Backward Projections Research Articles

Backward and forward Wasserstein projections in stochastic order

Motivated by applications of stochastic orders in statistics and economics, we study metric projections onto cones in the Wasserstein space of probability measures, defined by stochastic orders. Duality theorems for backward and forward projections are established under mild conditions, which lay a foundation for investigating sampling of measures in stochastic orders. To demonstrate applications of these theorems beyond practice, particular attention is given to mathematical properties of convex order and subharmonic order. While backward and forward cones possess distinct geometric properties, strong connections between backward and forward projections can be obtained in the convex order case. Compared with convex order, the study of subharmonic order is subtler, particularly the existence and regularity of optimal mappings. In all cases, Brenier-Strassen type polar factorization theorems are proved, thus providing a full picture of the decomposition of optimal couplings between probability measures given by deterministic contractions (resp. expansions) and stochastic couplings. The factorization of convex order projection completes the decomposition obtained by Gozlan and Juillet, which builds a connection with Caffarelli's contraction theorem. A further noteworthy addition to earlier results is the decomposition in the subharmonic order case where the optimal mappings are characterized by volume distortion properties. To our knowledge, this is the first time in this occasion such results are available in the literature.

A Novel Thanka Image Inpainting Method with Euler’s Elastica and Iterative Denoising and Backward Projections

This paper presents a brand-new Thanka picture inpainting technique based on Euler’s elastica, iterative denoising, and backward projections (EEIDBP). Specifically, a model of Euler’s elastica is introduced to estimate the original observation due to its lower staircasing effects and better approximation of natural images. A method for backward projection and iterative denoising is applied to achieve a more accurate estimate of the original signal by alternating iterations between the estimation of the original signal and the estimation of the original observation. The experimental findings demonstrate that, in terms of a subjective assessment, the quantitative peak signal-to-noise ratio (PSNR), and the structural similarity (SSIM), the proposed technique outperforms the state-of-the-art picture inpainting methods.

A deep network-based model of hippocampal memory functions under normal and Alzheimer's disease conditions.

We present a deep network-based model of the associative memory functions of the hippocampus. The proposed network architecture has two key modules: (1) an autoencoder module which represents the forward and backward projections of the cortico-hippocampal projections and (2) a module that computes familiarity of the stimulus and implements hill-climbing over the familiarity which represents the dynamics of the loops within the hippocampus. The proposed network is used in two simulation studies. In the first part of the study, the network is used to simulate image pattern completion by autoassociation under normal conditions. In the second part of the study, the proposed network is extended to a heteroassociative memory and is used to simulate picture naming task in normal and Alzheimer's disease (AD) conditions. The network is trained on pictures and names of digits from 0 to 9. The encoder layer of the network is partly damaged to simulate AD conditions. As in case of AD patients, under moderate damage condition, the network recalls superordinate words ("odd" instead of "nine"). Under severe damage conditions, the network shows a null response ("I don't know"). Neurobiological plausibility of the model is extensively discussed.

X-ray Tomography Reconstruction Accelerated on FPGA through High-Level Synthesis Tools.

Model-Based Iterative Reconstruction (MBIR) algorithms iteratively use expensive computational operators of forward and backward projections. The irregular memory access pattern of these operators makes them a memory-bound application. Their computation time must be reduced to meet clinical routine constraints. This paper proposes a hardware accelerator architecture based on Field Programmable Gate Arrays (FPGAs) through high-level language, as an alternative to GPU architecture. This acceleration is based on an offline memory access analysis to address the main bottleneck of the algorithm and maximize the data reuse rate. The offline analysis allows for the tuning of the architecture parameters so that they converge to an optimal solution. Then, the Berkeley Roofline model guides our optimization steps by iteratively analyzing the design performance. Our design flow significantly improved the algorithm's computational intensity and overcame the memory bottleneck. Thus, our architecture takes advantage of the FPGA local memory to achieve significant memory bandwidth and efficiently harness the pipeline without stalling the computation. Furthermore, we present the scaling-up strategy from mid-range FPGA to high-end FPGA and any concerns of portability. We used two Intel FPGA devices to implement the algorithm, and then we compared the results with our GPU implementation in terms of speedup and energy efficiency. Our experimental results show that our design has achieved better computational throughput than the works on FPGA architectures reported in the literature.

