (a) h, (b) r and (c) 71 4.7 Reconstruction from experimental data: the evolution of parameters (a) h, (b) r and (c) estimated through BS, DBS and DBS-B filters 72 4.8 The reconstructed gray-level images from experimental data, using (a) BS, (b) DBS and (c) DBS-B filters. 73 5.1 Reconstruction of absorption coefficient for a phantom with two inhomogeneities using (b) PD-EnKF and (c) Gauss-Newton method where (a) is the reference. 86 5.2 Parameters estimated through PD-EnKF and GN method for a phantom with two inhomogeneities (a) h, and (b) 0, 0 (sgn(())) abs c real c and (c) 1, 1 (sgn(())) abs c real c 86-87 5.3 Reconstruction of absorption coefficient for an annular ring shaped phantom using (b) PD-EnKF and (c) GN method where (a) is the reference. 87 5.4 Recovery of absorption coefficient for a dumbbell shaped phantom using (b) PD-EnKF and (c) GN method where (a) is the reference. 88 5.5 The reconstructed gray level image corresponding to the experimental data using (a) PD-EnKF and (b) GN method 89 5.6 The parameters estimated through PD-EnKF from the experimental data (a) h and (b) 0,abs c 90 5.7 Reference figure for Figs. 5.8 and 5.9. 91 5.8 (a) Reconstruction through EnKF; (b) reconstruction through GN algorithm; (c) CS through the insonified region along y-axis; (d) convergence of parameters (EnKF) (e) L 2 error norm, for =4 time samples when the GN algorithm is used with 1% noisy data. 91-92 5.9 (a) Reconstructed figure through EnKF, (b) Reconstructed figure through brute-force (c) CS through the insonification region along y-axis, (d) Convergence of parameters (EnKF) (e) L2 error norm, for = 4 time samples with CDE for 3% noise. 92-93 5.10 (a) Reference, (b) Reconstructed figure through EnKF, (c) Reconstructed figure through brute-force, (d) CS through the insonification region along yaxis, (e) Convergence of parameters (EnKF), (f) L2 error norm, for = 4 time samples with CDE for 1% noise. 93-94 6.1 Object and meshing used for 3D forward problem, showing the arrangement brute force EVP when solving a forward 2-D problem. 106
An overview of inverse problem regularization using sparsity
2009 16th IEEE International Conference on Image Processing (ICIP), 2009
Sparsity constraints are now very popular to regularized inverse problems. We review several approaches which have been proposed in the last ten years to solve inverse problems such as inpainting, deconvolution or blind source separation. We will focus especially on iterative thresholding methods.
Regularization Techniques for Inverse Problem in DOT Applications
Journal of Physics: Conference Series
Diffuse optical tomography (DOT) is an emerging diagnostic technique which uses near-infra-red light to investigate the optical coefficients distribution in biological tissues. The surface of the tissue is illuminated by light sources, then the outgoing light is measured by detectors placed at various locations on the surface itself. In order to reconstruct the optical coefficients, a mathematical model of light propagation is employed: such model leads to the minimization of the discrepancy between the detected data and the corresponding theoretical field. Due to severe ill-conditioning, regularization techniques are required: common procedures consider mainly 1-norm (LASSO) and 2-norm (Tikhonov) regularization. In the present work we investigate two original approaches in this context: the elastic-net regularization, previously used in machine learning problems, and the Bregman procedure. Numerical experiments are performed on synthetic 2D geometries and data, to evaluate the performance of these approaches. The results show that these techniques are indeed suitable choices for practical applications, where DOT is used as a cheap, first-level and almost real-time screening technique for breast cancer detection.
ICASSP, IEEE International Conference on Acoustics, Speech and Signal Processing - Proceedings, 2010
Ill-posed linear inverse problems (ILIP), such as restoration and reconstruction, are a core topic of signal/image processing. A standard approach to deal with ILIP uses a constrained optimization problem, where a regularization function is minimized under the constraint that the solution explains the observations sufficiently well. The regularizer and constraint are usually convex; however, several particular features of these problems (huge dimensionality, nonsmoothness) preclude the use of off-the-shelf optimization tools and have stimulated much research. In this paper, we propose a new efficient algorithm to handle one class of constrained problems (known as basis pursuit denoising) tailored to image recovery applications. The proposed algorithm, which belongs to the category of augmented Lagrangian methods, can be used to deal with a variety of imaging ILIP, including deconvolution and reconstruction from compressive observations (such as MRI). Experiments testify for the effectiveness of the proposed method.
A new approach for regularization of inverse problems in images processing
Optical flow motion estimation from two images is limited by the aperture problem. A method to deal with this problem is to use regularization techniques. Usually, one adds a regularization term with appriopriate weighting parameter to the optical flow cost funtion. Here, we suggest a new approach to regularization for optical flow motion estimation. In this approach, all the regularization informations are used in the definition of an appropriate norm for the cost function via a trust function to be defined, one don't ever need weighting parameter. A simple derivation of such a trust function from images is proposed and a comparison with usual approaches is presented. These results show the superiority of such approach over usual ones. RÉSUMÉ. L'estimation du mouvement par flot optique est sujet au problème d'ouverture. Pour cela, on a recours aux techniques de régularisation. De façon usuelle, Cela se caractérise par l'ajout d'un terme de régularisation pondéré à la fonction coût du flot optique. Dans ce papier, nous proposons une nouvelle approche pour la régularisation des méthodes de flot optique. Toute l'information de régularisation est utilisée pour définir une norme appropriée à la fonction coût par l'intermédiaire d'une fonction de confiance qui permet de se passer du paramètre de poids. Nous proposons une dérivation simplifiée de la fonction de confiance à partir des images et présentons les résultats comparés avec les méthodes usuelles. Ces résultats montrent la supériorité de la nouvelle approche