General Diffusion Model Research Articles

DiffMAR: A Generalized Diffusion Model for Metal Artifact Reduction in CT Images.

X-ray imaging frequently introduces varying degrees of metal artifacts to computed tomography (CT) images when metal implants are present. For the metal artifact reduction (MAR) task, existing end-to-end methods often exhibit limited generalization capabilities. While methods based on multiple iterations often suffer from accumulative error, resulting in lower-quality restoration outcomes. In this work, we innovatively present a generalized diffusion model for Metal Artifact Reduction (DiffMAR). The proposed method utilizes a linear degradation process to simulate the physical phenomenon of metal artifact formation in CT images and directly learn an iterative restoration process from paired CT images in the reverse process. During the reverse process of DiffMAR, a Time-Latent Adjustment (TLA) module is designed to adjust time embedding at the latent level, thereby minimizing the accumulative error during iterative restoration. We also designed a structure information extraction (SIE) module to utilize linear interpolation data in the image domain, guiding the generation of anatomical structures during the iterative restoring. This leads to more accurate and robust shadow-free image generation. Comprehensive analysis, including both synthesized data and clinical evidence, confirms that our proposed method surpasses the current state-of-the-art (SOTA) MAR methods in terms of both image generation quality and generalization.

Using a patient-specific diffusion model to generate CBCT-based synthetic CTs for CBCT-guided adaptive radiotherapy.

Cone beam computed tomography (CBCT) can be used to evaluate the inter-fraction anatomical changes during the entire course for image-guided radiotherapy (IGRT). However, CBCT artifacts from various sources restrict the full application of CBCT-guided adaptive radiation therapy (ART). Inter-fraction anatomical changes during ART, including variations in tumor size and normal tissue anatomy, can affect radiation therapy (RT) efficacy. Acquiring high-quality CBCT images that accurately capture patient- and fraction-specific (PFS) anatomical changes is crucial for successful IGRT. To enhance CBCT image quality, we proposed PFS lung diffusion models (PFS-LDMs). The proposed PFS models use a pre-trained general lung diffusion model (GLDM) as a baseline, which is trained on historical deformed CBCT (dCBCT)-planning CT (pCT) paired data. For a given patient, a new PFS model is fine-tuned on a CBCT-deformed pCT (dpCT) pair after each fraction to learn the PFS knowledge for generating personalized synthetic CT (sCT) with quality comparable to pCT or dpCT. The learned PFS knowledge is the specific mapping relationships, including personal inter-fraction anatomical changes between personalized CBCT-dpCT pairs. The PFS-LDMs were evaluated on an institutional lung cancer dataset, quantified by mean absolute error (MAE), peak signal-to-noise ratio (PSNR), normalized cross-correlation (NCC), and structural similarity index measure (SSIM) metrics. We also compared our PFS-LDMs with a mainstream GAN-based model, demonstrating that our PFS fine-tuning strategy could be applied to existing generative models. Our models showed remarkable improvements across all four evaluation metrics. The proposed PFS-LDMs outperformed the GLDM, demonstrating the effectiveness of our proposed fine-tuning strategy. The PFS model fine-tuned with CBCT images from four prior fractions, reduced the MAE from 103.95 to 15.96 Hounsfield units (HU), and increased the mean PSNR, NCC, and SSIM from 25.36dB to 33.57dB, 0.77 to 0.98, and 0.75 to 0.97, respectively. Applying our PFS fine-tuning strategy to a Cycle GAN model also showed improvements, with all four fine-tuned PFS Cycle GAN (PFS-CG) models outperforming the general Cycle GAN model. Overall, our proposed PFS fine-tuning strategy improved CBCT image quality compared to both the pre-correction and non-fine-tuned general models, with our proposed PFS-LDMs yielding better performance than the GAN-based model across all metrics. Our proposed PFS-LDMs significantly improve CBCT image quality with increased HU accuracy and fewer artifacts, thus better capturing inter-fraction anatomical changes. This lays the groundwork for enabling CBCT-based ART, which could enhance clinical efficiency and achieve personalized high-precision treatment by accounting for inter-fraction anatomical changes.

Optimal Market Completion through Financial Derivatives with Applications to Volatility Risk

This paper investigates the optimal choices of financial derivatives to complete a financial market in the framework of stochastic volatility (SV) models. We first introduce an efficient and accurate simulation-based method applicable to generalized diffusion models to approximate the optimal derivatives-based portfolio strategy. We build upon a double optimization approach, i.e., expected utility maximization and risk exposure minimization, already proposed in the literature, demonstrating that strangle options are the best choices for market completion within equity options. They lead to lower investors’ risk exposure for a wide range of strikes compared to the lesser flexibility of calls, puts, and strangles. Furthermore, we explore the benefit of using volatility index derivatives and conclude that they could be more convenient substitutes when short-term maturity equity options are not available.

Learning to reconstruct accelerated MRI through K-space cold diffusion without noise

Deep learning-based MRI reconstruction models have achieved superior performance these days. Most recently, diffusion models have shown remarkable performance in image generation, in-painting, super-resolution, image editing and more. As a generalized diffusion model, cold diffusion further broadens the scope and considers models built around arbitrary image transformations such as blurring, down-sampling, etc. In this paper, we propose a k-space cold diffusion model that performs image degradation and restoration in k-space without the need for Gaussian noise. We provide comparisons with multiple deep learning-based MRI reconstruction models and perform tests on a well-known large open-source MRI dataset. Our results show that this novel way of performing degradation can generate high-quality reconstruction images for accelerated MRI.

