Mathematical Regularization Research Articles

Quantification and mitigation of the effect of resynchronization errors in ultrasound sensor network for passive imaging in elastic plates.

The problem of signal desynchronization in passive imaging based on noise correlation for defect detection in elastic plates is investigated. Although a post-processing resynchronization process relying on the symmetry of noise correlation functions can be applied prior to the imaging algorithm, perfect synchronization might not be achieved experimentally. Effect of such residual synchronization errors on the defect detection performance is quantified as a function of their probability density function. A mathematical regularization process is then proposed to reduce the standard deviation of the resynchronization errors by a factor of N-1/N (N is the number of sensors), which results in a significant improvement in the detection performance. Finally, these theoretical results are validated through a simple flexural-wave propagation simulation.

A large deformation gradient theory for glassy polymers by means of micromorphic regularization

AbstractCold forming of polycarbonate films results in the formation of shear bands in the necking zone. The numerical results obtained from standard viscoplastic material models exhibit mesh size dependency, requiring mathematical regularization. For this purpose, we present in this work a large deformation gradient theory for a viscoplastic isotropic material model published before. We extend our model to a micromorphic model by introducing a new micromorphic variable as an additional degree of freedom along with its first gradient. This variable represents a microequivalent plastic strain. The relation between the macroequivalent plastic strain and the micromorphic variable is accomplished by a micromorphic coupling modulus. This coupling forces proximity between the macro- and microvariables, leading to the targeted regularization effect. The micromorphic model is implemented as a three-dimensional initial boundary value problem in an in-house finite element tool. The analysis is performed for both uniaxial and biaxial specimens. The provided numerical examples show the ability of our model to regularize shear bands within the specimens and address the issue of localization.

A finite strain gradient theory for viscoplasticity by means of micromorphic regularization

AbstractStretching of polycarbonate films leads to the formation of shear bands in the necking zone [1]. Standard viscoplastic material models render mesh size dependent results, which requires a mathematical regularization. To this end, we present a finite strain gradient theory for a viscoplastic, isotropic material model where we extend the model presented in [2] to a micromorphic model by introducing a new micromorphic variable as an additional degree of freedom with its first gradient [3, 4]. The variable here has the meaning of a micro plastic strain, and is coupled with the macro plastic by a micro penalty term, forcing the macro‐plastic strain to be close to the micro‐plastic strain for the targeted shear band regularization effect. We have implemented the model equations as a three dimensional initial boundary value problem in an in house FE‐tool, to simulate different geometries with different thickness and to compare it the experimental tests. The analysis is performed for a uniaxial tensile geometry as well as for a biaxial tensile geometry. The numerical examples show the ability of the model to regularize the shear bands and solve the problem of localization.

Hyperspectral Change Detection Using Semi-Supervised Graph Neural Network and Convex Deep Learning

With recent advances in hyperspectral remote sensing, hyperspectral change detection (HCD) methods have been developed for precision agriculture and land cover/use monitoring. Among the hyperspectral techniques, those based on convex optimization (CO) and deep learning (DE) have gained particular attentions. However, DE often relies on a vast amount of training data (big data) and time-consuming manual labeling tasks, in order to learn the inherent patterns of hyperspectral images (HSIs). On the other hand, CO typically requires sophisticated mathematical regularization terms for effectively and adaptively addressing the ill-posed HCD inverse problem. Considering these challenges, we employ the convex deep (CODE) small-data learning theory recently invented for hyperspectral satellite remote sensing, and propose a semi-supervised graph neural network to achieve very low labeling rates. Furthermore, the proposed CODE-HCD method exploits hyperspectral data affine geometry to design a convex <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q</i> -quadratic norm regularizer, whose mathematical form is very simple and thereby ensures a computationally efficient algorithm. The superiority of CODE-HCD over benchmark methods will be demonstrated on several real-world hyperspectral datasets.

ADMM-ADAM: A New Inverse Imaging Framework Blending the Advantages of Convex Optimization and Deep Learning

Alternating direction method of multipliers (ADMM) and adaptive moment estimation (ADAM) are two optimizers of paramount importance in convex optimization (CO) and deep learning (DL), respectively. Numerous state-of-the-art algorithms for solving inverse problems are achieved by carefully designing a convex criterion, typically composed of a data-fitting term and a regularizer. Even when the regularizer is convex, its mathematical form is often sophisticated, hence inducing a math-heavy optimization procedure and making the algorithm design a daunting task for software engineers. Probably for this reason, people turn to solve the inverse problems via DL, but this requires big data collection, quite time-consuming if not impossible. Motivated by these facts, we propose a new framework, termed as ADMM-ADAM, for solving inverse problems. As the key contribution, even just with small/single data, the proposed ADMM-ADAM is able to exploit DL to obtain a convex regularizer of very simple math form, followed by solving the regularized criterion using simple CO algorithm. As a side contribution, a state-of-the-art hyperspectral inpainting algorithm is designed under ADMM-ADAM, demonstrating its superiority even without the aid of big data or sophisticated mathematical regularization.

