Ill-posed Research Articles

Speech emotion recognition based on bi-directional acoustic–articulatory conversion

Acoustic and articulatory signals are naturally coupled and complementary. The challenge of acquiring articulatory data and the nonlinear ill-posedness of acoustic–articulatory conversions have resulted in previous studies on speech emotion recognition (SER) primarily relying on unidirectional acoustic–articulatory conversions. However, these studies have ignored the potential benefits of bi-directional acoustic–articulatory conversion. Addressing the problem of nonlinear ill-posedness and effectively extracting and utilizing these two modal features in SER remain open research questions. To bridge this gap, this study proposes a Bi-A2CEmo framework that simultaneously addresses the bi-directional acoustic–articulatory conversion for SER. This framework comprises three components: a Bi-MGAN that addresses the nonlinear ill-posedness problem, KCLNet that enhances the emotional attributes of the mapped features, and ResTCN-FDA that fully exploits the emotional attributes of the features. Another challenge is the absence of a parallel acoustic–articulatory emotion database. To overcome this issue, this study utilizes electromagnetic articulography (EMA) to create a multi-modal acoustic–articulatory emotion database for Mandarin Chinese called STEM-E2VA. A comparative analysis is then conducted between the proposed method and state-of-the-art models to evaluate the effectiveness of the framework. Bi-A2CEmo achieves an accuracy of 89.04% in SER, which is an improvement of 5.27% compared with the actual acoustic and articulatory features recorded by the EMA. The results for the STEM-E2VA dataset show that Bi-MGAN achieves a higher accuracy in mapping and inversion than conventional conversion networks. Visualization of the mapped features before and after enhancement reveals that KCLNet reduces the intra-class spacing while increasing the inter-class spacing of the features. ResTCN-FDA demonstrates high recognition accuracy on three publicly available datasets. The experimental results show that the proposed bi-directional acoustic–articulatory conversion framework can significantly improve the SER performance.

Tikhonov regularization with conjugate gradient least squares method for large-scale discrete ill-posed problem in image restoration

Image restoration is a large-scale discrete ill-posed problem, which can be transformed into a Tikhonov regularization problem that can approximate the original image. Kronecker product approximation is introduced into the Tikhonov regularization problem to produce an alternative problem of solving the generalized Sylvester matrix equation, reducing the scale of the image restoration problem. This paper considers solving this alternative problem by applying the conjugate gradient least squares (CGLS) method which has been demonstrated to be efficient and concise. The convergence of the CGLS method is analyzed, and it is demonstrated that the CGLS method converges to the least squares solution within the finite number of iteration steps. The effectiveness and superiority of the CGLS method are verified by numerical tests.

A New Regularization Method for a Class of Ill-Posed Cauchy Problems

A New Regularization Method for a Class of Ill-Posed Cauchy Problems

Iterative algorithm computational spectrometer based on a single-hidden-layer neural network

Computational spectrometers have great application prospects in hyperspectral detection, and fast and high-precision in situ measurement is an important development trend. The computational spectrometer based on iterative algorithms has low requirements for computational resources and is easy to achieve hardware integration and in situ measurement. However, iterative algorithms are difficult to achieve high reconstruction accuracy due to the ill-posed nature of problems. Neural networks have powerful learning capabilities and can achieve high-precision spectral reconstruction. However, solely relying on neural network algorithms for reconstruction requires higher storage space and computing power from hardware devices, which makes it difficult to integrate large-scale neural network models into embedded systems. We propose using neural networks to alleviate the effect of the problem ill-posedness on the reconstruction results of iterative algorithms, so as to improve the reconstruction accuracy of the iterative algorithm computational spectrometers. First, spectral reconstruction was performed with iterative algorithms using a public spectral dataset. Then, a single-hidden-layer neural network was trained to establish a fitting relationship between the iterative algorithm spectral reconstruction results and the original spectrum. Finally, simulation and experimental results show that the proposed application of neural networks to alleviate the ill-posed problem of the iterative algorithm spectral reconstruction can effectively improve the reconstruction accuracy of iterative algorithm computational spectrometers with low computational resources. The research results may have good potential in achieving fast and high-precision in situ measurements of computational spectrometers.

