Ill-posed Research Articles

Estimate of the Spectrum of Discrete Sequences in Ill-Posed Problems Based on the Study of the Numerical Rank of the Trajectory Matrix

Estimate of the Spectrum of Discrete Sequences in Ill-Posed Problems Based on the Study of the Numerical Rank of the Trajectory Matrix

Structural Tectonic Scheme Creation Based on Seismic-Gravity Models and Isostasy Usage: Ural Case

Process of Earth’s density models creation leads to the solution of direct and inverse gravimetry problems. The inverse problem of gravimetry is a classic example of an ill-posed problem: in the common statement, its solution is not unique and unstably depends on input data. Therefore, it is necessary to determine solutions belonging substantial sets of correctness, choosing reasonable models of an initial approximation. In this paper the application of complex interpretation methods of seismic and gravitational data for the creation of three-dimensional models of crust and the upper mantle are presented. Original algorithms and programs were developed for implementation of these methods. They contain solution of non-linear (structural) inverse problem and the solution of the linear three-dimensional inverse problem taking note of the side sources. Coefficients of the “density-velocity” correlation formulas for a number of geo-traverses were defined. Also, we suggest a technique of tectonic maps construction, which is based on the lithostatic pressure calculation. Its idea can be applied to both two- and three-dimensional cases. In the 2D case we show the way to split the mantle to blocks with vertical boundaries. If lithostatic compensation hypothesis is adopted, the method also allows one to calculate density value for each block. Such separation of the mantle helps to diminish discrepancy between model and observed fields. In 3D case we suggest a method, which can be used to construct tectonic structure maps with information about approximate depth and height of each tectonic block.

A modified algebraic method of mathematical signal processing in radar problems

A new method for approximate solution of ill-posed inverse problems of reconstructing images of objects with angular resolution exceeding the Rayleigh criterion is proposed and justified, i.e. with super resolution. Angular super-resolution allows you to obtain images of objects with increased clarity and distinguish previously invisible details of images of complex objects. In addition, on this basis, the probability of correct solutions to problems of object recognition and identification increases. Mathematically, the problem is reduced to solving the linear Fredholm integral equation of the first kind of convolution type. Solutions are sought with additional conditions in the form of restrictions on the location and size of the desired radiation source, which makes it possible to regularize the problem. The method is a development of one of the parameterization methods the algebraic method. The solution is sought in the form of a representation of the desired function in the area where the source is located in the form of a series expansion over the input sequence of orthogonal functions with unknown coefficients. Thus, the inverse problem is parameterized and reduced to searching for expansion coefficients. The presented method is based on the use of a priori information about the localization area of the radiation source, or on an estimate of the location and size of this area obtained by scanning the viewing sector with a goniometer system. Using zero values of the function describing the source outside this region, for systems based on antenna arrays it is possible to find tens and even hundreds of expansion coefficients of the desired function in a Fourier series. The solution is constructed in the form of an iterative process with a consistent increase in the number of functions used in the expansion until the solution remains stable. The adequacy and stability of the solutions was verified during numerical experiments using a mathematical model. The results of numerical studies show that the presented methods of digital processing of received signals make it possible to achieve an effective angular resolution 3–10 times higher than the Rayleigh criterion. The proposed method makes it possible to miniaturize the antenna system without degrading its characteristics. Compared to known ones, it is relatively simple, which allows it to be used by systems in real time.

Color constancy in real-world settings.

Color constancy denotes the ability to assign a particular and stable color percept to an object, irrespective of its surroundings and illumination. The light reaching the eye confounds illumination and spectral reflectance of the object, making the recovery of constant object color an ill-posed problem. How good the visual system is at accomplishing this task is still a matter of heated debate, despite more than a 100 years of research. Depending on the laboratory task and the specific cues available to observers, color constancy was found to be at levels ranging between 15% and 80%, which seems incompatible with the relatively stable color appearance of objects around us and the consistent usage of color names in real life. Here, we show close-to-perfect color constancy using real objects in a natural task and natural environmental conditions, chosen to mimic the role of color constancy in everyday life. Participants had to identify the color of a (non-present) item familiar to them in an office room under five different experimental illuminations. They mostly selected the same colored Munsell chip as their match to the absent object, even though the light reaching the eye in each case differed substantially. Our results demonstrate that color constancy under ideal conditions in the real world can indeed be exceptionally good. We found it to be as good as visual memory permits and not generally compromised by sensory uncertainty.

