Abstract In cone beam tomography, the saddle trajectory has been investigated for many years now, but usually in the framework of no truncation of the projections, or in the framework of axial truncation only. To handle larger patients or target objects, or for dose reduction purposes, transverse truncation can also arise. In this situation, the only known and suitable analytic method to study reconstruction feasibility is based on the differentiated backprojection method. We prove here that this method is suitable for nearly all situations of the saddle source trajectory with transversely truncated projections, and demonstrate analytically that unique and stable image reconstruction is possible in the feasible region. We present some simulations studies to illustrate these results.
Abstract This paper investigates the optimal geometric structure of affine quaternion phase retrieval (AQPR), which aims to recover quaternion-valued signals from magnitude-only measurements with a known bias. By generalized HR calculus, we rigorously characterize the second-order structure of the associated nonlinear least-squares loss function and prove that, when the number of measurements is of the order O ( n log n ) , this loss function exhibits strong convexity in the quaternion field with high probability, provided that mild conditions are imposed on the bias. This result reveals a fundamental difference between AQPR and classical quaternion phase retrieval—the latter employs a least-squares formulation that is nonconvex. Based on this strong convexity property, we employ a simple quaternion Wirtinger flow algorithm and prove its linear convergence to the ground truth signal. Numerical experiments verify the practical advantages of strong convexity in terms of convergence speed, recovery success rate, and computational efficiency.
Abstract Nonlinear inverse problems are typically solved by minimizing a data-misfit functional, which is often non-convex that leads minimization algorithms to stagnate at a local minimum. A typical example is the cycle-skipping phenomenon in full waveform inversion (FWI) of seismic reflection or transmission data. To overcome this difficulty, distance functionals are constructed by training neural networks to emulate a convex distance measure. Two construction strategies are discussed: (i) a data-converter network that simplifies the forward map so that a standard quadratic loss becomes convex, and (ii) a scalar-valued distance-network based on the data residual. Training samples can be generated from measured data exploiting an approximate invariance of the forward operator. Under a set of structural assumptions, it is proven that applying the Landweber iteration to the learned functional is a well-defined regularization method that guarantees monotone error reduction and convergence to the exact solution as the noise level vanishes. Given certain restrictions on the trained network and the nonlinear forward operator, it is validated that these structural assumptions are satisfied by the learned distance. This methodology is numerically validated on the Camembert benchmark model for FWI in the acoustic regime. Replacing the conventional L 2 -misfit with the learned convexified functional considerably mitigates cycle-skipping and enlarges the domain of convergence for gradient-based optimization. Numerical experiments show that the Landweber scheme with the learned misfit reaches a lower reconstruction error than the standard least-squares approach. For comparison, in some experiments convergence has been accelerated by using a limited-memory BFGS optimizer leading to slightly larger reconstruction errors compared to the Landweber scheme. Overall, the work provides a systematic way to embed learned, convex distance measures into inverse problem solvers, supplies a solid theoretical foundation for their use, and demonstrates practical gains in a seismic imaging benchmark model.
Abstract We correct the proof of lemma 2.1 in the above paper. The correction pertains exclusively to the logical derivation in the proof section. The revised proof corrects an intermediate step but arrives at the identical final result. Therefore, none of the conclusions of this work are affected.
Abstract Electrocardiographic imaging (ECGi) aims to reconstruct cardiac electrical activity from noninvasive measurements of torso surface potentials. This inverse problem is commonly formulated as a PDE-constrained optimization problem relying on a source model. One common way to overcome its severe ill-posedness is to add a regularization term to the cost function to be minimized in the optimization problem. Another way consists in adding constraints in the optimization problem, for instance based on physical informations or modeling. Following this idea, in this work, we introduce a novel source model, named the epicardial model (EPI), which provides a heuristic three-dimensional extension of a previously proposed depth-averaged two-dimensional model. This new model is formulated in the torso domain only and couples a bidomain-like surface equation on the heart with the standard Laplace equation in the torso. Thus, it can naturally be substituted into standard torso-based clinical workflows while offering more possibilities for incorporating physiologically relevant prior information on the modeled EPI surface. The EPI is first validated numerically against reference bidomain simulations. While amplitude discrepancies are observed, the model is able to reproduce the spatial distribution of the electrical potentials. The EPI is then evaluated within an ECGi reconstruction framework. Two EPI-constrained optimization formulations are considered: one using classical Tikhonov regularization and another using a physiological total variation (TV) regularization applied to the transmembrane voltage. Reconstructions are compared with those obtained from the classical Laplace-based Cauchy formulation. Without addition of information, the proposed EPI based framework yields reconstruction of comparable accuracy to standard approaches despite the presence of modeling approximations. The incorporation of a prior through the TV regularization, allowed by the model, improves signals reconstruction and reduces artificial lines of block in activation maps. These results highlight the potential of the EPI as a physiologically enriched alternative to classical ECGi formulations.