Multi-scale hyperspectral recovery networks: RGB-hyperspectral imaging consistency empowered deep spectral super-resolution

Hyperspectral imaging, which analyzes a spectrum of light instead of primary colors, can provide rich physical and chemical information not regularly available to traditional imaging modalities. To physically obtain hyperspectral images, various methods have been designed relying on dispersion elements, filters, and coded masks, inevitably making the system complicated and expensive. To address these challenges, emerging deep neural networks can achieve spectral super-resolution by tackling the ill-posed problem, thereby enhancing the spectral recovery capabilities of RGB cameras. However, most of them ignore the consistency in spatial structure between the RGB and hyperspectral images, which can provide additional physical prior knowledge for spectral super-resolution tasks. To increase the fidelity of hyperspectral recovery from RGB images, we propose multi-scale hyperspectral recovery networks (MHRNs), designed to take full consideration of the consistency. In addition, physical constraints are employed to train MHRNs for better performance and interpretability. The spectral super-resolution applicability of the MHRN on publicly available datasets as well as practical applications suggests that it has great potential for advancing spectral recovery across various practical applications such as remote sensing, medical imaging, agricultural testing, and industrial inspection.

Real-GDSR: Real-World Guided DSM Super-Resolution via Edge-Enhancing Residual Network

Abstract. A low-resolution digital surface model (DSM) features distinctive attributes impacted by noise, sensor limitations and data acquisition conditions, which failed to be replicated using simple interpolation methods like bicubic. This causes super-resolution models trained on synthetic data does not perform effectively on real ones. Training a model on real low and high resolution DSMs pairs is also a challenge because of the lack of information. On the other hand, the existence of other imaging modalities of the same scene can be used to enrich the information needed for large-scale super-resolution. In this work, we introduce a novel methodology to address the intricacies of real-world DSM super-resolution, named REAL-GDSR, breaking down this ill-posed problem into two steps. The first step involves the utilization of a residual local refinement network. This strategic approach departs from conventional methods that trained to directly predict height values instead of the differences (residuals) and utilize large receptive fields in their networks. The second step introduces a diffusion-based technique that enhances the results on a global scale, with a primary focus on smoothing and edge preservation. Our experiments underscore the effectiveness of the proposed method. We conduct a comprehensive evaluation, comparing it to recent state-of-the-art techniques in the domain of real-world DSM super-resolution (SR). Our approach consistently outperforms these existing methods, as evidenced through qualitative and quantitative assessments.

Image reconstruction based on TV-L1 model for planar array capacitance defect detection

Planar array capacitive imaging is an emerging nondestructive testing (NDT) technology with broad application prospects. The severe ill-posed inverse problem of planar array capacitive imaging reduces quality of reconstructed images. In this paper, a total variation regularization model with an L1-norm (TV-L1 model) of 3 × 4 planar array capacitance imaging is established for imaging the defects detected in the inner layers of multi-layer composite components. Considering the solution instability caused by the non-differentiability of the TV-L1 model, a variable splitting self-adaptive augmented Lagrange alternating direction algorithm (VS-SALAD) is proposed. An augmented Lagrange method enhances model splitability, and a series of subproblems are obtained by variable splitting operation. The subproblems are solved iteratively using alternating minimization method with self-adaptive penalty parameter. Experimental results demonstrate that TV-L1 model effectively alleviates the ill-posed problem, and the proposed algorithm significantly improves both the quality of reconstructed images and solving speed of the TV-L1 model.

