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Inverse Problems and Imaging

May 2015 , Volume 9 , Issue 2

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Recovery of the absorption coefficient in radiative transport from a single measurement
Sebastian Acosta
2015, 9(2): 289-300 doi: 10.3934/ipi.2015.9.289 +[Abstract](2338) +[PDF](377.1KB)
In this paper, we investigate the recovery of the absorption coefficient from boundary data assuming that the region of interest is illuminated at an initial time. We consider a sufficiently strong and isotropic, but otherwise unknown initial state of radiation. This work is part of an effort to reconstruct optical properties using unknown illumination embedded in the unknown medium.
    We break the problem into two steps. First, in a linear framework, we seek the simultaneous recovery of a forcing term of the form $\sigma(t,x,\theta) f(x)$ (with $\sigma$ known) and an isotropic initial condition $u_{0}(x)$ using the single measurement induced by these data. Based on exact boundary controllability, we derive a system of equations for the unknown terms $f$ and $u_{0}$. The system is shown to be Fredholm if $\sigma$ satisfies a certain positivity condition. We show that for generic term $\sigma$ and weakly absorbing media, this linear inverse problem is uniquely solvable with a stability estimate. In the second step, we use the stability results from the linear problem to address the nonlinearity in the recovery of a weak absorbing coefficient. We obtain a locally Lipschitz stability estimate.
A control approach to recover the wave speed (conformal factor) from one measurement
Sebastian Acosta
2015, 9(2): 301-315 doi: 10.3934/ipi.2015.9.301 +[Abstract](2629) +[PDF](396.2KB)
In this paper we consider the problem of recovering the conformal factor in a conformal class of Riemannian metrics from the boundary measurement of one wave field. More precisely, using boundary control operators, we derive an explicit equation satisfied by the contrast between two conformal factors (or wave speeds). This equation is Fredholm and generically invertible provided that the domain of interest is properly illuminated at an initial time. We also show locally Lipschitz stability estimates.
On the range of the attenuated magnetic ray transform for connections and Higgs fields
Gareth Ainsworth and Yernat M. Assylbekov
2015, 9(2): 317-335 doi: 10.3934/ipi.2015.9.317 +[Abstract](2609) +[PDF](438.1KB)
For a two-dimensional simple magnetic system, we study the attenuated magnetic ray transform $I_{A,\Phi}$, with attenuation given by a unitary connection $A$ and a skew-Hermitian Higgs field $\Phi$. We give a description for the range of $I_{A,\Phi}$ acting on $\mathbb{C}^n$-valued tensor fields.
Half-linear regularization for nonconvex image restoration models
Bartomeu Coll, Joan Duran and Catalina Sbert
2015, 9(2): 337-370 doi: 10.3934/ipi.2015.9.337 +[Abstract](3572) +[PDF](5201.3KB)
Image restoration is the problem of recovering an original image from an observation of it in order to extract the most meaningful information. In this paper, we study this problem from a variational point of view through the minimization of energies composed of a quadratic data-fidelity term and a nonsmooth nonconvex regularization term. In the discrete setting, existence of minimizer is proved for arbitrary linear operators. For this kind of problems, fully segmented solutions can be found by minimizing objective nonconvex functionals. We propose a dual formulation of the model by introducing an auxiliary variable with a double function. On one hand, it marks the edges and it ensures their preservation from smoothing. On the other hand, it makes the criterion half-linear in the sense that the dual energy depends linearly on the gradient of the image to be recovered. This leads to design an efficient optimization algorithm with wide applicability to several image restoration tasks such as denoising and deconvolution. Finally, we present experimental results and we compare them with TV-based image restoration algorithms.
Some geometric inverse problems for the linear wave equation
Anna Doubova and Enrique Fernández-Cara
2015, 9(2): 371-393 doi: 10.3934/ipi.2015.9.371 +[Abstract](3745) +[PDF](876.5KB)
In this paper we consider some geometric inverse problems for the linear wave equation. We prove uniqueness results, we present some reconstruction algorithms and we perform numerical experiments in dimensions one and two.
Dynamic linear inverse problems with moderate movements of the object: Ill-posedness and regularization
Bernadette N. Hahn
2015, 9(2): 395-413 doi: 10.3934/ipi.2015.9.395 +[Abstract](3838) +[PDF](1387.1KB)
One challenge for modern imaging methods is the investigation of objects which change during the data acquisition. This occurs in non-destructive testing as well as in medical applications, e.g. on account of patient or organ movements. Due to the object's deformations, the respective imaging modality is described by a dynamic inverse problem. In this paper, a classification scheme for linear dynamic problems depending on the object's motion is provided. Based on this scheme, we study the class in detail where the dynamic problem is still related to the operator in the static case, and where we call the deformations moderate. We proof important properties of the dynamic operator, derive a singular value decomposition and develop suitable regularization methods. The application of these methods to specific problems is illustrated at two examples including dynamic computerized tomography.
