ISSN:

1930-8337

eISSN:

1930-8345

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## Inverse Problems & Imaging

May 2014 , Volume 8 , Issue 2

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2014, 8(2): 339-359
doi: 10.3934/ipi.2014.8.339

*+*[Abstract](1854)*+*[PDF](936.8KB)**Abstract:**

Exterior inverse problem for the circular means transform (CMT) arises in the intravascular photoacoustic imaging (IVPA), in the intravascular ultrasound imaging (IVUS), as well as in radar and sonar. The reduction of the IPVA to the CMT is quite straightforward. As shown in the paper, in IVUS the circular means can be recovered from measurements by solving a certain Volterra integral equation. Thus, a tomography reconstruction in both modalities requires solving the exterior problem for the CMT.

Numerical solution of this problem usually is not attempted due to the presence of "invisible" wavefronts, which results in severe instability of the reconstruction. The novel inversion algorithm proposed in this paper yields a stable partial reconstruction: it reproduces the "visible" part of the image and blurs the "invisible" part. If the image contains little or no invisible wavefronts (as frequently happens in the IVPA and IVUS) the reconstruction is quantitatively accurate. The presented numerical simulations demonstrate the feasibility of tomography-like reconstruction in these modalities.

2014, 8(2): 361-387
doi: 10.3934/ipi.2014.8.361

*+*[Abstract](2723)*+*[PDF](1372.8KB)**Abstract:**

In this paper a new mathematically-founded method for the optimal partitioning of domains, with applications to the classification of greyscale and color images, is proposed. Since optimal partition problems are in general ill-posed, some regularization strategy is required. Here we regularize by a non-standard approximation of the total interface length, which does not involve the gradient of approximate characteristic functions, in contrast to the classical Modica-Mortola approximation. Instead, it involves a system of uncoupled linear partial differential equations and nevertheless shows $\Gamma$-convergence properties in appropriate function spaces. This approach leads to an alternating algorithm that ensures a decrease of the objective function at each iteration, and which always provides a partition, even during the iterations. The efficiency of this algorithm is illustrated by various numerical examples. Among them we consider binary and multilabel minimal partition problems including supervised or automatic image classification, inpainting, texture pattern identification and deblurring.

2014, 8(2): 389-408
doi: 10.3934/ipi.2014.8.389

*+*[Abstract](2402)*+*[PDF](1804.3KB)**Abstract:**

In this paper we propose an algorithm for the detection of edges in images that is based on topological asymptotic analysis. Motivated from the Mumford--Shah functional, we consider a variational functional that penalizes oscillations outside some approximate edge set, which we represent as the union of a finite number of thin strips, the width of which is an order of magnitude smaller than their length. In order to find a near optimal placement of these strips, we compute an asymptotic expansion of the functional with respect to the strip size. This expansion is then employed for defining a (topological) gradient descent like minimization method. As opposed to a recently proposed method by some of the authors, which uses coverings with balls, the usage of strips includes some directional information into the method, which can be used for obtaining finer edges and can also result in a reduction of computation times.

2014, 8(2): 409-420
doi: 10.3934/ipi.2014.8.409

*+*[Abstract](2083)*+*[PDF](384.3KB)**Abstract:**

Conjugate Gradient is widely used as a regularizing technique for solving linear systems with ill-conditioned coefficient matrix and right-hand side vector perturbed by noise. It enjoys a good convergence rate and computes quickly an iterate, say $x_{k_{opt}}$, which minimizes the error with respect to the exact solution. This behavior can be a disadvantage in the regularization context, because also the high-frequency components of the noise enter quickly the computed solution, leading to a difficult detection of $k_{opt}$ and to a sharp increase of the error after the $k_{opt}$th iteration. In this paper we propose an inner-outer algorithm based on a sequence of restarted Conjugate Gradients, with the aim of overcoming this drawback. A numerical experimentation validates the effectiveness of the proposed algorithm.