Self-supervised rigid object 3-D motion estimation from monocular video

Independent object 3D motion estimation is a fundamental problem in 3D computer vision. Directly segmenting and estimating rigid object 3D motion from a consistent video frame is an ill-posed problem. We present a self-supervised framework for segmenting moving independent rigid objects and estimating their motion information including location, driving direction, and speed from a monocular video. Specifically, we first estimate depth, optical flow, and camera pose between a pair of video frames, and then synthesize a new 3D viewpoint from this pair. Subsequently, the Motion Recurrent All-Pairs Field Transforms (MRAFT) module is introduced to extract 3D scene flow and a motion area binary mask from a pair of images and depth. After that, a Rigid Object Motion Estimation Module (ROMEM) with a slot attention mechanism is proposed to extract rigid object motion masks from a multi-layer motion field, including optical flow, depth changes, refined scene flow, and motion masks. Finally, 2D images and 3D scene reconstruction errors are used to facilitate self-supervised training for rigid object motion. Experiments on the FlyingThings3D and KITTI datasets demonstrate that our method outperforms other advanced algorithms in estimating depth, optical flow, scene flow, and rigid moving object masks, demonstrating the benefits of our approach.

Image reconstruction for electrostatic tomography with deep convolutional neural network

Electrostatic tomography (EST) has the characteristics of being radiation-free and high real-time imaging performance. However, due to its passive measurement method, EST only provides limited input data equal to the number of electrode plates, which results in more serious ill-posed inverse problem. As a result, the imaging accuracy of traditional algorithms is low. To address this problem, an improved deep learning model is proposed for image reconstruction of EST, which is based on a convolutional neural network (CNN). The proposed network is built upon the LeNet-5 deep network model, which incorporates two convolution layers to extract data features and two maximum pooling layers to perform dimension reduction. Furthermore, the LeakyReLU activation function is employed to solve the gradient vanishing problem and accelerate the training process by introducing the non-zero gradients. By constructing an EST model, 8 representative distributions were simulated and 15,930 samples were generated. After 250 iterations, more accurate reconstructed images were generated. To verify the noise immunity of the network model, Gaussian white noises of 20 dB, 30 dB, and 50 dB were added to the test set, and the proposed model could still accurately reconstruct the positions and shapes of the samples. The network is further compared with the traditional algorithms, which reconstructs better results. In addition, the reconstruction results by the proposed model are still relatively accurate for several simulated random samples of different number, shapes and positions. Finally, a 16-electrode sensor array experimental system is designed. The experimental results show that the improved CNN algorithm has a better imaging effect than the traditional algorithms and the CNN algorithm, which is consistent with the simulation results, proving that the model has good generalization ability.

On the well posedness and Steffensen’s based numerical approximation of an inverse Cauchy problem

This work is devoted to the study of an inverse Cauchy problem governed by Poisson’s equation, which is one of highly ill-posed problems in the Hadamard’s sense (Hadamard, 1953). We propose an efficient approach for solving this data completion problem, based on its reformulation into a root finding problem which is equivalent to a fixed point one involving a Steklov-Poincaré like operator (Chakib et al., 2018, Nachaoui et al., 2021). We propose first another proof of the existence of the fixed point using the Fredholm alternative. This allows us to establish also the uniqueness and the stability of the solution, in contrast to the previous work (Chakib et al., 2018, Nachaoui et al., 2021), where only the existence result was established. Then, the numerical approximation of the fixed point problem is investigated using an iterative process performed by a Steffensen’s approach (Greenbaum and Chartier, 2012) and the P1 finite element discretization. Finally, we present some numerical results showing the efficiency of the proposed approach in term of the solution accuracy and the CPU execution time.