An interface-regularizing model for compressible three-fluid flows with interfacial tensions

We present a diffuse-interface model for simulating compressible three-fluid flows with surface tension. In this model different fluids are separated with resolved interfaces. We first deduce the jump conditions across the material interfaces under surface tension. The interfaces are represented with the advection equations for volume fractions. Traditional methods for solving these equations result in serious numerical smearing of interface surfaces. To minimize this numerical effect, we introduce mathematical regularization of the governing equations by implementing an artificial compression method. The mathematical regularization is introduced in such a way that the jump conditions across the material interface maintain unaltered. Further we demonstrate that for the three-fluid formulation the direct implementation of the artificial compression method for each volume fraction can violate the non-negativity constraints for volume fractions. To cope with this problem, we propose a simple and robust method to ensure the volume fraction constraints. In this method instead of the original advection equations for volume fractions, we solve the advection equation with an additional compression term for new characteristic functions in the form of non-linear combinations of volume fractions. Moreover, a HLLC solver for solving the model equations is developed. The numerical results of several benchmark problems display the efficiency of the proposed algorithm.

The effect of plasticity on damage evolution using a relaxation-based material model

Abstract As damage occurs in the context of high stresses that are also related to the presence of plastic strains, it is natural to investigate the effect of plasticity on damage evolution and to thus achieve a more realistic model. In this work, the existing and new damage model presented in [Junker P, Schwarz S, Makowski J, Hackl K. Continuum Mech. Therm. 2017, 29 (1), 291–310] is enhanced with plasticity and isotropic hardening. The damage model is based on a relaxation-based approach and does not require additional complex regularization techniques besides considering viscous effects. The benefit of the model are mesh-independent results for the rate-dependent case, even without considering, e.g. gradient terms for mathematical regularization. The enhancement with plasticity and isotropic hardening was investigated for a representative volume element that considerd a damaging matrix material and non-damaging hard precipitates. Two different loading types, pure tension and pure shear, yielded the homogenized stress/strain response for the material at various loading rates. Hereto, several finite discretizations in terms of finite-element meshes were used. The results underline the mesh-independence for physically reasonable loading rates and viscosities.

A parameter-free method to extract the superconductor’s Jc(B,θ) field-dependence from in-field current–voltage characteristics of high temperature superconductor tapes

The estimation of the critical current (Ic) and AC losses of high-temperature superconductor devices through modeling and simulation requires the knowledge of the critical current density (Jc) of the superconducting material. This Jc is in general not constant and depends both on the magnitude (Bloc) and the direction (θ, relative to the tape) of the local magnetic flux density. In principle, Jc(Bloc,θ) can be obtained from the experimentally measured critical current Ic(Ba,θ), where Ba is the magnitude of the applied magnetic field. However, for applications where the superconducting materials experience a local field that is close to the self-field of an isolated conductor, obtaining Jc(Bloc,θ) from Ic(Ba,θ) is not a trivial task. It is necessary to solve an inverse problem to correct for the contribution derived from the self–field. The methods presented in the literature comprise a series of approaches dealing with different degrees of mathematical regularization to fit the parameters of preconceived nonlinear formulas by means of brute force or optimization methods. In this contribution, we present a parameter-free method that provides excellent reproduction of experimental data and requires no human interaction or preconception of the Jc dependence with respect to the magnetic field. In particular, it allows going from the experimental data to a ready-to-run Jc(Bloc,θ) model in a few minutes.

Complex optimization for big computational and experimental neutron datasets

We present a framework to use high performance computing to determine accurate solutions to the inverse optimization problem of big experimental data against computational models. We demonstrate how image processing, mathematical regularization, and hierarchical modeling can be used to solve complex optimization problems on big data. We also demonstrate how both model and data information can be used to further increase solution accuracy of optimization by providing confidence regions for the processing and regularization algorithms. We use the framework in conjunction with the software package SIMPHONIES to analyze results from neutron scattering experiments on silicon single crystals, and refine first principles calculations to better describe the experimental data.