Iterative Reconstruction: State-of-the-Art and Future Perspectives.

Image reconstruction processing in computed tomography (CT) has evolved tremendously since its creation, succeeding at optimizing radiation dose while maintaining adequate image quality. Computed tomography vendors have developed and implemented various technical advances, such as automatic noise reduction filters, automatic exposure control, and refined imaging reconstruction algorithms.Focusing on imaging reconstruction, filtered back-projection has represented the standard reconstruction algorithm for over 3 decades, obtaining adequate image quality at standard radiation dose exposures. To overcome filtered back-projection reconstruction flaws in low-dose CT data sets, advanced iterative reconstruction algorithms consisting of either backward projection or both backward and forward projections have been developed, with the goal to enable low-dose CT acquisitions with high image quality. Iterative reconstruction techniques play a key role in routine workflow implementation (eg, screening protocols, vascular and pediatric applications), in quantitative CT imaging applications, and in dose exposure limitation in oncologic patients.Therefore, this review aims to provide an overview of the technical principles and the main clinical application of iterative reconstruction algorithms, focusing on the strengths and weaknesses, in addition to integrating future perspectives in the new era of artificial intelligence.

A mass conservative, well balanced and positivity-preserving central scheme for shallow water equations

Based on the invariant-region-preserving (IRP) reconstruction method introduced in [Yan, Tong and Chen, Appl. Math. Comput., 436 (2023) 127500], a second order unstaggered central scheme is proposed to solve the shallow water equations with bottom topography in the framework that the bottom is discretized by a continuous, piecewise linear approximation. The reconstruction applies a modification locally on a preliminary reconstructed surface gradient in every cell to yield a convexity property of the sampled point value in the forward and backward projections. The water mass conservation is proved by rewriting the scheme in a conservation form. The modification does not change the preliminary reconstructed slope of water surface for the lake-at-rest steady state and then keeps the well-balancing property of the surface gradient method. The convexity property ensures the nonnegativity of the updated water depth under a large CFL number which yields a considerable speed-up. The numerical experiments are shown to demonstrate the robustness of the scheme.

Preterm neonates distinguish rhythm violation through a hierarchy of cortical processing

Rhythm is a fundamental component of the auditory world, present even during the prenatal life. While there is evidence that some auditory capacities are already present before birth, whether and how the premature neural networks process auditory rhythm is yet not known. We investigated the neural response of premature neonates at 30–34 weeks gestational age to violations from rhythmic regularities in an auditory sequence using high-resolution electroencephalography and event-related potentials. Unpredicted rhythm violations elicited a fronto-central mismatch response, indicating that the premature neonates detected the rhythmic regularities. Next, we examined the cortical effective connectivity underlying the elicited mismatch response using dynamic causal modeling. We examined the connectivity between cortical sources using a set of 16 generative models that embedded alternate hypotheses about the role of the frontal cortex as well as backward fronto-temporal connection. Our results demonstrated that the processing of rhythm violations was not limited to the primary auditory areas, and as in the case of adults, encompassed a hierarchy of temporo-frontal cortical structures. The result also emphasized the importance of top-down (backward) projections from the frontal cortex in explaining the mismatch response. Our findings demonstrate a sophisticated cortical structure underlying predictive rhythm processing at the onset of the thalamocortical and cortico-cortical circuits, two months before term.