Multiscale reweighted smoothing regularization in curvelet domain for hyperspectral image denoising

ABSTRACT Hyperspectral images are easily contaminated by multi-modal noise in the process of data acquisition. Generally, hyperspectral signatures and noise appear unexpectedly scattered in the spatial-spectral domain. Conversely, in the transform domain, the principle components of an image are gathered in the low-frequency bands, while the details and noise often occur in high-frequency parts. The existing denoising methods usually divide the whole frequency domain into two parts, that is, high frequency and low frequency, which can not make the most of the knowledge contained at different frequencies or scales, especially in the multiscale transform domain. Furthermore, aiming to restore a clean image from the noise-corrupted data, hyperspectral image denoising often amounts to an ill-posed problem, in which reshaping cubic data into matrix form results in the loss of structure information. To address the aforementioned issues, this paper incorporates the reweighted smoothing regularization in the transform domain into the noise-perturbed degradation model to reformulate a novel optimization problem. First, the multiscale curvelet is leveraged to transform the spatial matrix into frequency domain, where the coefficients are carefully designed to weight the components at different scales and orientations. Second, l 1 -norm is imposed on the abundance term to promote sparsity in the transform domain for performance enhancement. Finally, using proximal alternating optimization, an efficient algorithm is well developed to obtain the closed-form solutions to the proposed denoising problem. The experimental results on several synthetic and real datasets, illustrate that the proposed method improves PSNR by at least 1.5 dB and reduces ERGAS error by more than 35% than the state-of-the-art approaches under the multi-modal noise mixed environments, which verifies the validity of regularization terms in hyperspectral images denoising.

Feasibility of acousto-electric tomography

In acousto-electric tomography (AET) the goal is to reconstruct the electric conductivity in a domain from electrostatic boundary measurements of corresponding currents and voltages, while the domain is perturbed by a time-dependent acoustic wave, thus taking advantage of the acousto-electric effect. We approach the AET reconstruction in two steps: First, the interior power density is obtained from boundary measurements by solving a linear inverse and ill-posed problem; second, the interior conductivity is reconstructed from the power density by solving a non-linear and well-posed problem. Mathematically these inverse problems are fairly well understood, and reconstruction methods work well on synthetic data. This is in contrast to experimental findings. An effect can indeed be observed and data can be collected. However, the acousto-electric coupling is very weak, and consequently, the change in the measured voltage due to the acoustic perturbation might be too small compared to the background noise for viable reconstructions. In this paper, we take one step towards understanding the feasibility of AET. We provide an in-silico model of the coupled physics scenario based on standard models for the individual phenomena. Moreover, we formulate and implement numerically a full reconstruction method for the inverse problem via the two steps. We perform computational experiments with realistically chosen parameters from the context of medical imaging. The focus is on understanding the role of the acousto-electric coupling parameter and the signal-to-noise ratio (SNR). The critical signal strength is analyzed and the omnipresent Johnson–Nyquist noise is estimated. We obtain both positive and negative findings; we can reconstruct features even under severe noise conditions, but we also find that the SNR one is likely to face in practice is too low to obtain useful reconstructions.

Multi-Branch Network for Color Image Denoising Using Dilated Convolution and Attention Mechanisms.

Image denoising is regarded as an ill-posed problem in computer vision tasks that removes additive noise from imaging sensors. Recently, several convolution neural network-based image-denoising methods have achieved remarkable advances. However, it is difficult for a simple denoising network to recover aesthetically pleasing images owing to the complexity of image content. Therefore, this study proposes a multi-branch network to improve the performance of the denoising method. First, the proposed network is designed based on a conventional autoencoder to learn multi-level contextual features from input images. Subsequently, we integrate two modules into the network, including the Pyramid Context Module (PCM) and the Residual Bottleneck Attention Module (RBAM), to extract salient information for the training process. More specifically, PCM is applied at the beginning of the network to enlarge the receptive field and successfully address the loss of global information using dilated convolution. Meanwhile, RBAM is inserted into the middle of the encoder and decoder to eliminate degraded features and reduce undesired artifacts. Finally, extensive experimental results prove the superiority of the proposed method over state-of-the-art deep-learning methods in terms of objective and subjective performances.

