# American Institute of Mathematical Sciences

ISSN:
1930-8337

eISSN:
1930-8345

All Issues

## Inverse Problems and Imaging

November 2012 , Volume 6 , Issue 4

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2012, 6(4): 565-598 doi: 10.3934/ipi.2012.6.565 +[Abstract](3469) +[PDF](2791.8KB)
Abstract:
Cone-Beam Computerized Tomography (CBCT) and Positron Emission Tomography (PET) are two complementary medical imaging modalities providing respectively anatomic and metabolic information on a patient. In the context of public health, one must address the problem of dose reduction of the potentially harmful quantities related to each exam protocol : X-rays for CBCT and radiotracer for PET. Two demonstrators based on a technological breakthrough (acquisition devices work in photon-counting mode) have been developed. It turns out that in this low-dose context, i.e. for low intensity signals acquired by photon counting devices, noise should not be approximated anymore by a Gaussian distribution, but is following a Poisson distribution. We investigate in this paper the two related tomographic reconstruction problems. We formulate separately the CBCT and the PET problems in two general frameworks that encompass the physics of the acquisition devices and the specific discretization of the object to reconstruct. We propose various fast numerical schemes based on proximal methods to compute the solution of each problem. In particular, we show that primal-dual approaches are well suited in the PET case when considering non differentiable regularizations such as Total Variation. Experiments on numerical simulations and real data are in favor of the proposed algorithms when compared with well-established methods.
2012, 6(4): 599-621 doi: 10.3934/ipi.2012.6.599 +[Abstract](2653) +[PDF](459.3KB)
Abstract:
Consider an infinite linear mass-spring system and a modification of it obtained by changing the first mass and spring of the system. We give results on the interplay of the spectra of such systems and on the reconstruction of the system from its spectrum and the one of the modified system. Furthermore, we provide necessary and sufficient conditions for two sequences to be the spectra of the mass-spring system and the perturbed one.
2012, 6(4): 623-644 doi: 10.3934/ipi.2012.6.623 +[Abstract](3171) +[PDF](1795.2KB)
Abstract:
Many problems in image processing can be posed as non-convex minimization problems. For certain classes of non-convex problems involving scalar-valued functions, it is possible to recast the problem in a convex form using a functional lifting'' technique. In this paper, we present a variational functional lifting technique that can be viewed as a generalization of previous works by Pock et. al and Ishikawa. We then generalize this technique to the case of minimization over vector-valued problems, and discuss a condition which allows us to determine when the solution to the convex problem corresponds to a global minimizer. This generalization allows functional lifting to be applied to a wider range of problems then previously considered. Finally, we present a numerical method for solving the convexified problems, and apply the technique to find global minimizers for optical flow image registration.
2012, 6(4): 645-661 doi: 10.3934/ipi.2012.6.645 +[Abstract](2997) +[PDF](561.3KB)
Abstract:
Spherical means are a widespread model in modern imaging modalities like photoacoustic tomography. Besides direct inversion methods for specific geometries, iterative methods are often used as reconstruction scheme such that each iteration asks for the efficient and accurate computation of spherical means. We consider a spectral discretization via trigonometric polynomials such that the computation can be done via nonequispaced fast Fourier transforms. Moreover, a recently developed sparse fast Fourier transform is used in the three dimensional case and gives optimal arithmetic complexity. All theoretical results are illustrated by numerical experiments.
2012, 6(4): 663-679 doi: 10.3934/ipi.2012.6.663 +[Abstract](3038) +[PDF](277.8KB)
Abstract:
We consider the inverse problem of determining both an unknown diffusion and an unknown absorption coefficient from knowledge of (partial) Cauchy data in an elliptic boundary value problem. For piecewise analytic coefficients, we prove a complete characterization of the reconstructible information. It is shown to consist of a combination of both coefficients together with the jumps in the leading order diffusion coefficient and its derivative.
2012, 6(4): 681-695 doi: 10.3934/ipi.2012.6.681 +[Abstract](3478) +[PDF](375.2KB)
Abstract:
In this paper we justify the Factorization Method for a coupled acoustic-elastic medium. Under natural assumptions on the data we prove an explicit form of the characteristic function of the scattering medium $D$ where only the spectral data of the far field operator enter. This information is provided by the knowledge of the far field patterns for all incident plane waves. In the last section we investigate the corresponding interior transmission eigenvalue problem and prove that the eigenvalues form a discrete set.
2012, 6(4): 697-708 doi: 10.3934/ipi.2012.6.697 +[Abstract](4593) +[PDF](517.5KB)
Abstract:
A feature of the human visual system (HVS) is color constancy, namely, the ability to determine the color under varying illumination conditions. Retinex theory, formulated by Edwin H. Land, aimed to simulate and explain how the HVS perceives color. In this paper, we establish a total variation (TV) and nonlocal TV regularized model of Retinex theory that can be solved by a fast computational approach based on Bregman iteration. We demonstrate the performance of our method by numerical results.
2012, 6(4): 709-747 doi: 10.3934/ipi.2012.6.709 +[Abstract](3629) +[PDF](1845.3KB)
Abstract:
We consider the problem of locating a penetrable obstacle in an acoustic waveguide from measurements of pressure waves due to point sources inside the waveguide. More precisely, we assume that we are given the scattered field and its normal derivative for any source point and receiver placed on a pair of surfaces known as the source and the measurement surfaces, respectively. A novel feature of this work is that the obstacle is allowed to touch the boundary of the pipe.
We first analyze the associated interior transmission problem. Then, we adapt and analyze the Reciprocity Gap Method (RGM) and the Linear Sampling Method (LSM) to deal with the inverse problem. We also study the relationship between these two methods and provide numerical results.
2012, 6(4): 749-773 doi: 10.3934/ipi.2012.6.749 +[Abstract](3668) +[PDF](617.3KB)
Abstract:
In this paper, we investigate the problem of reconstructing sound-soft acoustic obstacles using multifrequency far field measurements corresponding to one direction of incidence. The idea is to obtain a rough estimate of the obstacle's shape at the lowest frequency using the least-squares approach, then refine it using a recursive linearization algorithm at higher frequencies. Using this approach, we show that an accurate reconstruction can be obtained without requiring a good initial guess. The analysis is divided into three steps. Firstly, we give a quantitative estimate of the domain in which the least-squares objective functional, at the lowest frequency, has only one extreme (minimum) point. This result enables us to obtain a rough approximation of the obstacle at the lowest frequency from initial guesses in this domain using convergent gradient-based iterative procedures. Secondly, we describe the recursive linearization algorithm and analyze its convergence for noisy data. We qualitatively explain the relationship between the noise level and the resolution limit of the reconstruction. Thirdly, we justify a conditional asymptotic Hölder stability estimate of the illuminated part of the obstacle at high frequencies. The performance of the algorithm is illustrated with numerical examples.

2020 Impact Factor: 1.639
5 Year Impact Factor: 1.720
2020 CiteScore: 2.6