Abstract Full waveform inversion (FWI) enables high-resolution reconstruction of the speed of sound (SOS) in medical ultrasound imaging, but conventional FWI does not quantify reconstruction uncertainty. We propose a Stein variational gradient descent (SVGD)-based FWI framework for posterior approximation and uncertainty quantification, in which the posterior distribution of SOS is represented by interacting particles. The particle updates are driven by the log-posterior gradient, combining an adjoint-state likelihood term with a Gaussian prior. The converged particle ensemble provides both the posterior mean SOS and the posterior standard deviation. The method is validated on synthetic models and a realistic breast-tissue model. Numerical results show that SVGD yields accurate SOS reconstructions, with mean estimates comparable to or slightly better than those of conventional FWI, while also providing informative uncertainty maps. Reliable mean reconstructions can be obtained with relatively few particles, whereas uncertainty estimates become more stable as the particle number increases. In non-ideal cases with source-signature mismatch and source–receiver positioning error, the posterior standard deviation remains useful for identifying regions associated with inversion failure or acquisition error. The uncertainty estimates produced by SVGD are consistent with the Metropolis–Hasting–Markov Chain Monte Carlo benchmark along selected profiles. Overall, SVGD provides a practical compromise between reconstruction accuracy, computational feasibility, and uncertainty quantification for medical ultrasound FWI.
Abstract Multi-temporal image reconstruction under cloud corruption is a profoundly ill-posed inverse problem. The existing low-rank regularization methods mainly rely on a single-subspace assumption, which inadequately captures the intrinsic structure of the multi-temporal image that inherently lies in a union of multiple tensor subspaces. To break this fundamental limitation, we propose a multi-subspace tensor dictionary learning model. The model learns a self-representation tensor dictionary to uncover the underlying multi-subspace structure of the image data. Moreover, a data-adaptive tensor nuclear norm is introduced to characterize the distinct low-rank properties of both the dictionary and the coefficient tensor. Compared to the single-subspace model, the proposed model provides a more promising regularization paradigm to exploit the complex tensor structures. An efficient proximal alternating minimization algorithm is developed to solve the resulting nonconvex optimization problem, with theoretical guarantees establishing convergence to a critical point. Experimental evaluations on multiple multi-temporal image datasets validate the advantages of the proposed method, demonstrating its efficacy and robustness as a regularization strategy for ill-posed multi-temporal inverse problems.
Abstract In this paper, we consider the parameter determination problem for the Brinkman equation using boundary measurements. By analyzing the precise characterization of the singularity in the H m -norm of singular solutions and the exploding Sobolev norms of the volume potential with the fundamental solution, we prove a uniqueness theorem, that the constant viscosity coefficient μ and any order derivative of the parameter χ 2 at the boundary can be uniquely determined from boundary measurements.
In this paper, we consider the inverse spectral problem of determining the spherically symmetric refractive index in a bounded spherical region of radius b . Instead of the usual case of the refractive index ρ ∈ W 2 2 , by using singular Sturm–Liouville theory, we discuss the case when the refractive index ρ is a piecewise W 2 1 function. We prove that if ∫ 0 b ρ ( r ) d r < b , then ρ is uniquely determined by all special transmission eigenvalues; if ∫ 0 b ρ ( r ) d r = b , then all special transmission eigenvalues with some additional information can uniquely determine ρ . We also consider the mixed spectral problem and obtain that ρ is uniquely determined from partial information of ρ and the ‘almost real subspectrum’.
In this paper, we propose a non-convex total variation image restoration model based on the hyper-Laplacian gradient prior. The proposed model enhances sparsity while preserving image edges. The regularization term is defined as a linear combination of the & ell;p quasi-norm and the squared & ell;2 norm. It also balances the sparsity promotion in the image gradients and the smoothness in the restored image. We develop an efficient numerical algorithm using the alternating direction method of multipliers (ADMM) combined with a Chebyshev iterative scheme. The proposed method converges rapidly and provides an accurate solution. To further enhance the overall convergence, the Nesterov acceleration method is incorporated into the proposed ADMM framework. We also establish theoretical convergence analysis of the proposed method. Extensive experiments are conducted on grayscale images with varying resolutions. Different blur kernels and noise levels are considered to show the robustness of the proposed methods. The results indicate that the proposed method achieves superior performance among the compared methods, with improved edge preservation and faster convergence.