Novel methods to select environmental variables in MaxEnt: A case study using invasive crayfish

The popularity of MaxEnt in species distribution modeling has been driven by several factors including its high degree of accuracy, and flexibility to tailor efforts to species-specific situations. Although many recent studies have identified the importance of adjusting mathematical transformation (feature class) and regularization of coefficient values, collectively known as tuning, few studies have addressed the need to customize the variables used in species distribution modeling, and use unselected variable sets. This study presents two novel methods to select for environmental variables in MaxEnt. The first involves selecting from a priori determined environmental variable sets (pre-selected based on ecological or biological knowledge), and the second utilizes a reiterative process of model formation and stepwise removal of least contributing variables. Both methods were tested on eight known species of invasive crayfish, with results reinforcing the need for species-specific environmental variable sets. While the reiterative process generally performs better than the a priori selected variables, selection of method can be based on information availability. These techniques appear to outperform the current practice of utilizing unselected variable sets and is especially important considering the increasing application of species distribution modeling (across spatial and temporal barriers) in conservation and management efforts whereby inaccurate predictions might have adverse effects.

A PDE-Based Regularization Algorithm Toward Reducing Speckle Tracking Noise: A Feasibility Study for Ultrasound Breast Elastography.

Obtaining accurate ultrasonically estimated displacements along both axial (parallel to the acoustic beam) and lateral (perpendicular to the beam) directions is an important task for various clinical elastography applications (e.g., modulus reconstruction and temperature imaging). In this study, a partial differential equation (PDE)-based regularization algorithm was proposed to enhance motion tracking accuracy. More specifically, the proposed PDE-based algorithm, utilizing two-dimensional (2D) displacement estimates from a conventional elastography system, attempted to iteratively reduce noise contained in the original displacement estimates by mathematical regularization. In this study, tissue incompressibility was the physical constraint used by the above-mentioned mathematical regularization. This proposed algorithm was tested using computer-simulated data, a tissue-mimicking phantom, and in vivo breast lesion data. Computer simulation results demonstrated that the method significantly improved the accuracy of lateral tracking (e.g., a factor of 17 at 0.5% compression). From in vivo breast lesion data investigated, we have found that, as compared with the conventional method, higher quality axial and lateral strain images (e.g., at least 78% improvements among the estimated contrast-to-noise ratios of lateral strain images) were obtained. Our initial results demonstrated that this conceptually and computationally simple method could be useful for improving the image quality of ultrasound elastography with current clinical equipment as a post-processing tool.

Damage Detection in Long Suspension Bridges Using Stress Influence Lines

AbstractNumerous long-span cable-supported bridges have been built throughout the world in recent years. These bridges begin to deteriorate once built and continuously accumulate damage during their long service life. The growing popularity of comprehensive structural health monitoring systems (SHMSs) in recently built long-span bridges has started a new trend of integrating SHMS and damage detection technology for real-time condition assessment of these bridges. This paper explores a novel damage detection technique based on stress influence lines (SILs) of bridge components and validates the efficacy of the technique through a case study of the Tsing Ma suspension bridge. A mathematical regularization method is first introduced to identify SILs based on the in situ measurement of train information and train-induced stress responses in local bridge components. Good agreement between the identified and baseline SILs demonstrates the effectiveness of the proposed identification method. Damage indexes based...

Radiation dose reduction in medical x‐ray CT via Fourier‐based iterative reconstruction

A Fourier-based iterative reconstruction technique, termed Equally Sloped Tomography (EST), is developed in conjunction with advanced mathematical regularization to investigate radiation dose reduction in x-ray CT. The method is experimentally implemented on fan-beam CT and evaluated as a function of imaging dose on a series of image quality phantoms and anonymous pediatric patient data sets. Numerical simulation experiments are also performed to explore the extension of EST to helical cone-beam geometry. EST is a Fourier based iterative algorithm, which iterates back and forth between real and Fourier space utilizing the algebraically exact pseudopolar fast Fourier transform (PPFFT). In each iteration, physical constraints and mathematical regularization are applied in real space, while the measured data are enforced in Fourier space. The algorithm is automatically terminated when a proposed termination criterion is met. Experimentally, fan-beam projections were acquired by the Siemens z-flying focal spot technology, and subsequently interleaved and rebinned to a pseudopolar grid. Image quality phantoms were scanned at systematically varied mAs settings, reconstructed by EST and conventional reconstruction methods such as filtered back projection (FBP), and quantified using metrics including resolution, signal-to-noise ratios (SNRs), and contrast-to-noise ratios (CNRs). Pediatric data sets were reconstructed at their original acquisition settings and additionally simulated to lower dose settings for comparison and evaluation of the potential for radiation dose reduction. Numerical experiments were conducted to quantify EST and other iterative methods in terms of image quality and computation time. The extension of EST to helical cone-beam CT was implemented by using the advanced single-slice rebinning (ASSR) method. Based on the phantom and pediatric patient fan-beam CT data, it is demonstrated that EST reconstructions with the lowest scanner flux setting of 39 mAs produce comparable image quality, resolution, and contrast relative to FBP with the 140 mAs flux setting. Compared to the algebraic reconstruction technique and the expectation maximization statistical reconstruction algorithm, a significant reduction in computation time is achieved with EST. Finally, numerical experiments on helical cone-beam CT data suggest that the combination of EST and ASSR produces reconstructions with higher image quality and lower noise than the Feldkamp Davis and Kress (FDK) method and the conventional ASSR approach. A Fourier-based iterative method has been applied to the reconstruction of fan-bean CT data with reduced x-ray fluence. This method incorporates advantageous features in both real and Fourier space iterative schemes: using a fast and algebraically exact method to calculate forward projection, enforcing the measured data in Fourier space, and applying physical constraints and flexible regularization in real space. Our results suggest that EST can be utilized for radiation dose reduction in x-ray CT via the readily implementable technique of lowering mAs settings. Numerical experiments further indicate that EST requires less computation time than several other iterative algorithms and can, in principle, be extended to helical cone-beam geometry in combination with the ASSR method.