Analysis of Combination Algorithms for Denoising and Contrast Enhancement Images

Reducing noise and increasing image contrast is part of the purpose of enhancing image quality; instead, it will impact change the diversity of information in the image based on the Shannon entropy value. Decrease quality caused by noise salt and pepper in this research or abnormal contrast in the image causes objects in the image to become unclear. Low contrast has a major impact on image quality, including noise reduction processes affecting image information so that the quality of the reduced image becomes something to consider for large noise. Iterative Denoising and Backward Projections with CNN (IDBP-CNN) and Different Applied Median Filter (DAMF) is a good solution for denoising a large percentage of noise with good quality results image. In other research for contrast enhancement, Triangular Fuzzy Membership-Contrast Limited Adaptive Histogram Equalization (TFM-CLAHE) and Adaptive Fuzzy Contrast Enhancement Algorithm with Details Preserving (AFCEDP) is claimed to a good solution to solve low contrast of the image. Therefore, this study is to find the best combination of denoising and contrast enhancement to get good image results with step denoising followed by contrast enhancement. Based on the experimental testing is got the best combination is the DAMF + AFCEDP algorithm with an average of PSNR 35dB and an average difference Shannon entropy of 0.0130.

The Quixotic Task of Forecasting Peaks of COVID-19: Rather Focus on Forward and Backward Projections.

Over the last months, mathematical models have been extensively used to help control the COVID-19 pandemic worldwide. Although extremely useful in many tasks, most models have performed poorly in forecasting the pandemic peaks. We investigate this common pitfall by forecasting four countries' pandemic peak: Austria, Germany, Italy, and South Korea. Far from the peaks, our models can forecast the pandemic dynamics 20 days ahead. Nevertheless, when calibrating our models close to the day of the pandemic peak, all forecasts fail. Uncertainty quantification and sensitivity analysis revealed the main obstacle: the misestimation of the transmission rate. Inverse uncertainty quantification has shown that significant changes in transmission rate commonly precede a peak. These changes are a key factor in forecasting the pandemic peak. Long forecasts of the pandemic peak are therefore undermined by the lack of models that can forecast changes in the transmission rate, i.e., how a particular society behaves, changes of mitigation policies, or how society chooses to respond to them. In addition, our studies revealed that even short forecasts of the pandemic peak are challenging. Backward projections have shown us that the correct estimation of any temporal change in the transmission rate is only possible many days ahead. Our results suggest that the distance between a change in the transmission rate and its correct identification in the curve of active infected cases can be as long as 15 days. This is intrinsic to the phenomenon and how it affects epidemic data: a new case is usually only reported after an incubation period followed by a delay associated with the test. In summary, our results suggest the phenomenon itself challenges the task of forecasting the peak of the COVID-19 pandemic when only epidemic data is available. Nevertheless, we show that exciting results can be obtained when using the same models to project different scenarios of reduced transmission rates. Therefore, our results highlight that mathematical modeling can help control COVID-19 pandemic by backward projections that characterize the phenomena' essential features and forward projections when different scenarios and strategies can be tested and used for decision-making.

Nondestructive Transmission Tomography of Pipeline Surface Defects

Based on results of theoretical analysis, a new method of transmission tomography of pipeline surface defects using ultrasound waves propagating along the pipeline walls is proposed. The waves are excited by a source and received by sensors that are located on ring-shaped lines and spaced along the tube surface. These waves propagating along the pipe surface represent either left or right handed rectilinear cylindrical waves like torsion waves. When a defect arises on the pipeline surface, the resulting waves are accompanied by changes in their shadow projections at the controlled pipe end. The problem of defect tomography is to determine locations and forms of defects using the backward projections. The physical feature of this method is that unlike traditional tomography, these projections arise on mutually intersecting left and right handed trajectories. The problem is solved by multiple joining of cylindrical pipe sections unfolded onto the plane in which all wave paths become rectilinear. A number of examples given in the paper confirm the possibility of restoration of locations and forms of various defects by the proposed method.

