# American Institute of Mathematical Sciences

ISSN:
1930-8337

eISSN:
1930-8345

All Issues

## Inverse Problems & Imaging

August 2007 , Volume 1 , Issue 3

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2007, 1(3): 437-442 doi: 10.3934/ipi.2007.1.437 +[Abstract](1299) +[PDF](123.9KB)
Abstract:
We prove uniqueness and stability for an inverse boundary problem associated to an anisotropic elliptic equation arising in the modelling of prestressed elastic membranes.
2007, 1(3): 443-456 doi: 10.3934/ipi.2007.1.443 +[Abstract](1928) +[PDF](195.6KB)
Abstract:
The interior transmission problem plays a basic role in the study of inverse scattering problems for inhomogeneous medium. In this paper we study the interior transmission problem for the Maxwell equations in the electromagnetic scattering problem for an anisotropic inhomogeneous object. We use a variational approach which extends the method developed in [15] to the case when the index of refraction is less than one as well as for partially coated scatterers. In addition, we also describe the structure of the transmission eigenvalues.
2007, 1(3): 457-479 doi: 10.3934/ipi.2007.1.457 +[Abstract](1482) +[PDF](300.5KB)
Abstract:
This paper presents a new method for the exponential Radon transform inversion based on the harmonic analysis of the Euclidean motion group of the plane. The proposed inversion method is based on the observation that the exponential Radon transform can be modified to obtain a new transform, defined as the modified exponential Radon transform, that can be expressed as a convolution on the Euclidean motion group. The convolution representation of the modified exponential Radon transform is block diagonalized in the Euclidean motion group Fourier domain. Further analysis of the block diagonal representation provides a class of relationships between the spherical harmonic decompositions of the Fourier transforms of the function and its exponential Radon transform. These relationships and the block diagonalization lead to three new reconstruction algorithms. The proposed algorithms are implemented using the fast implementation of the Euclidean motion group Fourier transform and their performances are demonstrated in numerical simulations. Our study shows that convolution representation and harmonic analysis over groups motivates novel solutions for the inversion of the exponential Radon transform.
2007, 1(3): 481-506 doi: 10.3934/ipi.2007.1.481 +[Abstract](1290) +[PDF](386.9KB)
Abstract:
We consider a thin elastic plate subjected to a couple field applied at its boundary and we study the inverse problem consisting in determining an unknown cavity inside the plate by measuring the transversal displacement and its normal derivative at the boundary of the plate. We prove uniqueness with two measurements.
2007, 1(3): 507-523 doi: 10.3934/ipi.2007.1.507 +[Abstract](1785) +[PDF](2310.5KB)
Abstract:
In part I we introduced modified Landweber--Kaczmarz methods and established a convergence analysis. In the present work we investigate three applications: an inverse problem related to thermoacoustic tomography, a nonlinear inverse problem for semiconductor equations, and a nonlinear problem in Schlieren tomography. Each application is considered in the framework established in the previous part. The novel algorithms show robustness, stability, computational efficiency and high accuracy.
2007, 1(3): 525-535 doi: 10.3934/ipi.2007.1.525 +[Abstract](1480) +[PDF](156.2KB)
Abstract:
The inverse scattering problem for multidimensional Schrödinger operator is studied. More exactly we prove a new formula for the first nonlinear term to estimate more accurately this term. This estimate allows to conclude that all singularities and jumps of the unknown potential can be recovered from the Born approximation. Especially, we show for the potentials in $L^p$ for certain values of $p$ that the approximation agrees with the true potential up to the continuous function.% Text of abstract
2007, 1(3): 537-556 doi: 10.3934/ipi.2007.1.537 +[Abstract](1358) +[PDF](238.7KB)
Abstract:
We consider the stable dependence of solutions to wave equations on metrics in $C^{1,1}$ class. The main result states that solutions depend uniformly continuously on the metric, when the Cauchy data is given in a range of Sobolev spaces. The proof is constructive and uses the wave packet approach to hyperbolic equations.
2007, 1(3): 557-575 doi: 10.3934/ipi.2007.1.557 +[Abstract](1727) +[PDF](516.9KB)
Abstract:
Closely inspired by the total variation (TV) model of Rudin, Osher and Fatemi [Physica D, 60:259-268,1992], we propose the quantized or quantum TV model (either with a preassigned quanta set $Q$ or without), and study the associated mathematical properties and computational algorithms. An algorithm based on stochastic or Markovian gradient descent is proposed to handle the discrete programming nature of the quantum TV model, which further leads to a two-step iterative algorithm for the computationally more challenging free quantum TV model. We also demonstrate several major applications of the proposed models and algorithms in bar code scanning, image quantization, and image segmentation.
2007, 1(3): 577-592 doi: 10.3934/ipi.2007.1.577 +[Abstract](1648) +[PDF](215.8KB)
Abstract:
We derive an optimal transmit waveform for filtered backprojection-based synthetic-aperture imaging. The waveform is optimal in terms of minimising the mean square error (MSE) in the resulting image. Our optimization is performed in two steps: First, we consider the minimum-mean-square-error (MMSE) for an arbitrary but fixed waveform, and derive the corresponding filter for the filtered backprojection reconstruction. Second, the MMSE is further reduced by finding an optimal transmit waveform. The transmit waveform is derived for stochastic models of the scattering objects of interest (targets), other scattering objects (clutter), and additive noise. We express the waveform in terms of spatial spectra for the random fields associated with target and clutter, and the spectrum for the noise process. This approach results in a constraint that involves only the amplitude of the Fourier transform of the transmit waveform. Therefore, considerable flexibility is left for incorporating additional requirements, such as minimal variation of transmit amplitude and phase-coding.

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