A new nonlocal variational setting for image processing
Yan Jin, Jürgen Jost and Guofang Wang
2015, 9(2): 415-430 doi: 10.3934/ipi.2015.9.415 +[Abstract](3945) +[PDF](456.3KB)
We introduce a new nonlocal variational scheme for image denoising. This scheme is motivated by, but different from the nonlocal means filter of Buades et al [9] and the nonlocal TV model proposed by Gilboa-Osher by using nonlocal operators. Our approach is based on general geometric considerations. Experiments show that the corresponding TV model yields denoising results that can compare favorably with those obtained by other methods.
Empirical average-case relation between undersampling and sparsity in X-ray CT
Jakob S. Jørgensen, Emil Y. Sidky, Per Christian Hansen and Xiaochuan Pan
2015, 9(2): 431-446 doi: 10.3934/ipi.2015.9.431 +[Abstract](3325) +[PDF](505.7KB)
In X-ray computed tomography (CT) it is generally acknowledged that reconstruction methods exploiting image sparsity allow reconstruction from a significantly reduced number of projections. The use of such reconstruction methods is inspired by recent progress in compressed sensing (CS). However, the CS framework provides neither guarantees of accurate CT reconstruction, nor any relation between sparsity and a sufficient number of measurements for recovery, i.e., perfect reconstruction from noise-free data. We consider reconstruction through 1-norm minimization, as proposed in CS, from data obtained using a standard CT fan-beam sampling pattern. In empirical simulation studies we establish quantitatively a relation between the image sparsity and the sufficient number of measurements for recovery within image classes motivated by tomographic applications. We show empirically that the specific relation depends on the image class and in many cases exhibits a sharp phase transition as seen in CS, i.e., same-sparsity images require the same number of projections for recovery. Finally we demonstrate that the relation holds independently of image size and is robust to small amounts of additive Gaussian white noise.
4D-CT reconstruction with unified spatial-temporal patch-based regularization
Daniil Kazantsev, William M. Thompson, William R. B. Lionheart, Geert Van Eyndhoven, Anders P. Kaestner, Katherine J. Dobson, Philip J. Withers and Peter D. Lee
2015, 9(2): 447-467 doi: 10.3934/ipi.2015.9.447 +[Abstract](4282) +[PDF](10881.1KB)
In this paper, we consider a limited data reconstruction problem for temporarily evolving computed tomography (CT), where some regions are static during the whole scan and some are dynamic (intensely or slowly changing). When motion occurs during a tomographic experiment one would like to minimize the number of projections used and reconstruct the image iteratively. To ensure stability of the iterative method spatial and temporal constraints are highly desirable. Here, we present a novel spatial-temporal regularization approach where all time frames are reconstructed collectively as a unified function of space and time. Our method has two main differences from the state-of-the-art spatial-temporal regularization methods. Firstly, all available temporal information is used to improve the spatial resolution of each time frame. Secondly, our method does not treat spatial and temporal penalty terms separately but rather unifies them in one regularization term. Additionally we optimize the temporal smoothing part of the method by considering the non-local patches which are most likely to belong to one intensity class. This modification significantly improves the signal-to-noise ratio of the reconstructed images and reduces computational time. The proposed approach is used in combination with golden ratio sampling of the projection data which allows one to find a better trade-off between temporal and spatial resolution scenarios.
Increasing stability for the inverse problem of the Schrödinger equation with the partial Cauchy data
Li Liang
2015, 9(2): 469-478 doi: 10.3934/ipi.2015.9.469 +[Abstract](3264) +[PDF](341.1KB)
To show increasing stability in the problem of recovering potential $c \in C^1(\Omega)$ in the Schrödinger equation with the given partial Cauchy data when energy frequency $k$ is growing, we will obtain some bounds for $c$ which can be viewed as an evidence of such phenomenon. The proof uses almost exponential solutions and methods of reflection.