Linearized maximum rank correlation estimation when covariates are functional

This paper extends the linearized maximum rank correlation (LMRC) estimation proposed by Shen et al. (2023) to the setting where the covariate is a function. However, this extension is nontrivial due to the difficulty of inverting the covariance operator, which may raise the ill-posed inverse problem, for which we integrate the functional principal component analysis to the LMRC procedure. The proposed estimator is robust to outliers in response and computationally efficient. We establish the rate of convergence of the proposed estimator, which is minimax optimal under certain smoothness assumptions. Furthermore, we extend the proposed estimation procedure to handle discretely observed functional covariates, including both sparse and dense sampling designs, and establish the corresponding rate of convergence. Simulation studies demonstrate that the proposed estimators outperform the other existing methods for some examples. Finally, we apply our method to a real data to illustrate its usefulness.

Untrained Metamaterial-Based Coded Aperture Imaging Optimization Model Based on Modified U-Net

Metamaterial-based coded aperture imaging (MCAI) is a forward-looking radar imaging technique based on wavefront modulation. The scattering coefficients of the target can resolve as an ill-posed inverse problem. Data-based deep-learning methods provide an efficient, but expensive, way for target reconstruction. To address the difficulty in collecting paired training data, an untrained deep radar-echo-prior-based MCAI (DMCAI) optimization model is proposed. DMCAI combines the MCAI model with a modified U-Net for predicting radar echo. A joint loss function based on deep-radar echo prior and total variation is utilized to optimize network weights through back-propagation. A target reconstruction strategy by alternatively using the imaginary and real part of the radar echo signal (STAIR) is proposed to solve the DMCAI. It makes the target reconstruction task turn into an estimation from an input image by the U-Net. Then, the optimized weights serve as a parametrization that bridges the input image and the target. The simulation and experimental results demonstrate the effectiveness of the proposed approach under different SNRs or compression measurements.

Adaptive anisotropic Bayesian meshing for inverse problems

We consider inverse problems estimating distributed parameters from indirect noisy observations through discretization of continuum models described by partial differential or integral equations. It is well understood that errors arising from the discretization can be detrimental for ill-posed inverse problems, as discretization error behaves as correlated noise. While this problem can be avoided with a discretization fine enough to decrease the modeling error level below that of the exogenous noise that is addressed, e.g. by regularization, the computational resources needed to deal with the additional degrees of freedom may increase so much as to require high performance computing environments. Following an earlier idea, we advocate the notion of the discretization as one of the unknowns of the inverse problem, which is updated iteratively together with the solution. In this approach, the discretization, defined in terms of an underlying metric, is refined selectively only where the representation power of the current mesh is insufficient. In this paper we allow the metrics and meshes to be anisotropic, and we show that this leads to significant reduction of memory allocation and computing time.

Convergent Data-Driven Regularizations for CT Reconstruction

The reconstruction of images from their corresponding noisy Radon transform is a typical example of an ill-posed linear inverse problem as arising in the application of computerized tomography (CT). As the (naïve) solution does not depend on the measured data continuously, regularization is needed to reestablish a continuous dependence. In this work, we investigate simple, but yet still provably convergent approaches to learning linear regularization methods from data. More specifically, we analyze two approaches: one generic linear regularization that learns how to manipulate the singular values of the linear operator in an extension of our previous work, and one tailored approach in the Fourier domain that is specific to CT-reconstruction. We prove that such approaches become convergent regularization methods as well as the fact that the reconstructions they provide are typically much smoother than the training data they were trained on. Finally, we compare the spectral as well as the Fourier-based approaches for CT-reconstruction numerically, discuss their advantages and disadvantages and investigate the effect of discretization errors at different resolutions.