Image Inpainting by Low-Rank Prior and Iterative Denoising

To reconstruct the missing or damaged parts of images from observed incomplete data, some traditional methods have been researched in recent years. The iterative denoising and backward projections(IDBP)algorithm with a simple parameter mechanism have been recently introduced, which solves the typical inverse problem by utilizing the existing 3D transform-domain collaborative filtering denoising algorithm(BM3D). While this algorithm has simple parameter tuning, the collaborative hard-thresholding applied to the 3D group is greatly restricted in the procedure of denoising. In this paper, we remedy this deficiency using an iteration reweighted shrinkage denoising method. First, the model is obtained by a Plug and Play(P&P) framework. Then, we solve the optimization problem by using a proposed denoising model based on low rank prior and reweighted shrinkage and obtain a closed-form solution. Finally, the closed-form solution is operated iteratively by using the adaptive backward projection technique. Utilizing this novel strategy, the proposed algorithm not only removes the image noise and effectively recovers the degraded image, but also preserves fine structure and texture information of the image. Experimental results indicate that the proposed algorithm is competitive with some state-of-the-art inpainting algorithms in terms of both numerical evaluation and visual quality.

Back-Projection based Fidelity Term for Ill-Posed Linear Inverse Problems.

Ill-posed linear inverse problems appear in many image processing applications, such as deblurring, superresolution and compressed sensing. Many restoration strategies involve minimizing a cost function, which is composed of fidelity and prior terms, balanced by a regularization parameter. While a vast amount of research has been focused on different prior models, the fidelity term is almost always chosen to be the least squares (LS) objective, that encourages fitting the linearly transformed optimization variable to the observations. In this paper, we examine a different fidelity term, which has been implicitly used by the recently proposed iterative denoising and backward projections (IDBP) framework. This term encourages agreement between the projection of the optimization variable onto the row space of the linear operator and the pseudoinverse of the linear operator ("back-projection") applied on the observations. We analytically examine the difference between the two fidelity terms for Tikhonov regularization and identify cases (such as a badly conditioned linear operator) where the new term has an advantage over the standard LS one. Moreover, we demonstrate empirically that the behavior of the two induced cost functions for sophisticated convex and non-convex priors, such as total-variation, BM3D, and deep generative models, correlates with the obtained theoretical analysis.

Ordered subset expectation maximization algorithm for positron emission tomographic image reconstruction using belief kernels.

The aim of this study is to investigate the benefits of incorporating prior information in list mode, time-of-flight (TOF) positron emission tomography (PET) image reconstruction using the ordered subset expectation maximization (OSEM) algorithm. This investigation consists of an IEC phantom study and a patient study. For the image under reconstruction, the activity profile along a line of response is treated as a priori and is combined with the TOF measurement to define a belief kernel used for forward and backward projections during the OSEM image reconstruction. Activity profiles are smoothed and combined with the TOF kernels to control the adverse impact of noise, and different levels of smoothness are attempted. The standard TOF OSEM reconstruction is used as a baseline for comparison. Image quality is assessed using a combination of visual assessment and quantitative measurement including contrast recovery coefficients (CRC) and background variability. On the IEC phantom study, the reconstruction using belief kernels converges faster and the reconstructed images are more appealing. The CRCs for all sizes of regions of interest on images reconstructed with belief kernels are higher than those of the baseline. The background variability, measured as a coefficient of variation, is generally lower for the images reconstructed using belief kernels. Similar observations occur on the patient study. Particularly, the images reconstructed using belief kernels have better defined lesions, improved contrast, and reduced background noise. OSEM PET image reconstruction using belief kernels that combine the information from prior images and TOF measurements seems promising and worth further investigation.

Image Restoration by Iterative Denoising and Backward Projections.