We revisit the direct method for reconstructing coefficients in second-order elliptic partial differential equations. Two model problems are considered: first, the reconstruction of the diffusion coefficient in a scalar elliptic problem, and second, the reconstruction of the shear modulus in the elastography problem. To highlight the versatility of the framework, different notions of stability are exploited in the two situations. In the scalar case, the system is interpreted as a hyperbolic transport equation and an inf-sup condition on the discrete level is leveraged for the analysis of the numerical method. We obtain error estimates on the reconstruction coefficient that are suboptimal by half an order, which is known to be sharp on general meshes. In the vector case, the minimization of the residual in dual norm and a stability result on the continuous problem lead to error estimates that are optimal compared to the approximation. For both problems, the theoretical results are illustrated by some numerical examples.
We develop a mixed variational formulation for inverse problems governed by first order hyperbolic differential equations with possibly discontinuous coefficients and unknowns. This formulation is a variant of the equation error (EE) approach, which has been used extensively to solve inverse problems. Standard formulations of the EE approach tend to slow down very quickly as the size of the computational mesh used to solve the discrete problem is increased. This motivated us to develop a EE formulation that uses a Lagrange multiplier to enforce the differential equation constraint. We prove that this new EE formulation is well-posed, show that it is equivalent to prior EE formulations, and use it to reconstruct both continuous and discontinuous parameter fields. Finally, we show that the solution time for the new EE formulation increases at a rate approximately three times slower than the corresponding rate for prior EE formulations as the mesh size increases.
Medical image registration is an inverse problem and a fundamental task in medical imaging, aiming to align two or more images into a common spatial coordinate system. Traditional image registration methods, such as variational frameworks, provide mathematically interpretable solutions but often suffer from high computational complexity, making them impractical for real-time applications. Though deep learning-based approaches have significantly improved registration speed, they still face challenges in preserving anatomical topology and maintaining interpretability. To address these limitations, we propose a cascade interpretable diffeomorphic image registration network (C-I-DIRNet), which integrates deep learning with diffeomorphic theory to achieve highly accurate, topology-preserving, large deformation and real-time registration. Theoretical analysis demonstrates how C-I-DIRNet guarantees diffeomorphic transformations, ensuring smooth and invertible deformations. Several numerical experiments are performed to show the superior performance of C-I-DIRNet in eliminating physical mesh folding phenomenon when compared with other existing learning based registration methods. The code of this paper is available at: https://github.com/hrhao166/C-I-DIRNet.
Over the past few decades, various numerical methods have been developed to solve linear inverse problems with sparse solutions. However, there remains a shortage of efficient algorithms specifically tailored for large-scale problems. In this paper, we work toward bridging this gap by developing the so-called General Decomposition Pursuit algorithm, which is designed to directly tackle a large-scale problem. This algorithmic framework is notably distinct from any existing sparsity-aware method. The main mechanism of the algorithm involves decomposing a large problem into several mutually coupled small subproblems and then combining the (inexact) solutions of these subproblems to generate a sparse solution to the original problem. The global convergence of the algorithm is shown under the restricted isometry property. Simulations with synthetic data and applications in medical image reconstruction indicate that the proposed algorithm, when equipped with an appropriate inner solver for subproblems, can outperform several mainstream algorithms in success rates for locating the sparse solution to the problem.
We investigate density results for solutions of the non-local heat equation at a fixed time slice, considering two models: one with homogeneous Dirichlet boundary conditions and another with singular boundary data. In both cases, the non-local exponent satisfies a is an element of(12,1). We study both qualitative and quantitative approximation properties. For the model with singular boundary data, we assume the potential is non-negative, sufficiently small, and exhibits mild growth. The smallness condition is explicit and depends only on the domain and the spatial dimension. We also address Calder & oacute;n-type inverse problems for these parabolic models, recovering the potential from solution data measured either on the boundary or at a fixed time slice. The Pohozaev identity is a key tool in establishing both the density results and the inverse problem analysis. Finally, we apply the Pohozaev identity to an elliptic eigenvalue problem and show that the corresponding eigenfunctions, when divided by a suitable power of the distance to the boundary, cannot vanish on any non-empty open subset of the boundary. This result holds without restrictions on the non-local exponent.
