# American Institute of Mathematical Sciences

ISSN:
1930-8337

eISSN:
1930-8345

All Issues

## Inverse Problems & Imaging

2007 , Volume 1 , Issue 2

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2007, 1(2): 229-245 doi: 10.3934/ipi.2007.1.229 +[Abstract](622) +[PDF](1036.1KB)
Abstract:
We consider the inverse problem to recover a part $\Gamma_c$ of the boundary of a simply connected planar domain $D$ from a pair of Cauchy data of a harmonic function $u$ in $D$ on the remaining part $\partial D\setminus \Gamma_c$ when $u$ satisfies a homogeneous impedance boundary condition on $\Gamma_c$. Our approach extends a method that has been suggested by Kress and Rundell [17] for recovering the interior boundary curve of a doubly connected planar domain from a pair of Cauchy data on the exterior boundary curve and is based on a system of nonlinear integral equations. As a byproduct, these integral equations can also be used for the problem to extend incomplete Cauchy data and to solve the inverse problem to recover an impedance profile on a known boundary curve. We present the mathematical foundation of the method and illustrate its feasibility by numerical examples.
2007, 1(2): 247-263 doi: 10.3934/ipi.2007.1.247 +[Abstract](443) +[PDF](572.1KB)
Abstract:
Cellular metabolism can be modelled as a multi-compartment dynamical system, the compartments representing the circulatory system consisting of blood and interstitial fluid, and different subcellular structures. The inverse problem in cellular metabolism is to obtain information about the state of the system based on few measured concentrations of metabolites or intermediates either in the blood or in the tissue. In this article, we first discuss a new three compartment metabolic model for human skeletal muscle metabolism and the corresponding inverse problem of determining the metabolic reaction and transport rates given blood concentration data under sustained exercise. We introduce the concept of a metabolic stationary state, describe a Bayesian methodology to analyze it and apply it to study the stationary state of human leg skeletal muscles under exercise. Our analysis demonstrates that the system is fairly well identified if the concentrations of certain species in the blood are known, and that the lack of oxygen concentration data can be replaced by prescribing either the ATP hydrolysis level or the glycogen depletion rate.
2007, 1(2): 265-287 doi: 10.3934/ipi.2007.1.265 +[Abstract](399) +[PDF](1403.5KB)
Abstract:
We present a method for the automatic estimation of the minimum set of colors needed to describe an image. We call this minimal set ''color palette''. The proposed method combines the well-known K-Means clustering technique with a thorough analysis of the color information of the image. The initial set of cluster seeds used in K-Means is automatically inferred from this analysis. Color information is analyzed by studying the 1D histograms associated to the hue, saturation and intensity components of the image colors. In order to achieve a proper parsing of these 1D histograms a new histogram segmentation technique is proposed. The experimental results seem to endorse the capacity of the method to obtain the most significant colors in the image, even if they belong to small details in the scene. The obtained palette can be combined with a dictionary of color names in order to provide a qualitative image description.
2007, 1(2): 289-298 doi: 10.3934/ipi.2007.1.289 +[Abstract](614) +[PDF](172.5KB)
Abstract:
In this article we develop and analyze novel iterative regularization techniques for the solution of systems of nonlinear ill-posed operator equations. The basic idea consists in considering separately each equation of this system and incorporating a loping strategy. The first technique is a Kaczmarz-type method, equipped with a novel stopping criteria. The second method is obtained using an embedding strategy, and again a Kaczmarz-type approach. We prove well-posedness, stability and convergence of both methods.
2007, 1(2): 299-317 doi: 10.3934/ipi.2007.1.299 +[Abstract](645) +[PDF](459.9KB)
Abstract:
In electrical impedance tomography, one tries to recover the spatial conductivity distribution inside a body from boundary measurements of current and voltage. In many practically important situations, the object has known background conductivity but it is contaminated by inhomogeneities. The factorization method of Andreas Kirsch provides a tool for locating such inclusions. In earlier work, it has been shown, both theoretically and numerically, that the inhomogeneities can be characterized by the factorization technique if the input current can be controlled and the potential can be measured everywhere on the object boundary. However, in real-world electrode applications, one can only control the net currents through certain surface patches and measure the corresponding potentials on the electrodes. In this work, the factorization method is translated to the framework of the complete electrode model of electrical impedance tomography and its functionality is demonstrated through two-dimensional numerical experiments. Special attention is paid to the efficient implementation of the algorithm in polygonal domains.
2007, 1(2): 319-348 doi: 10.3934/ipi.2007.1.319 +[Abstract](353) +[PDF](1124.5KB)
Abstract:
In this work we propose a new automatic methodology for computing accurate digital elevation models (DEMs) in urban environments from low baseline stereo pairs that shall be available in the future from a new kind of earth observation satellite. This setting makes both views of the scene very similar, thus avoiding occlusions and illumination changes, which are the main disadvantages of the commonly accepted wide-baseline configuration. There still remain two crucial technological challenges: (i) precisely estimating DEMs with strong discontinuities and (ii) providing a statistically proven result, automatically. The first one is solved here by a piecewise affine representation that is well adapted to man-made landscapes, whereas the application of computational Gestalt theory introduces reliability and automation. In fact this theory allows us to reduce the number of parameters to be adjusted, and to control the number of false detections. This leads to the selection of a suitable segmentation into affine regions (whenever possible) by a novel and completely automatic perceptual grouping method. It also allows us to discriminate e.g. vegetation-dominated regions, where such an affine model does not apply and a more classical correlation technique should be preferred. In addition we propose here an extension of the classical "quantized" Gestalt theory to continuous measurements, thus combining its reliability with the precision of variational robust estimation and fine interpolation methods that are necessary in the low baseline case. Such an extension is very general and will be useful for many other applications as well.
2007, 1(2): 349-369 doi: 10.3934/ipi.2007.1.349 +[Abstract](487) +[PDF](283.4KB)
Abstract:
We extend results of Dos Santos Ferreira-Kenig-Sjöstrand-Uhlmann (Comm. Math. Phys., 2007) to less smooth coefficients, and we show that measurements on part of the boundary for the magnetic Schrödinger operator determine uniquely the magnetic field related to a Hölder continuous potential. We give a similar result for determining a convection term. The proofs involve Carleman estimates, a smoothing procedure, and an extension of the Nakamura-Uhlmann pseudodifferential conjugation method to logarithmic Carleman weights.
2007, 1(2): 371-389 doi: 10.3934/ipi.2007.1.371 +[Abstract](489) +[PDF](1610.1KB)
Abstract:
Numerical realization of mathematical models always induces errors to the computational models, thus affecting both predictive simulations and related inversion results. Especially, inverse problems are typically sensitive to modeling and measurement errors, and therefore the accuracy of the numerical model is a crucial issue in inverse computations. For instance, in problems related to partial differential equation models, the implementation of a numerical model with high accuracy necessitates the use of fine discretization and realistic boundary conditions. However, in some cases realistic boundary conditions can be posed only for very large or even unbounded computational domains. Fine discretization and large domains lead to very high-dimensional models that may be of prohibitive computational cost. Therefore, it is often necessary in practice to use coarser discretization and smaller computational domains with more or less incorrect boundary conditions in order to decrease the dimensionality of the model. In this paper we apply the recently proposed approximation error approach to the problem of incorrectly posed boundary conditions. As a specific computational example we consider the imaging of conductivity distribution of soil using electrical resistance tomography. We show that the approximation error approach can also be applied to domain truncation problems and that it allows one to use significantly smaller scale forward models in the inversion.
2007, 1(2): 391-398 doi: 10.3934/ipi.2007.1.391 +[Abstract](418) +[PDF](145.0KB)
Abstract:
We construct the modified wave operator for the nonlinear Schrödinger type equations