Inverse problems appear in many applications, such as image deblurring and inpainting. The common approach to address them is to design a specific algorithm for each problem. The Plug-and-Play (P&P) framework, which has been recently introduced, allows solving general inverse problems by leveraging the impressive capabilities of existing denoising algorithms. While this fresh strategy has found many applications, a burdensome parameter tuning is often required in order to obtain high-quality results. In this paper, we propose an alternative method for solving inverse problems using off-the-shelf denoisers, which requires less parameter tuning. First, we transform a typical cost function, composed of fidelity and prior terms, into a closely related, novel optimization problem. Then, we propose an efficient minimization scheme with a P&P property, i.e., the prior term is handled solely by a denoising operation. Finally, we present an automatic tuning mechanism to set the method's parameters. We provide a theoretical analysis of the method and empirically demonstrate its competitiveness with task-specific techniques and the P&P approach for image inpainting and deblurring.

Actual Situation of Asbestos in Tract Drinking-Water in Korean and Japanese Local Cities

Although every year, thousands of people die from asbestos-related diseases, many people disregard things that are past. Meanwhile, a lot of people, in particular, in many Asia countries that have no guideline values for waterborne asbestos are currently in a fever of anxiety about the possibility of its health hazards. This study focused on a grasping the actual situation of asbestos in household tap water. An intensive collection of the tract drinking-water was conducted in Iksan, Korea (at six homes), and Fukuoka, Japan (at nine homes). After pretreatment, both morphological observation and elemental analysis were simultaneously carried out using a scanning electron microscopy (SEM) with an energy dispersive X-ray spectrometer (EDX). The concentrations of waterborne asbestos fiber varied from place to place in both local cities. Their average concentrations at all sites in Iksan and Fukuoka were 213.3 and 181.1 f/L, respectively. Although the measured values in this study were the sum of chrysotile, amosite, and crocidolite fibers with a high risk of cancer, they were found to be consistently below the MCL (the Maximum Contaminant Level recommended by the U.S. toxicological profile for Asbestos). A significant link was found between open water supply channels and waterborne asbestos. The backward wind trajectory projections indicated that the present water sampling sites might be affected by the airborne asbestos fiber in the upwind atmosphere.

An efficient iterative CBCT reconstruction approach using gradient projection sparse reconstruction algorithm.

The purpose of this study is to develop a fast and convergence proofed CBCT reconstruction framework based on the compressed sensing theory which not only lowers the imaging dose but also is computationally practicable in the busy clinic. We simplified the original mathematical formulation of gradient projection for sparse reconstruction (GPSR) to minimize the number of forward and backward projections for line search processes at each iteration. GPSR based algorithms generally showed improved image quality over the FDK algorithm especially when only a small number of projection data were available. When there were only 40 projections from 360 degree fan beam geometry, the quality of GPSR based algorithms surpassed FDK algorithm within 10 iterations in terms of the mean squared relative error. Our proposed GPSR algorithm converged as fast as the conventional GPSR with a reasonably low computational complexity. The outcomes demonstrate that the proposed GPSR algorithm is attractive for use in real time applications such as on-line IGRT.

Quantitative damage imaging using Lamb wave diffraction tomography**Project supported by the National Natural Science Foundation of China (Grant Nos. 11474195, 11274226, 11674214, and 51478258).

In this paper, we investigate the diffraction tomography for quantitative imaging damages of partly through-thickness holes with various shapes in isotropic plates by using converted and non-converted scattered Lamb waves generated numerically. Finite element simulations are carried out to provide the scattered wave data. The validity of the finite element model is confirmed by the comparison of scattering directivity pattern (SDP) of circle blind hole damage between the finite element simulations and the analytical results. The imaging method is based on a theoretical relation between the one-dimensional (1D) Fourier transform of the scattered projection and two-dimensional (2D) spatial Fourier transform of the scattering object. A quantitative image of the damage is obtained by carrying out the 2D inverse Fourier transform of the scattering object. The proposed approach employs a circle transducer network containing forward and backward projections, which lead to so-called transmission mode (TMDT) and reflection mode diffraction tomography (RMDT), respectively. The reconstructed results of the two projections for a non-converted S0 scattered mode are investigated to illuminate the influence of the scattering field data. The results show that Lamb wave diffraction tomography using the combination of TMDT and RMDT improves the imaging effect compared with by using only the TMDT or RMDT. The scattered data of the converted A0 mode are also used to assess the performance of the diffraction tomography method. It is found that the circle and elliptical shaped damages can still be reasonably identified from the reconstructed images while the reconstructed results of other complex shaped damages like crisscross rectangles and racecourse are relatively poor.