Tikhonov regularization as a nonparametric method for uncertainty quantification in aggregate data problems

Dynamical systems are widely used in mathematical models in engineering and in the applied sciences. However, the parameters of these systems are usually unknown and they must be estimated from measured data. Performing parameter estimation and quantifying the uncertainty in these estimates becomes critical to realize the predictive power of dynamical systems. In some applications, only measurements about the states of an ensemble of systems are available, where each one evolves according to the same model but for different parameter values, leading to aggregate data problems. In this case, an approach proposed in the literature is to estimate a density that is consistent with some estimates of the expected value of some quantities of interest. To solve the problem in practice, the density is discretized and, to solve the resulting ill-posed problem, Tikhonov regularization is used. In this work, we propose a Bayesian model that shows this approach can be interpreted as a maximum a posteriori estimate for the density. We show that the infinite-dimensional problem defining the MAP can be reformulated as a finite-dimensional problem that does not require discretizing the density. In several cases of interest, this problem is convex, unconstrained, and the objective function is smooth. Thus, it can be solved using algorithms with optimal convergence rates. The trade-off is that the objective is defined by an integral. However, our results characterize the regularity of the integrand, allowing the use of tailored numerical schemes to approximate it. Furthermore, our theoretical results characterize the form of the optimal density, whereas our numerical results illustrate the performance of our method and confirm our theoretical findings.

Improved SQP and SLSQP algorithms for feasible path-based process optimisation

Feasible path algorithms have been widely used for process optimisation due to its good convergence. The sequential quadratic programming (SQP) algorithm is usually used to drive the feasible path algorithms towards optimality. However, existing SQP algorithms may suffer from inconsistent quadratic programming (QP) subproblems and numerical noise, especially for ill-conditioned process optimisation problems, leading to a suboptimal or infeasible solution. In this work, we propose an improved SQP algorithm (I-SQP) and an improved sequential least squares programming algorithm (I-SLSQP) that solves a least squares (LSQ) subproblem at each major iteration. A hybrid method through the combination of two existing relaxations is proposed to solve the inconsistent subproblems for better convergence and higher efficiency. We find that a certain part of the dual LSQ algorithm suffers from serious cancellation errors, resulting in an inaccurate search direction or no viable search direction generated. Therefore, the QP solver is used to solve LSQ subproblems in such a situation. The computational results indicates that I-SLSQP is more robust than fmincon in MATLAB, IPOPT, Py-SLSQP and I-SQP. It is also shown that I-SLSQP and Py-SLSQP is superior to I-SQP for ill-conditioned process optimisation problems, whilst I-SQP is more computationally efficient than I-SLSQP and Py-SLSQP for well-conditioned problems.

THREE-COMPONENT VERSION OF THE TIKHONOV REGULARIZATION METHOD FOR OPERATOR EQUATIONS OF THE FIRST KIND

t An ill-posed problem in the form of a linear operator equation is considered. It is assumed that the solution to the equation in the one-dimensional case can be represented in the form of a sum of three components: the first component contains discontinuities, the second contains discontinuities in the derivative, and the third is continuous. To construct a stable approximate solution, the three-component Tikhonov method is used. In this case, the stabilizer is the sum of three functionals: BVp-norm of the first component, BVp-norm of the derivative for the second component and the norm of the Sobolev space for the third component, and each functional depends on only one component. The convergence of the sum of regularized components to the solution of the original equation is proved. In addition, piecewise uniform convergence of approximate solutions is established. The results of numerical experiments on reconstructing a three-component model solution for the Fredholm equation of the first kind are presented.