$u_{t}-\frac{i}{\beta }\| partial _{x} |^{\beta }u=i\lambda \ |u| ^{\rho -1}u,$

for $\( t,x ) \in \mathbf{R}\times \mathbf{R.}$ We find the solutions in the neighborhood of suitable approximate solutions provided that $\beta \geq 2$, $\Im \lambda >0$ and $\rho <3$ is sufficiently close to $3.$ Also we prove the time decay estimate of solutions

$\ ||u ( t )| |\ _{\mathbf{L}^{2}}\leq Ct^{\frac{1}{2} -\frac{1}{\rho -1}}.$

When we prove the existence of a modified scattering operator, then a natural inverse problem arises to reconstruct the parameters $\beta ,\lambda$ and $\rho$ from the modified scattering operator.

2007, 1(2): 399-422 doi: 10.3934/ipi.2007.1.399 +[Abstract](518) +[PDF](495.3KB)
Abstract:
The Bayesian approach and especially the maximum a posteriori (MAP) estimator is most widely used to solve various problems in signal and image processing, such as denoising and deblurring, zooming, and reconstruction. The reason is that it provides a coherent statistical framework to combine observed (noisy) data with prior information on the unknown signal or image which is optimal in a precise statistical sense. This paper presents an objective critical analysis of the MAP approach. It shows that the MAP solutions substantially deviate from both the data-acquisition model and the prior model that underly the MAP estimator. This is explained theoretically using several analytical properties of the MAP solutions and is illustrated using examples and experiments. It follows that the MAP approach is not relevant in the applications where the data-observation and the prior models are accurate. The construction of solutions (estimators) that respect simultaneously two such models remains an open question.
2007, 1(2): 423-435 doi: 10.3934/ipi.2007.1.423 +[Abstract](596) +[PDF](349.2KB)
Abstract:
In this paper, we consider the electrical impedance tomography problem in a computational approach. This inverse problem is the recovery of the electrical conductivity $\sigma$ in a domain from boundary measurements, given in the form of the Neumann-to-Dirichlet map. We formulate the inverse problem as a variational one, with a fitting term and a regularization term. We restrict the minimization with respect to the unknown $\sigma$ to piecewise-constant functions defined on rectangular domains in two dimensions. We borrow image segmentation techniques to solve the minimization problem. Several experimental results of conductivity reconstruction from synthetic data are shown, with and without noise, that validate the proposed method.

2017  Impact Factor: 1.465