2014, 8(2): 421-457
doi: 10.3934/ipi.2014.8.421

*+*[Abstract](2442)*+*[PDF](2037.6KB)**Abstract:**

Optical Deflectometric Tomography (ODT) provides an accurate characterization of transparent materials whose complex surfaces present a real challenge for manufacture and control. In ODT, the refractive index map (RIM) of a transparent object is reconstructed by measuring light deflection under multiple orientations. We show that this imaging modality can be made

*compressive*, i.e., a correct RIM reconstruction is achievable with far less observations than required by traditional minimum energy (ME) or Filtered Back Projection (FBP) methods. Assuming a cartoon-shape RIM model, this reconstruction is driven by minimizing the map Total-Variation under a fidelity constraint with the available observations. Moreover, two other realistic assumptions are added to improve the stability of our approach: the map positivity and a frontier condition. Numerically, our method relies on an accurate ODT sensing model and on a primal-dual minimization scheme, including easily the sensing operator and the proposed RIM constraints. We conclude this paper by demonstrating the power of our method on synthetic and experimental data under various compressive scenarios. In particular, the potential compressiveness of the stabilized ODT problem is demonstrated by observing a typical gain of 24 dB compared to ME and of 30 dB compared to FBP at only 5% of 360 incident light angles for moderately noisy sensing.

2014, 8(2): 459-473
doi: 10.3934/ipi.2014.8.459

*+*[Abstract](2387)*+*[PDF](1538.9KB)**Abstract:**

In this paper we combine a few techniques to label blood vessels in the matched filter (MF) response image by using a finite element based binary level set method. An operator-splitting method is applied to numerically solve the Euler-Lagrange equation from minimizing an energy functional. Unlike the traditional MF methods, where a threshold is difficult to be selected, our method can automatically get more precise blood vessel segmentation using an enhanced edge information. In order to demonstrate the good performance, we compare our method with a few other methods when they are applied to a publicly available standard database of coloured images (with manual segmentations available too).

2014, 8(2): 475-489
doi: 10.3934/ipi.2014.8.475

*+*[Abstract](2474)*+*[PDF](444.1KB)**Abstract:**

An analogue of Rellich's theorem is proved for discrete Laplacians on square lattices, and applied to show unique continuation properties on certain domains as well as non-existence of embedded eigenvalues for discrete Schrödinger operators.

2014, 8(2): 491-505
doi: 10.3934/ipi.2014.8.491

*+*[Abstract](2077)*+*[PDF](1349.6KB)**Abstract:**

This paper presents a new computer-aided method for detection of brain metastases at early-stage (diameter less than $6$mm) on MR images. The proposed detection method has a high level of sensitivity with a relatively low number of false-positives. The strong detection capability of the method is possible due to a size filtering function that sorts out metastases based on the geometry and size. In experiments, we used whole-brain MR data acquired with a contrast-enhanced black-blood type MR imaging technique, which enables distinction of brain metastases from blood vessels. The proposed method performed highly in analysis of the results of experimental MR data and numerical simulation. Because the proposed method has unique features, it could be used in combination with a complementary pre-existing technique.

2014, 8(2): 507-535
doi: 10.3934/ipi.2014.8.507

*+*[Abstract](2728)*+*[PDF](2618.2KB)**Abstract:**

Image restoration plays an important role in image processing, and numerous approaches have been proposed to tackle this problem. This paper presents a modified model for image restoration, that is based on a combination of Total Variation and Dictionary approaches. Since the well-known TV regularization is non-differentiable, the proposed method utilizes its dual formulation instead of its approximation in order to exactly preserve its properties. The data-fidelity term combines the one commonly used in image restoration and a wavelet thresholding based term. Then, the resulting optimization problem is solved via a first-order primal-dual algorithm. Numerical experiments demonstrate the good performance of the proposed model. In a last variant, we replace the classical TV by the nonlocal TV regularization, which results in a much higher quality of restoration.