Combination of voxel-based and projection-based methods in terms of convergence for CT reconstruction.

Recent applications of iterative image reconstruction algorithms to multislice helical CT have shown that iterative reconstruction can significantly improve image quality and reduce artifacts. In this paper, the authors introduce a combination of two different algorithms with different convergence properties: ordered subsets separable paraboloidal surrogates (OS-SPS) and iterative coordinate descent (ICD). The first one updates image voxels simultaneously, slightly changing attenuation values iteration by iteration. The second algorithm updates image voxel by voxel, each time performing full forward and backward projections of the voxel. It has been shown that ICD converges better at high-frequency areas and requires more iterations to reconstruct low-frequency components of the image. In contrast to ICD, SPS requires multiple iterations to reconstruct high-frequency areas. In this paper, the authors introduce an algorithm which leverages the benefits of both ICD and SPS. The idea is to update the entire image with SPS, determine high-frequency components, and focus ICD computations on it using nonhomogeneous ICD update. The authors have successfully implemented OS-SPS, ICD, their hybrid approach, and few variations of ICD based on spatially nonuniform updates. The authors have examined the convergence of different algorithms and found that proposed algorithm converges better than OS-SPS, ICD, as well as various improved variants of ICD.

The Temporal Pole Top-Down Modulates the Ventral Visual Stream During Social Cognition.

The temporal pole (TP) has been associated with diverse functions of social cognition and emotion processing. Although the underlying mechanism remains elusive, one possibility is that TP acts as domain-general hub integrating socioemotional information. To test this, 26 participants were presented with 60 empathy-evoking film clips during fMRI scanning. The film clips were preceded by a linguistic sad or neutral context and half of the clips were accompanied by sad music. In line with its hypothesized role, TP was involved in the processing of sad context and furthermore tracked participants' empathic concern. To examine the neuromodulatory impact of TP, we applied nonlinear dynamic causal modeling to a multisensory integration network from previous work consisting of superior temporal gyrus (STG), fusiform gyrus (FG), and amygdala, which was extended by an additional node in the TP. Bayesian model comparison revealed a gating of STG and TP on fusiform-amygdalar coupling and an increase of TP to FG connectivity during the integration of contextual information. Moreover, these backward projections were strengthened by emotional music. The findings indicate that during social cognition, TP integrates information from different modalities and top-down modulates lower-level perceptual areas in the ventral visual stream as a function of integration demands.

Using a Cellular Automata-Markov Model to Reconstruct Spatial Land-Use Patterns in Zhenlai County, Northeast China

Decadal to centennial land use and land cover change has been consistently singled out as a key element and an important driver of global environmental change, playing an essential role in balancing energy use. Understanding long-term human-environment interactions requires historical reconstruction of past land use and land cover changes. Most of the existing historical reconstructions have insufficient spatial and thematic detail and do not consider various land change types. In this context, this paper explored the possibility of using a cellular automata-Markov model in 90 m × 90 m spatial resolution to reconstruct historical land use in the 1930s in Zhenlai County, China. Then the three-map comparison methodology was employed to assess the predictive accuracy of the transition modeling. The model could produce backward projections by analyzing land use changes in recent decades, assuming that the present land use pattern is dynamically dependent on the historical one. The reconstruction results indicated that in the 1930s most of the study area was occupied by grasslands, followed by wetlands and arable land, while other land categories occupied relatively small areas. Analysis of the three-map comparison illustrated that the major differences among the three maps have less to do with the simulation model and more to do with the inconsistencies among the land categories during the study period. Different information provided by topographic maps and remote sensing images must be recognized.