Model-augmented methods for estimation of contact pressure distribution

On-line measurement of the contact pressure distribution (CPD) at the tool–workpiece interface during sheet metal stamping is critical to advancing the state-of-art in tool wear and product quality monitoring. Since the number of sensors that can be integrated into a tool structure is limited by concerns of structural integrity and cost, estimation of CPD through a small number of sparsely located sensors has created unique challenges in information acquisition and representation. Specifically, the problem of determining continuous CPD from discrete sensor measurements is under-constrained and thus ill-posed. A mathematical framework is needed for treating such a problem. This paper presents three mathematical approaches Regularization, Kriging, and Spatial Blending to address this problem and discusses their relative merits and limitations.

Efficient modeling and compensation of ultrasound attenuation losses in photoacoustic imaging

Wave attenuation can be considered from the dual viewpoints of assuming a complex wavenumber or a complex frequency. Experimentally, the first viewpoint is preferable if the wave signal can be measured over time, and the second is preferable if the wave signal can be measured over space. These two approaches are discussed in the context of photoacoustic imaging where a short laser pulse excites a broadband ultrasound signal in a sample (e.g., some biological tissue) which can be recorded by detectors configured around the target. Reconstruction of the initial pressure distribution from the detector signals clearly poses an inverse problem. For the complex frequency viewpoint the damping rates of the spatial Fourier modes are calculated using Szabo's wave equation which describes ultrasound propagation in attenuating media obeying a frequency power law. For a symmetric sample problem, a mathematical regularization method is applied to compensate for attenuation losses. It is shown for this important special case that with the complex frequency approach regularization can be performed faster and with more accurate results.

Low-dose x-ray phase-contrast and absorption CT using equally sloped tomography

Tomographic reconstruction from undersampled and noisy projections is often desirable in transmission CT modalities for purposes of low-dose tomography and fast acquisition imaging. However under such conditions, due to the violation of the Nyquist sampling criteria and the presence of noise, reconstructions with acceptable accuracy may not be possible. Recent experiments in transmission electron tomography and coherent diffraction microscopy have shown that the technique of equally sloped tomography (EST), an exact tomographic method utilizing an oversampling iterative Fourier-based reconstruction, provides more accurate image reconstructions when the number of projections is significantly undersampled relative to filtered back projection and algebraic iterative methods. Here we extend this technique by developing new reconstruction algorithms which allow for the incorporation of advanced mathematical regularization constraints, such as the nonlocal means total variational model, in a manner that is consistent with experimental projections. We then evaluate the resulting image quality of the developed algorithm through simulations and experiments at the European Synchrotron Radiation Facility on image quality phantoms using the x-ray absorption and phase contrast CT modalities. Both our simulation and experimental results have indicated that the method can reduce the number of projections by 60–75% in parallel beam modalities, while achieving comparable or better image quality than the conventional reconstructions. As large-scale and compact synchrotron radiation facilities are currently under rapid development worldwide, the implementation of low-dose x-ray absorption and phase-contrast CT can find broad applications in biology and medicine using these advanced x-ray sources.

