Andreas Rieder

On the regularization of nonlinear ill-posed problems via inexact Newton iterations

Inexact Newton methods for the stable solution of nonlinear ill-posed problems are considered. The corresponding inner scheme can be chosen to be any linear regularization with a sufficient modulus of convergence. The regularization property of these Newton-type algorithms is verified, that is, the iterates converge to a solution of the nonlinear problem with exact data when the noise level tends to zero. Moreover, convergence rates are given. Finally, implementation issues are discussed and the algorithm is applied to a parameter identification problem for an elliptic PDE. The numerical results reproduce nicely theoretical predictions and show the efficiency of the proposed method.

Incomplete data problems in X-ray computerized tomography

SummaryIn the present paper we study truncated projections for the fanbeam geometry in computerized tomography. First we derive consistency conditions for the divergent beam transform. Then we study a singular value decomposition for the case where only the interior rays in the fan are provided, as for example in region-of-interest tomography. We show that the high angular frequency components of the searched-for densities are well determined and we present reconstructions from real data where the missing information is approximated based on the singular value decomposition.

On convergence rates of inexact Newton regularizations

Summary.REGINN is an algorithm of inexact Newton type for the regularization of nonlinear ill-posed problems [Inverse Problems 15 (1999), pp. 309–327]. In the present article convergence is shown under weak smoothness assumptions (source conditions). Moreover, convergence rates are established. Some computational illustrations support the theoretical results.

Newton regularizations for impedance tomography: convergence by local injectivity

In Lechleiter et al (2006 Inverse Problems 22 1967–87) we demonstrated experimentally that the Newton-like regularization method CG-REGINN is a competitive solver for the inverse problem of the complete electrode model in 2D-electrical impedance tomography. Here we establish rigorously the observed convergence of CG-REGINN (and related schemes). To this end we prove that the underlying nonlinear operator has an injective Frechet derivative whenever the number of electrodes is sufficiently large and the discretization step size is sufficiently small. Though injectivity of the Frechet derivative is an interesting new result on its own, it is only a secondary issue here. We namely rely on it to obtain a so-called tangential cone condition in the fully discrete setting which is the main ingredient in a well-developed convergence theory for Newton-like regularization schemes.

Newton regularizations for impedance tomography: a numerical study

The inexact Newton iteration REGINN for regularizing nonlinear ill-posed problems consists of two components: the (outer) Newton iteration, stopped by a discrepancy principle, and the inner iteration, which computes the Newton update by solving approximately a linearized system. The second author proved the convergence of REGINN furnished with the conjugate gradients method as inner iteration (Rieder 2005 SIAM J. Numer. Anal. 43 604–22). Amongst others the following feature distinguishes REGINN from other Newton-like regularization schemes: the regularization level for the locally linearized systems can be adapted dynamically incorporating information on the local degree of ill-posedness gained during the iteration. Of course, the potential of this feature can be fully explored only by meaningful numerical experiments in a realistic setting. Therefore, we apply REGINN to the 2D-electrical impedance tomography problem with the complete electrode model. This inverse problem is known to be severely ill-posed. The achieved reconstructions are compared qualitatively and quantitatively with reconstructions from a one-step method which is closely related to the NOSER algorithm (Cheney et al 1990 Int. J. Imag. Syst. Technol. 2 66–75), an often used solver in impedance tomography. Our detailed numerical comparison reveals REGINN to be a competitive solver outperforming the one-step method when noise corrupts the data and/or a moderately large number of electrodes is used.

Shape from Specular Reflection and Optical Flow

Abstract Inferring scene geometry from a sequence of camera images is one of the central problems in computer vision. While the overwhelming majority of related research focuses on diffuse surface models, there are cases when this is not a viable assumption: in many industrial applications, one has to deal with metal or coated surfaces exhibiting a strong specular behavior. We propose a novel and generalized constrained gradient descent method to determine the shape of a purely specular object from the reflection of a calibrated scene and additional data required to find a unique solution. This data is exemplarily provided by optical flow measurements obtained by small scale motion of the specular object, with camera and scene remaining stationary. We present a non-approximative general forward model to predict the optical flow of specular surfaces, covering rigid body motion as well as elastic deformation, and allowing for a characterization of problematic points. We demonstrate the applicability of our method by numerical experiments on synthetic and real data.

Inexact Newton Regularization Using Conjugate Gradients as Inner Iteration

In our papers [Inverse Problems, 15 (1999), pp. 309--327] and [Numer. Math., 88 (2001), pp. 347--365] we proposed algorithm {\tt REGINN}, an inexact Newton iteration for the stable solution of nonlinear ill-posed problems. {\tt REGINN} consists of two components: the outer iteration, which is a Newton iteration stopped by the discrepancy principle, and an inner iteration, which computes the Newton correction by solving the linearized system. The convergence analysis presented in both papers covers virtually any linear regularization method as inner iteration, especially Landweber iteration,

The Approximate Inverse in Action with an Application to Computerized Tomography

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Towards a general convergence theory for inexact Newton regularizations

-methods, and Tikhonov--Phillips regularization. In the present paper we prove convergence rates for {\tt REGINN} when the conjugate gradient method, which is nonlinear, serves as inner iteration. Thereby we add to a convergence analysis of {Hanke}, who had previously investigated {\tt REGINN} furnished with the conjugate gradient method [Numer. Funct. Anal. Optim., 18 (1997), pp. 971--993]. By numerical experiments we illustrate that the conjugate gradient method outperforms the

Local Inversion Of The Sonar Transform Regularized By The Approximate Inverse

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