Single image super-resolution with denoising diffusion GANS

Single image super-resolution (SISR) refers to the reconstruction from the corresponding low-resolution (LR) image input to a high-resolution (HR) image. However, since a single low-resolution image corresponds to multiple high-resolution images, this is an ill-posed problem. In recent years, generative model-based SISR methods have outperformed conventional SISR methods in performance. However, the SISR methods based on GAN, VAE, and Flow have the problems of unstable training, low sampling quality, and expensive computational cost. These models also struggle to achieve the trifecta of diverse, high-quality, and fast sampling. In particular, denoising diffusion probabilistic models have shown impressive variety and high quality of samples, but their expensive sampling cost prevents them from being well applied in the real world. In this paper, we investigate the fundamental reason for the slow sampling speed of the SISR method based on the diffusion model lies in the Gaussian assumption used in the previous diffusion model, which is only applicable for small step sizes. We propose a new Single Image Super-Resolution with Denoising Diffusion GANS (SRDDGAN) to achieve large-step denoising, sample diversity, and training stability. Our approach combines denoising diffusion models with GANs to generate images conditionally, using a multimodal conditional GAN to model each denoising step. SRDDGAN outperforms existing diffusion model-based methods regarding PSNR and perceptual quality metrics, while the added latent variable Z solution explores the diversity of likely HR spatial domain. Notably, the SRDDGAN model infers nearly 11 times faster than diffusion-based SR3, making it a more practical solution for real-world applications.

Non-orthogonal optical multiplexing empowered by deep learning

Orthogonality among channels is a canonical basis for optical multiplexing featured with division multiplexing, which substantially reduce the complexity of signal post-processing in demultiplexing. However, it inevitably imposes an upper limit of capacity for multiplexing. Herein, we report on non-orthogonal optical multiplexing over a multimode fiber (MMF) leveraged by a deep neural network, termed speckle light field retrieval network (SLRnet), where it can learn the complicated mapping relation between multiple non-orthogonal input light field encoded with information and their corresponding single intensity output. As a proof-of-principle experimental demonstration, it is shown that the SLRnet can effectively solve the ill-posed problem of non-orthogonal optical multiplexing over an MMF, where multiple non-orthogonal input signals mediated by the same polarization, wavelength and spatial position can be explicitly retrieved utilizing a single-shot speckle output with fidelity as high as ~ 98%. Our results resemble an important step for harnessing non-orthogonal channels for high capacity optical multiplexing.

Semantic and Geometric-Aware Day-to-Night Image Translation Network.

Autonomous driving systems heavily depend on perception tasks for optimal performance. However, the prevailing datasets are primarily focused on scenarios with clear visibility (i.e., sunny and daytime). This concentration poses challenges in training deep-learning-based perception models for environments with adverse conditions (e.g., rainy and nighttime). In this paper, we propose an unsupervised network designed for the translation of images from day-to-night to solve the ill-posed problem of learning the mapping between domains with unpaired data. The proposed method involves extracting both semantic and geometric information from input images in the form of attention maps. We assume that the multi-task network can extract semantic and geometric information during the estimation of semantic segmentation and depth maps, respectively. The image-to-image translation network integrates the two distinct types of extracted information, employing them as spatial attention maps. We compare our method with related works both qualitatively and quantitatively. The proposed method shows both qualitative and qualitative improvements in visual presentation over related work.

Integrating L1 and weighted L2 regularization for moving force identification from combined response measurements

Existing regularization methods have been proven effective in the ill-posed problem of moving force identification (MFI). However, both the constant and time-varying components of moving forces cannot be accurately identified at the same time, especially under higher measurement noises. To address this issue, a new MFI method is proposed by integrating the L1 and weighted L2 regularization methods in this study, where the presented weighted L2 regularization strategy achieves a more nuanced balance of moving force components through the incorporation of weight coefficients. Furthermore, the inherent solving characteristics of L1 and weighted L2 regularization methods are fully utilized to solve different components in moving forces separately. Numerical simulations and experimental verifications are carried out to assess the effectiveness of the proposed method. The results show that the proposed method can accurately identify moving forces even under 25% higher noise levels and it can effectively evaluate the axle loads and gross vehicle weight of the experimental vehicle. Besides, comparative studies further show that the proposed method outperforms the existing L1 and L2 regularization methods in the identification of both constant and time-varying forces, which provides a potential MFI tool for a higher accuracy and noise robustness to fix the MFI problem in practice.