2014, 8(2): 537-560
doi: 10.3934/ipi.2014.8.537

*+*[Abstract](2148)*+*[PDF](580.4KB)**Abstract:**

We consider the alternating method of Kozlov and Maz'ya for solving the Cauchy problem for elliptic boundary-value problems. Considering the case of the Laplacian, we show that this method can be recast as a form of Landweber iteration. In addition to conceptual advantages, this observation leads to some practical improvements. We show how to accelerate Kozlov-Maz'ya iteration using the conjugate gradient algorithm, and we show how to modify the method to obtain a more practical stopping criterion.

2014, 8(2): 561-586
doi: 10.3934/ipi.2014.8.561

*+*[Abstract](3977)*+*[PDF](2107.2KB)**Abstract:**

We study flexible and proper smoothness priors for Bayesian statistical inverse problems by using Whittle-Matérn Gaussian random fields. We review earlier results on finite-difference approximations of certain Whittle-Matérn random field in $\mathbb{R}^2$. Then we derive finite-element method approximations and show that the discrete approximations can be expressed as solutions of sparse stochastic matrix equations. Such equations are known to be computationally efficient and useful in inverse problems with a large number of unknowns.

The presented construction of Whittle-Matérn correlation functions allows both isotropic or anisotropic priors with adjustable parameters in correlation length and variance. These parameters can be used, for example, to model spatially varying structural information of unknowns.

As numerical examples, we apply the developed priors to two-dimensional electrical impedance tomography problems.

2014, 8(2): 587-610
doi: 10.3934/ipi.2014.8.587

*+*[Abstract](2090)*+*[PDF](795.9KB)**Abstract:**

Differential optical absorption spectroscopy (DOAS) is a powerful tool for detecting and quantifying trace gases in atmospheric chemistry [22]. DOAS spectra consist of a linear combination of complex multi-peak multi-scale structures. Most DOAS analysis routines in use today are based on least squares techniques, for example, the approach developed in the 1970s [18,19,20,21] uses polynomial fits to remove a slowly varying background (broad spectral structures in the data), and known reference spectra to retrieve the identity and concentrations of reference gases [23]. An open problem [22] is that fitting residuals for complex atmospheric mixtures often still exhibit structure that indicates the presence of unknown absorbers.

In this work, we develop a novel three step semi-blind source separation method. The first step uses a multi-resolution analysis called empirical mode decomposition (EMD) to remove the slow-varying and fast-varying components in the DOAS spectral data matrix ${\bf X}$. This has the advantage of avoiding user bias in fitting the slow varying signal. The second step decomposes the preprocessed data $\hat{{\bf X}}$ in the first step into a linear combination of the reference spectra plus a remainder, or $\hat{{\bf X}} = {\bf A}\,{\bf S} + {\bf R}$, where columns of matrix ${\bf A}$ are known reference spectra, and the matrix ${\bf S}$ contains the unknown non-negative coefficients that are proportional to concentration. The second step is realized by a convex minimization problem ${\bf S} = \mathrm{arg} \min \mathrm{norm}\,(\hat{{\bf X}} - {\bf A}\,{\bf S})$, where the norm is a hybrid $\ell_1/\ell_2$ norm (Huber estimator) that helps to maintain the non-negativity of ${\bf S}$. Non-negative coefficients are necessary in order for the derived proportional concentrations to make physical sense. The third step performs a blind independent component analysis of the remainder matrix ${\bf R}$ to extract remnant gas components. This step demonstrates the ability of the new fitting method to extract orthogonal components without the use of reference spectra.

We illustrate utility of the proposed method in processing a set of DOAS experimental data by a satisfactory blind extraction of an a-priori unknown trace gas (ozone) from the remainder matrix. Numerical results also show that the method can identify trace gases from the residuals.

2019 Impact Factor: 1.373

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