In this paper, we provide a unified framework for understanding how and why artifacts arise in limited data tomography, using precise microlocal analysis of the underlying generalized Radon transforms (gRT). Our insights facilitate a more accurate assessment of reconstructions and eventually enable practical artifact reduction strategies in real-world imaging applications. To this end, we consider gRTs as Fourier integral operators whose propagation of singularities is governed by their canonical relations. Thus, we can discern various types of singularities in the reconstructions, including visible singularities (present in both the ground truth and its reconstruction), invisible singularities (present in the ground truth but not in the reconstruction), and artifacts (singularities present in the reconstruction but not in the ground truth). Several concrete examples illustrate the theoretical results: in photoacoustic tomography, mirror artifacts, a special kind of artifacts, arise due a non-injectivity property of the canonical relation (the Bolker condition is violated). In classical X-ray computerized tomography, we consider region of interest (ROI) tomography, where object independent artifacts, a further subtype of artifacts, may manifest at the boundaries of the ROI. Lastly, we investigate artifact formation in linearized seismic imaging (Kirchhoff migration), which depends on the underlying pressure wave velocity model used in the linearization process.
Abstract This paper studies convergence rates of the inexact Newton regularization method for nonlinear ill-posed problems in Banach spaces, where the forward operator maps from a Banach space into a Hilbert space. By employing the Landweber iteration as the inner scheme and terminating the outer Newton iteration by the discrepancy principle, under a benchmark source condition, we establish order-optimal convergence rates of the method via a Lyapunov-type argument. Our analysis allows for nonsmooth regularization functionals and substantial flexibility in the choice of step sizes, including variable step-size strategies. Numerical experiments are provided to validate the theoretical convergence rate results.
Abstract Electroencephalography (EEG) current localization aims to reconstruct the neuronal current that gives rise to measured scalp potentials. Although this inverse EEG problem is inherently ill-posed, under the minimal-energy assumption, it admits a unique solution. One of the authors has shown that EEG measurements are sensitive only to a scalar function, which represents the irrotational component of the primary neuronal current. This paper presents a complete numerical implementation in realistic head geometries, along with validation of a generalized inverse EEG minimization problem that was previously introduced only at the theoretical level. The approach combines a linear algebraic formulation with machine learning techniques. In particular, a deep neural network is trained to approximate forward EEG solutions, enabling the efficient computation of the gradients of an auxiliary function that appears in the proposed inverse formulation. The irrotational component of the neuronal current is represented in terms of an inverse multiquadratic radial basis function expansion, with the shape parameter serving as an implicit regularization mechanism. The new method is evaluated using synthetic EEG data generated from realistic head geometries and is compared with the established inverse methods sLORETA and eLORETA. Quantitative results demonstrate significant improvement in localization accuracy, reduced spatial dispersion, and increased robustness to measurement noise. These findings indicate that the new method provides a computationally efficient alternative for EEG current localization.
Abstract We establish uniqueness in an exterior Calderón problem for nonlocal Schrödinger operators Φ ( − Δ ) + q on bounded Lipschitz domains of R n , n ⩾ 2 . The unknown is a real-valued potential q ∈ L ∞ , and the data are localized exterior Dirichlet-to-Neumann measurements. We identify an explicit class A CBF of complete Bernstein functions such that, for arbitrary nonempty open sets W 1 , W 2 in the unbounded exterior component, equality of the measurements generated and tested in W 1 and W 2 implies equality of the potentials. The sets may be disjoint. The admissible symbols need not be homogeneous or of a fixed fractional order; the class includes fractional powers, finite positive sums and compact mixtures of powers, logarithmic symbols and their compact mixtures, and relativistic stable symbols. The main new ingredient is an exterior unique continuation principle obtained from a separated-support jump-kernel identity, off-diagonal analyticity of subordinated pure-jump kernels, the Paley–Wiener theorem, and a branch-rigidity argument excluding global meromorphic continuation of Φ . Runge approximation and the Alessandrini identity then yield exact identifiability of q . In the subordinate-diffusion interpretation, Φ specifies a multiscale jump law and q acts as an absorption or reaction coefficient, so the result provides a uniqueness mechanism for remote excitation and observation outside an inaccessible region.
In this work, we propose a novel two-stage image segmentation method addressing the issue of intensity inhomogeneities in images. In the first stage, a non-convex variational decomposition model based on a tight wavelet frame is proposed. The proposed non-convex variational decomposition model is a weighted sum of three terms: the data fitting term, the non-convex regularization term based on the tight wavelet frame for the structural component (piecewise constant), and the second-order gradient regularization term for the intensity component (smooth). The existence of a solution for the proposed decomposition model is established, and the uniqueness is guaranteed under certain conditions. In the second stage, a thresholding technique is applied to the piecewise constant component obtained in the first stage to achieve the final image segmentation result. This two-stage strategy focuses on the structural features of the image during image segmentation, thereby mitigating the impact of intensity inhomogeneities. In addition, this paper develops an effective algorithm based on variational Bayesian inference that simultaneously segments images and selects model parameters, reducing the burden of manual parameter adjustment. Numerical experimental results demonstrate that the proposed method not only effectively reduces illumination and shadow interference but also exhibits significant competitiveness in processing images with intensity inhomogeneities.