Sparse Regularization Based on Orthogonal Tensor Dictionary Learning for Inverse Problems

In seismic data processing, data recovery including reconstruction of the missing trace and removal of noise from the recorded data are the key steps in improving the signal-to-noise ratio (SNR). The reconstruction of seismic data and removal of noise becomes a sparse optimization problem that can be solved by using sparse regularization. Sparse regularization is a key tool in the solution of inverse problems. They are used to introduce prior knowledge and make the approximation of ill-posed inverses feasible. It deals with ill-posedness by replacing an ill-posed inverse problem with a well-posed problem that has a solution close to the true solution. In the last 2 decades, interest has shifted from linear toward nonlinear regularization methods even for linear inverse problems. In inverse problems, regularizations serve as stabilizing the solution of ill-posed inverse problems and give a solution that adequately fits measurements without producing unjustifiably complex artifacts. In this paper, we present a novel sparse regularization based on a tensor-based dictionary method for inverse problems (seismic data interpolation and denoising). This regularization avoids the vectorization step for sparse representation of seismic data during the reconstruction process. The key step in sparsifying signals is the choice of sparsity-promoting dictionary learning. The learning-based approach can adaptively sparsify datasets but has high computational complexity and involves no prior-constraint pattern information for the dataset. Many existing dictionary learning methods would be computationally infeasible for the high dimensional seismic data processing. These methods also destroy the essential information as well as it reduces the discriminability and expressibility of the signal, since they deal with vectorization. In this paper, the orthogonal tensor dictionary learning that learns a dictionary from the input data by employing orthogonality and separability is proposed as sparse regularization for the inverse problems. The performance of the proposed method was validated in seismic data interpolation and denoising individually as well as simultaneously. Numerical examples of synthetic and real seismic datasets demonstrate the validity of the proposed method. The SNR of the recovered data confirms that the proposed method is the most effective method than K-singular value decomposition and orthogonal dictionary learning methods.

A spatially adaptive regularization based three-dimensional reconstruction network for quantitative susceptibility mapping

Objective. Quantitative susceptibility mapping (QSM) is a new imaging technique for non-invasive characterization of the composition and microstructure of in vivo tissues, and it can be reconstructed from local field measurements by solving an ill-posed inverse problem. Even for deep learning networks, it is not an easy task to establish an accurate quantitative mapping between two physical quantities of different units, i.e. field shift in Hz and susceptibility value in ppm for QSM. Approach. In this paper, we propose a spatially adaptive regularization based three-dimensional reconstruction network SAQSM. A spatially adaptive module is specially designed and a set of them at different resolutions are inserted into the network decoder, playing a role of cross-modality based regularization constraint. Therefore, the exact information of both field and magnitude data is exploited to adjust the scale and shift of feature maps, and thus any information loss or deviation occurred in previous layers could be effectively corrected. The network encoding has a dynamic perceptual initialization, which enables the network to overcome receptive field intervals and also strengthens its ability to detect features of various sizes. Main results. Experimental results on the brain data of healthy volunteers, clinical hemorrhage and simulated phantom with calcification demonstrate that SAQSM can achieve more accurate reconstruction with less susceptibility artifacts, while perform well on the stability and generalization even for severe lesion areas. Significance. This proposed framework may provide a valuable paradigm to quantitative mapping or multimodal reconstruction.

Posterior sampling with convolutional neural network-based plug-and-play regularization with applications to poststack seismic inversion

Uncertainty quantification is a crucial component in any geophysical inverse problem, as it provides decision makers with valuable information about the inversion results. Seismic inversion is a notoriously ill-posed inverse problem, due to the band-limited and noisy nature of seismic data; as such, quantifying the uncertainties associated with the ill-posed nature of this inversion process is essential for qualifying the subsequent interpretation and decision-making processes. Selecting appropriate prior information is a crucial — yet nontrivial — step in probabilistic inversion because it influences the ability of sampling-based inference algorithms to provide geologically plausible posterior samples. However, the necessity to encapsulate prior knowledge into a probability distribution can greatly limit our ability to define expressive priors. To address this limitation and following in the footsteps of the plug-and-play (PnP) methodology for deterministic inversion, we develop a regularized variational inference framework that performs posterior inference by implicitly regularizing the Kullback-Leibler divergence loss — a measure of the distance between the approximated and target probabilistic distributions — with a convolutional neural network-based denoiser. We call this new algorithm PnP Stein variational gradient descent and determine its ability to produce high-resolution trustworthy samples that realistically represent subsurface structures. Our method is validated on synthetic and field poststack seismic data.

