Inverse Problems & Imaging
November 2008 , Volume 2 , Issue 4
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An approximate optimal control formulation of the Cauchy problem for elliptic equations is considered. A cost functional adding a fading through the iterations regularizing term borrowed from the domain decomposition communauty is proposed. Convergence of the descretized finite elements solution to the continuous one is proved. Numerical experiments involving smooth, non-smooth geometries as well as anisotropy highlight the capability of the present missing boundary data recovering process.
We study the stability of the reconstruction of the scattering and absorption coefficients in a stationary linear transport equation from knowledge of the full albedo operator in dimension $n\geq3$. The albedo operator is defined as the mapping from the incoming boundary conditions to the outgoing transport solution at the boundary of a compact and convex domain. The uniqueness of the reconstruction was proved in [2, 3] and partial stability estimates were obtained in  for spatially independent scattering coefficients. We generalize these results and prove an $L^1$-stability estimate for spatially dependent scattering coefficients.
We propose a regularization algorithm for color/vectorial images which is fast, easy to code and mathematically well-posed. More precisely, the regularization model is based on the dual formulation of the vectorial Total Variation (VTV) norm and it may be regarded as the vectorial extension of the dual approach defined by Chambolle in  for gray-scale/scalar images. The proposed model offers several advantages. First, it minimizes the exact VTV norm whereas standard approaches use a regularized norm. Then, the numerical scheme of minimization is straightforward to implement and finally, the number of iterations to reach the solution is low, which gives a fast regularization algorithm. Finally, and maybe more importantly, the proposed VTV minimization scheme can be easily extended to many standard applications. We apply this $L^1$ vectorial regularization algorithm to the following problems: color inverse scale space, color denoising with the chromaticity-brightness color representation, color image inpainting, color wavelet shrinkage, color image decomposition, color image deblurring, and color denoising on manifolds. Generally speaking, this VTV minimization scheme can be used in problems that required vector field (color, other feature vector) regularization while preserving discontinuities.
An algorithm for two dimensional histogram modal analysis is presented. A major challenge in two dimensional histogram analysis is to provide an accurate location and description of the extended modal shape. The approach presented in this paper combines the Fast Level Set Transform of the histogram and the Helmholtz principle to find the location and shape of the modes. Furthermore, the algorithm is devoid of any a priori assumptions about the underlying density or the number of modes. At the core, this approach is a new way to manage and search the number of regions that must be examined to identify meaningful sets. Computational issues required a new tail sum bound on the multinomial distribution to be stated and proven. This bound reduces to the Höeffding inequality for the binomial distribution. The histogram segmentation procedure was applied to the two problems of image color segmentation and correlation pattern recognition. With no a priori knowledge about the color image assumed, the two dimensional modal analysis is applied to the CIELAB color space to find perceptually uniform dominant colors. The modal analysis is also extended to correlation pattern recognition to find multiple targets in a single correlation plane.
We study the inverse problem of deducing the dynamical characteristics (such as the potential field) of large systems from kinematic observations. We show that, for a class of steady-state systems, the solution is unique even with fragmentary data, dark matter, or selection (bias) functions. Using spherically symmetric models for simulations, we investigate solution convergence and the roles of data noise and regularization in the inverse problem. We also present a method, analogous to tomography, for comparing the observed data with a model probability distribution such that the latter can be determined.
We consider uniqueness of three-dimensional parallel beam tomography in which both the object being imaged and the projection orientations are unknown. This problem occurs in certain practical applications, for example in cryo electron microscopy of viral particles, where the projection orientations may be completely unknown due to the random orientations of the particles being imaged. We show that only three projections are needed to guarantee unique recovery of the unknown projection orientations (up to a common orthogonal transformation), if the object belongs to a certain generic set of objects. In particular, the uniqueness holds for almost all objects. We also show that, if the object belongs to that generic set, $k+1$ projections at unknown orientations suffice to determine uniquely also the geometric moments of the object of order less or equal to $k$. As a consequence, the object belonging to that generic set, is uniquely determined (up to an orthogonal transformation) by almost any infinitely many projections at unknown orientations. We show that the uniqueness problem is related to some properties of certain homogeneous polynomials that depend on the projection orientations and the geometric moments of objects. Here certain theorems of algebraic geometry turn out to be useful. We also provide a system of equations, that is uniquely solvable and gives the desired projection orientations and object's geometric moments.
We present a constructive algorithm, the stationary waves method, to get approximative reconstructions in the inverse scattering problems. The method is a version of the singular sources/probe method but the singular fields are replaced with the stationary waves. The suggested indicator function compromises in getting exactly the obstacle, but it is very easy and fast to compute, robust with noise, and seems to give excellent reconstruction images. The method is introduced in the context of the acoustic inverse obstacle scattering problems.
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