SUSHI: An algorithm for source separation of hyperspectral images with non-stationary spectral variation

Context. Hyperspectral images are data cubes with two spatial dimensions and a third spectral dimension, providing a spectrum for each pixel, and thus allowing the mapping of extended sources’ physical properties. Aims. In this article, we present the Semi-blind Unmixing with Sparsity for Hyperspectral Images (SUSHI), an algorithm for non-stationary unmixing of hyperspectral images with spatial regularization of spectral parameters. The method allows for the disentangling of physical components without the assumption of a unique spectrum for each component. Thus, unlike most source separation methods used in astrophysics, all physical components obtained by SUSHI vary in spectral shape and in amplitude across the data cube. Methods. Non-stationary source separation is an ill-posed inverse problem that needs to be constrained. We achieve this by training a spectral model and applying a spatial regularization constraint on its parameters. For the spectral model, we used an Interpolatory Auto-Encoder, a generative model that can be trained with limited samples. For spatial regularization, we applied a sparsity constraint on the wavelet transform of the model parameter maps. Results. We applied SUSHI to a toy model meant to resemble supernova remnants in X-ray astrophysics, though the method may be used on any extended source with any hyperspectral instrument. We compared this result to the one obtained by a classic 1D fit on each individual pixel. We find that SUSHI obtains more accurate results, particularly when it comes to reconstructing physical parameters. We then applied SUSHI to real X-ray data from the supernova remnant Cassiopeia A and to the Crab Nebula. The results obtained are realistic and in accordance with past findings but have a much better spatial resolution. Thanks to spatial regularization, SUSHI can obtain reliable physical parameters at fine scales that are out of reach for pixel-by-pixel methods.

ON THE PROPERTIES OF A FAST ALGORITHM FOR SOLVING A THREE-DIMENSIONAL INVERSE PROBLEM OF SCALAR ACOUSTICS

t Theinverse single-frequency problem of scalar acoustics in three-dimensional space is considered. It consists in determining the characteristics of acoustic inhomogeneities lying in a flat layer from the distribution of the complex amplitude of the acoustic field in a flat recording layer. To effectively solve it, a special numerical algorithm is used, previously proposed and justified by the authors. As its component part, the algorithm uses solutions to special one-dimensional Fredholm equations of the 1st kind, and in the case of ambiguity in the solution of the latter, their normal solutions (solutions with a minimum norm) are sought. To regularize this ill-posed inverse problem, Tikhonov regularization and the TSVD method are used. A systematic numerical study is carried out of the influence of various parameters in the data recording scheme on the accuracy of the approximate solutions obtained using the algorithm. In particular, we study the dependence of this accuracy on the position of point sources and composite sources that cause sound vibrations, its dependence on the position of the data recording layer, and on the number of planes in which the recording sensors lie. It is shown that acceptable accuracy of approximate solutions can be obtained even with two such layers. In addition, an approach to the numerical study of the ambiguity of the solution to the inverse problem under consideration is proposed. It is based on special theoretical statements given in the article, and on the same numerical algorithm for solving the inverse problem. The approach is focused on comparing several normal solutions of one-dimensional integral equations from the problem under consideration. These normal solutions are calculated with respect to various elements. An example of using this theory to numerically estimate the non-uniqueness of a solution is given.

The Regularized Block GMERR Method and Its Simpler Version for Solving Large-Scale Linear Discrete Ill-Posed Problems

The Regularized Block GMERR Method and Its Simpler Version for Solving Large-Scale Linear Discrete Ill-Posed Problems

Existence and uniqueness of well-posed fractional boundary value problem.

In this paper, the existence and uniqueness of solution for a fractional differential model involving well-posed boundary conditions and implicit fractional differential equation is considered. The desired goals are achieved by using Banach contraction principle and Scheafer's fixed point theorem. To show the results applicability some examples are presented. The basic mathematical concept of well-posed fractional boundary value issues is investigated in this study. It dives into the existence and uniqueness of these difficulties, offering light on the conditions that allow for both practical and singular solutions. This study contributes to a better knowledge of fractional calculus and its applications in a variety of scientific and technical areas, giving significant insights for both scholars and practitioners.