Application of the Two-sided difference quotient in the solution of nonlinear Ill-posed inverse self-adjoint elliptic problem

When we use a discretization by finite differences, to solve differential equations we find problems at the border of the domain of the solution. If the solution is also immersed in a ill-posed inverse problem; we can find very bad solutions. In this paper we apply a discretization of two - sided difference quotients method to solve Ill-posed inverse self-adjoint elliptic problem [Kirsch(2011)]. Some numerical examples showing the effectiveness of this method and we will use mollification techniques to smooth the solutions.

Nighttime Thermal Infrared Image Translation Integrating Visible Images

Nighttime Thermal InfraRed (NTIR) image colorization, also known as the translation of NTIR images into Daytime Color Visible (DCV) images, can facilitate human and intelligent system perception of nighttime scenes under weak lighting conditions. End-to-end neural networks have been used to learn the mapping relationship between temperature and color domains, and translate NTIR images with one channel into DCV images with three channels. However, this mapping relationship is an ill-posed problem with multiple solutions without constraints, resulting in blurred edges, color disorder, and semantic errors. To solve this problem, an NTIR2DCV method that includes two steps is proposed: firstly, fuse Nighttime Color Visible (NCV) images with NTIR images based on an Illumination-Aware, Multilevel Decomposition Latent Low-Rank Representation (IA-MDLatLRR) method, which considers the differences in illumination conditions during image fusion and adjusts the fusion strategy of MDLatLRR accordingly to suppress the adverse effects of nighttime lights; secondly, translate the Nighttime Fused (NF) image to DCV image based on HyperDimensional Computing Generative Adversarial Network (HDC-GAN), which ensures feature-level semantic consistency between the source image (NF image) and the translated image (DCV image) without creating semantic label maps. Extensive comparative experiments and the evaluation metrics values show that the proposed algorithms perform better than other State-Of-The-Art (SOTA) image fusion and translation methods, such as FID and KID, which decreased by 14.1 and 18.9, respectively.

Bi-level iterative regularization for inverse problems in nonlinear PDEs

Abstract We investigate the ill-posed inverse problem of recovering unknown spatially dependent parameters in nonlinear evolution PDEs. We propose a bi-level Landweber scheme, where the upper-level parameter reconstruction embeds a lower-level state approximation. This can be seen as combining the classical reduced setting and the newer all-at-once setting, allowing us to, respectively, utilize well-posedness of the parameter-to-state map, and to bypass having to solve nonlinear PDEs exactly. Using this, we derive stopping rules for lower- and upper-level iterations and convergence of the bi-level method. We discuss application to parameter identification for the Landau-Lifshitz-Gilbert equation in magnetic particle imaging.

Imaging of nonlinear materials via the Monotonicity Principle

Inverse problems, which are related to Maxwell’s equations, in the presence of nonlinear materials is a quite new topic in the literature. The lack of contributions in this area can be ascribed to the significant challenges that such problems pose. Retrieving the spatial behavior of some unknown physical property, from boundary measurements, is a nonlinear and highly ill-posed problem even in the presence of linear materials. Furthermore, this complexity grows exponentially in the presence of nonlinear materials. In the tomography of linear materials, the Monotonicity Principle (MP) is the foundation of a class of non-iterative algorithms able to guarantee excellent performances and compatibility with real-time applications. Recently, the MP has been extended to nonlinear materials under very general assumptions. Starting from the theoretical background for this extension, we develop a first real-time inversion method for the inverse obstacle problem in the presence of nonlinear materials. The proposed method is intendend for all problems governed by the quasilinear Laplace equation, i.e. static problems involving nonlinear materials. In this paper, we provide some preliminary results which give the foundation of our method and some extended numerical examples.

Two-step reconstruction framework of fluorescence molecular tomography based on energy statistical probability.

Fluorescence molecular tomography (FMT), as a promising technique for early tumor detection, can non-invasively visualize the distribution of fluorescent marker probe three-dimensionally. However, FMT reconstruction is a severely ill-posed problem, which remains an obstacle to wider application of FMT. In this paper, a two-step reconstruction framework was proposed for FMT based on the energy statistical probability. First, the tissue structural information obtained from computed tomography (CT) is employed to associate the tissue optical parameters for rough solution in the global region. Then, according to the global-region reconstruction results, the probability that the target belongs to each region can be calculated. The region with the highest probability is delineated as region of interest to realize accurate and fast source reconstruction. Numerical simulations and in vivo experiments were carried out to evaluate the effectiveness of the proposed framework. The encouraging results demonstrate the significant effectiveness and potential of our method for practical FMT applications.