TU-B-201B-01: Radiation Dose Reduction in Medical X-Ray CT Using Equally-Sloped Tomography

Purpose: An exact Fourier‐based iterative reconstruction technique termed Equally‐Sloped Tomography (EST) has recently been developed to allow a reduction of radiation dose while maintaining image quality. Here we optimize this technique by incorporating a Gradient Descent algorithm with mathematical regularization constraints, including the Non‐Local Means Total Variation model. Using this technique, we conducted a series of comparative experiments to quantify the image quality, and explore radiation dose reduction in CT through reduction of X‐ray flux. Method and Materials: A standard quality control phantom containing multiple contrastvisibility and resolution inserts was imaged using a Siemens Somatom Sensation 64 scanner. The X‐ray flux was systematically lowered from 583mAs to 39mAs. All scans were performed under axial mode with constant tube current. Comparison reconstructions were produced by Filtered‐Back Projection (FBP) with a standard uncropped ramp filter. The resolution, signal‐to‐noise ratio (SNR) and contrast‐to‐noise ratio (CNR) were measured and compared as a function of flux. Results: Compared to the FBP reconstruction of 500mAs, the optimized EST reconstruction of 50mAs has comparable SNRs and CNRs. The low‐contrast regions of the phantom exhibit similar visibility. In addition, no resolution degradation is observed in the low‐dose EST reconstruction.Conclusion: As radiation dose is linearly proportional to the X‐ray flux, our results indicate that the optimized EST technique may potentially reduce the CT dose up to 90%, while producing comparable image resolution, quality and contrast as the full‐dose FBP reconstruction. The reasons that EST allows for significant reduction of the CT dose are because (i) an exact Fourier‐based iterative algorithm is used in the reconstruction where both physical and mathematical constraints can be naturally incorporated, (ii) the algorithm is guided towards a lowest‐possible noisy state, which is also strictly consistent with the measured data. Conflict of Interest: Research sponsored by UC Discovery / TomoSoft Technologies Grant #ITL07‐10166.

A multipole-expansion based linear sampling method for solving inverse scattering problems

Linear sampling method (LSM) is a qualitative method used to reconstruct the support of scatterers. This paper presents a modification of the LSM approach. The proposed method analyses the multipole expansion of the scattered field. Only monopole and dipole terms are used for the reconstruction of the scatterer support and all other higher order multipoles are truncated. It is shown that such modification performs better than the mathematical regularization typically used in LSM. The justification for truncation of higher order multipoles is presented. Various examples are presented to demonstrate the performance of the proposed method for dielectric as well as perfectly conducting scatterers in presence of significant amount of additive Gaussian noise.

Regularization of ill-conditioned problems in geometrical measurements

In measurements of small segments of geometrical objects such as circles, spheres, cylinders etc., object parameters are defined with large errors, that can essentially exceed both errors in the part manufacturing and errors in measurements. That's why such measurements could not be performed on coordinate measurement machines (CMM), but in the same time there is a great requirement in them in industry. Large errors are caused by ill-conditioned problem of small segments measurements. The method allowing to improve conditionality of considered problems by their mathematical regularization and to raise decision accuracy is described in this article. Offered method expands CMM possibilities, allowing to perform measurements that were not performed on them earlier.

An advanced regularization methodology for use in watershed model calibration

A calibration methodology based on an efficient and stable mathematical regularization scheme is described. This scheme is a variant of so-called Tikhonov regularization in which the parameter estimation process is formulated as a constrained minimization problem. Use of the methodology eliminates the need for a modeler to formulate a parsimonious inverse problem in which a handful of parameters are designated for estimation prior to initiating the calibration process. Instead, the level of parameter parsimony required to achieve a stable solution to the inverse problem is determined by the inversion algorithm itself. Where parameters, or combinations of parameters, cannot be uniquely estimated, they are provided with values, or assigned relationships with other parameters, that are decreed to be realistic by the modeler. Conversely, where the information content of a calibration dataset is sufficient to allow estimates to be made of the values of many parameters, the making of such estimates is not precluded by preemptive parsimonizing ahead of the calibration process. White Tikhonov schemes are very attractive and hence widely used, problems with numerical stability can sometimes arise because the strength with which regularization constraints are applied throughout the regularized inversion process cannot be guaranteed to exactly complement inadequacies in the information content of a given calibration dataset. A new technique overcomes this problem by allowing relative regularization weights to be estimated as parameters through the calibration process itself. The technique is applied to the simultaneous calibration of five subwatershed models, and it is demonstrated that the new scheme results in a more efficient inversion, and better enforcement of regularization constraints than traditional Tikhonov regularization methodologies. Moreover, it is argued that a joint calibration exercise of this type results in a more meaningful set of parameters than can be achieved by individual subwatershed model calibration. (c) 2005 Elsevier B.V. All rights reserved.