The acoustic inverse scattering problem is of critical importance in a number of fields, including medical imaging, sonar, and non-destructive evaluation. The problem of interest can vary from the detection of the shape to the properties of an obstacle. The challenge is that this problem is severely ill-posed and highly nonlinear. Significant efforts have been expended over the years to develop solutions to this problem. However, existing fast data-driven methods primarily focus on the two-dimensional scattering case. This paper explores the potential of using machine learning to accelerate the solution to the three-dimensional (3D) version of the problem. To this end, we develop inverse scattering shape reconstruction network (ISSRNet), a deep learning framework for 3D shape reconstruction using phaseless far-field data. The framework is implemented by (a) using a compact probabilistic shape latent space learned by a 3D variational auto-encoder, and (b) a convolutional neural network trained to extract far-field features due to multiple incident waves and map the acoustic scattering information to this shape representation. We demonstrate ISSRNet's 3D shape reconstruction capabilities on random rock-like particles, and airplane objects from the popular ShapeNet data set. We also evaluate the framework's performance when trained on lower-resolution scattering data and when receiver locations include uncertainty. Our experiments show that the proposed framework is able to capture both global and local details, differentiate between different types of shapes and performs several orders of magnitude faster than its numerical iterative counterparts.
This paper introduces and analyzes a generalized spherical mean operator that unifies the classical spherical mean and the wave forward operator. It is shown that this generalized transform satisfies a generalized Euler-Poisson-Darboux equation, and its analytical structure is investigated through Fourier techniques and explicit representation formulas. Several inversion formulas are derived for different cases: when the spatial variable lies on a hyperplane, through an analogue of the Fourier slice theorem; when it lies on the unit sphere, through spherical harmonics; and when it lies on the boundary of an arbitrary bounded domain, through an analogue of the Kirchhoff integral representation for the wave forward operator. In each case, the derivation extends methods from earlier works on the classical spherical mean or the wave forward operator, and the resulting formulas unify previously known inversion results. Together, these results form a unified analytical foundation for inversion in wave-based imaging across diverse geometric configurations.
We investigate the problem of recovering unknown inclusions in a heat conductor from boundary measurements, formulated mathematically as an inverse boundary value problem for the heat equation. A sampling-type approach, the linear sampling method (LSM), has been successfully applied to tackle this problem. This method is easy to implement and generates reasonable reconstructions using a well-designed indicator function. However, several practical limitations of the LSM remain evident. First, the decision boundary determined by the threshold of the indicator function is ambiguous and the threshold often varies with different inclusions. Second, high-resolution reconstructions require the highly accurate Neumann-to-Dirichlet map on the entire boundary, especially in cases involving multiple inclusions. Finally, although the time-dependent setting naturally provides multi-temporal indicator functions, the conventional LSM lacks a systematic strategy to integrate them effectively. To address these issues, we propose a learning-based hybrid strategy to realize high-quality reconstructions. Specifically, the LSM is first used to obtain rough reconstructions, which serve as noisy versions of true inclusions. Motivated by the remarkable success of deep learning in image processing, we then utilize the U-net neural network, which extracts the dominant features of inclusions, to refine the reconstructions. The network incorporates the total variation regularization and supports multi-temporal indicator functions as multi-channel inputs to exploit temporal information. Various numerical experiments demonstrate that the proposed method successfully overcomes the fundamental limitations of the LSM. We highlight that this framework, which combines physics-based feature extraction with learning-based post-processing, is extendable to various inverse problems for PDEs models. It is particularly applicable for time-dependent equations, where multi-temporal measurements naturally provide complementary information for data-driven refinement.
In time-harmonic elastography, the shear modulus is typically inferred from full-field displacement data by solving an inverse problem based on the time-harmonic elastodynamic equation. In this paper, we focus on nearly incompressible media, which pose robustness challenges, especially in the presence of noisy data. Restricting ourselves to 2D and considering an isotropic, linearly deforming medium, we reformulate the problem as a non-autonomous hyperbolic system and, through theoretical analysis, establish existence, uniqueness, and stability of the inverse problem. To ensure robustness with noisy data, we propose a least-squares approach with regularization. The convergence properties of the method are verified numerically using in silico data.