An improved fast local level set method for three-dimensional inverse gravimetry
Wangtao Lu, Shingyu Leung and Jianliang Qian
2015, 9(2): 479-509 doi: 10.3934/ipi.2015.9.479 +[Abstract](3147) +[PDF](1516.6KB)
We propose an improved fast local level set method for the inverse problem of gravimetry by developing two novel algorithms: one is of linear complexity designed for computing the Frechet derivative of the nonlinear domain inverse problem, and the other is designed for carrying out numerical continuation rapidly so as to obtain fictitious full measurement data from partial measurement. Since the Laplacian kernel is symmetric and translationally invariant, we design certain affine transformations to speed up the computational process in evaluating the Frechet derivative; since it decays rapidly away from diagonal, we carry out low-rank matrix approximation to reduce storage requirements. These properties are eventually translated into an algorithm of linear complexity and linear storage requirement for computing the derivative. Since the single layer density function, used in representing the potential, is smooth and periodic on an artificial hypersurface enclosing the target domain, the spectral expansion is allowed to approximate this density function, which eventually leads to rapid algorithms for carrying out the numerical continuation in both 2-D and 3-D cases. 2-D and 3-D numerical examples illustrate that this improved level-set method is capable of computing high-resolution inversions and handling 3-D large-scale inverse gravimetry problems.
A study of the one dimensional total generalised variation regularisation problem
Konstantinos Papafitsoros and Kristian Bredies
2015, 9(2): 511-550 doi: 10.3934/ipi.2015.9.511 +[Abstract](3302) +[PDF](990.1KB)
In this paper we study the one dimensional second order total generalised variation regularisation (TGV) problem with $L^{2}$ data fitting term. We examine the properties of this model and we calculate exact solutions using simple piecewise affine functions as data terms. We investigate how these solutions behave with respect to the TGV parameters and we verify our results using numerical experiments.
A fast edge detection algorithm using binary labels
Yuying Shi, Ying Gu, Li-Lian Wang and Xue-Cheng Tai
2015, 9(2): 551-578 doi: 10.3934/ipi.2015.9.551 +[Abstract](3512) +[PDF](3388.8KB)
Edge detection (for both open and closed edges) from real images is a challenging problem. Developing fast algorithms with good accuracy and stability for noisy images is difficult yet and in demand. In this work, we present a variational model which is related to the well-known Mumford-Shah functional and design fast numerical methods to solve this new model through a binary labeling processing. A pre-smoothing step is implemented for the model, which enhances the accuracy of detection. Ample numerical experiments on grey-scale as well as color images are provided. The efficiency and accuracy of the model and the proposed minimization algorithms are demonstrated through comparing it with some existing methodologies.
Modulated luminescence tomography
Plamen Stefanov, Wenxiang Cong and Ge Wang
2015, 9(2): 579-589 doi: 10.3934/ipi.2015.9.579 +[Abstract](2777) +[PDF](374.2KB)
We propose and analyze a mathematical model of Modulated Luminescence Tomography. We show that when single X-rays or focused X-rays are used as an excitation, the problem is similar to the inversion of weighted X-ray transforms. In particular, we give an explicit inversion in the case of Dual Cone X-ray excitation.
Determining an obstacle by far-field data measured at a few spots
Qi Wang and Yanren Hou
2015, 9(2): 591-600 doi: 10.3934/ipi.2015.9.591 +[Abstract](2713) +[PDF](470.5KB)
We consider the inverse scattering problem of determining an acoustic sound-soft obstacle by using the far-field data. It is shown that if the shape of the obstacle is known in advance, then the far-field data measured at four different spots can uniquely determine the location and size of the obstacle. If the shape of the obstacle is unknown, we show that the location of the obstacle can be approximately determined by using the far-field data measured at four appropriately chosen spots.
Parallel matrix factorization for low-rank tensor completion
Yangyang Xu, Ruru Hao, Wotao Yin and Zhixun Su
2015, 9(2): 601-624 doi: 10.3934/ipi.2015.9.601 +[Abstract](7124) +[PDF](839.8KB)
Higher-order low-rank tensors naturally arise in many applications including hyperspectral data recovery, video inpainting, seismic data reconstruction, and so on. We propose a new model to recover a low-rank tensor by simultaneously performing low-rank matrix factorizations to the all-mode matricizations of the underlying tensor. An alternating minimization algorithm is applied to solve the model, along with two adaptive rank-adjusting strategies when the exact rank is not known.
    Phase transition plots reveal that our algorithm can recover a variety of synthetic low-rank tensors from significantly fewer samples than the compared methods, which include a matrix completion method applied to tensor recovery and two state-of-the-art tensor completion methods. Further tests on real-world data show similar advantages. Although our model is non-convex, our algorithm performs consistently throughout the tests and gives better results than the compared methods, some of which are based on convex models. In addition, subsequence convergence of our algorithm can be established in the sense that any limit point of the iterates satisfies the KKT condtions.

2021 Impact Factor: 1.483
5 Year Impact Factor: 1.462
2021 CiteScore: 2.6




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