CoreDiff: Contextual Error-Modulated Generalized Diffusion Model for Low-Dose CT Denoising and Generalization.

Low-dose computed tomography (CT) images suffer from noise and artifacts due to photon starvation and electronic noise. Recently, some works have attempted to use diffusion models to address the over-smoothness and training instability encountered by previous deep-learning-based denoising models. However, diffusion models suffer from long inference time due to a large number of sampling steps involved. Very recently, cold diffusion model generalizes classical diffusion models and has greater flexibility. Inspired by cold diffusion, this paper presents a novel COntextual eRror-modulated gEneralized Diffusion model for low-dose CT (LDCT) denoising, termed CoreDiff. First, CoreDiff utilizes LDCT images to displace the random Gaussian noise and employs a novel mean-preserving degradation operator to mimic the physical process of CT degradation, significantly reducing sampling steps thanks to the informative LDCT images as the starting point of the sampling process. Second, to alleviate the error accumulation problem caused by the imperfect restoration operator in the sampling process, we propose a novel ContextuaL Error-modulAted Restoration Network (CLEAR-Net), which can leverage contextual information to constrain the sampling process from structural distortion and modulate time step embedding features for better alignment with the input at the next time step. Third, to rapidly generalize the trained model to a new, unseen dose level with as few resources as possible, we devise a one-shot learning framework to make CoreDiff generalize faster and better using only one single LDCT image (un)paired with normal-dose CT (NDCT). Extensive experimental results on four datasets demonstrate that our CoreDiff outperforms competing methods in denoising and generalization performance, with clinically acceptable inference time. Source code is made available at https://github.com/qgao21/CoreDiff.

Effectiveness of imperfect quarantine in controlling infectious diseases: A mathematical analysis of a general diffusive epidemic model

Effectiveness of imperfect quarantine in controlling infectious diseases: A mathematical analysis of a general diffusive epidemic model

PrideDiff: Physics-Regularized Generalized Diffusion Model for CT Reconstruction

PrideDiff: Physics-Regularized Generalized Diffusion Model for CT Reconstruction

An alternative general model for the effective longitudinal diffusion in chromatographic beds filled with ordered porous particles

The two-zone moment-analysis method for the determination of the dispersion tensor in hierarchical retentive porous media has been adopted to compute and model the effective longitudinal diffusion Deff, or equivalently the B-term band broadening, in chromatographic beds filled with ordered porous particles. On the one hand, this approach offers accurate numerical results for Deff while keeping computational expenses low. On the other hand, it also gives direct insight for the analytical modelling, readily revealings the two main essential quantities (resp. referred to as the mobile-zone and stationary-zone effective diffusion factors γm and γs) that contribute to Deff. Modelling these two main parameters provided us with two new analytical models for Deff: a general one, valid for diluted and concentrated packings and accurate in the whole range of relevant intra-particle diffusion coefficient Dpz, and an approximate one, reliable for diluted packings and accurate also for concentrated packings with low to intermediate values of Dpz. The large advantage of both models is that they do not need any fitting parameter because all the required information is incorporated into the experimentally accessible geometric obstruction factor in the mobile phase originating from the tortuosity of the through-pore space (limiting case of fully solid particles without any retention). These models hence serve as an alternative to the Effective Medium Theory (EMT) models used so far in the literature. To validate the theory, five ordered geometries have been investigated. The accuracy of the general model proposed has been quantified and found to be comparable with that of the 3rd order approximate Torquato model for four geometries, even for macro-porosities close to the close-packing limit. The case of a 2-d triangular array of ellipsoidal particles with different elongations is also investigated to show the general validity and applicability of the models.

Spread and control of medical rumors in a social network: A generalized diffusion model with a highly asymmetric network structure

Medical rumors have become a threat to modern society. To study the spread and control of rumors, nonlinear differential equations modeling with the well‐mixed assumption is commonly used. However, this approach ignores the underlying network structure which plays an important role in information spreading. We establish a generalized differential equations model to study the spread and control of medical rumors in a highly asymmetric social network. In our model, each node represents a group of people and a “weighted” and “directed” network describes the communications between these nodes. This network can be generated from real‐world data by community detection algorithms. We provide methods to numerically calculate the final size of a rumor in each node and its derivatives with respect to each parameter. With these methods, if the government has resources to influence the parameters subject to certain constraints or cost functions, one can obtain the optimal resources allocation easily through nonlinear programming algorithms. We show that the implications on the government's resources allocation from the well‐mixed special case in the literature or conventional wisdom may become inapplicable in the general situation. Therefore, the underlying network should not be ignored. Because the final size of a medical rumor is not always the best measure of its damage, we extend our results to a wide class of objectives and show that different objectives result in very different implications. While the lack of a rule of thumb may sound negative, our flexible framework provides a powerful workhorse for interested parties to work out the details in their specific situations. Finally, we provide a sufficient condition for no outbreak of rumors. This condition can serve as a heuristic that a government with abundant resources can use to prevent the outbreak of rumors.

On the apparent dispersion coefficient of the equilibrium dispersion model: An asymptotic analysis

To model chromatography, researchers have developed several approaches. These cover a broad range of applications and, depending on the assumptions adopted, have different levels of accuracy. In general, the most suitable modelling approach is the simplest that can describe a process with the desired accuracy. A model that often meets this criterion is the equilibrium dispersion model (EDM). This features one mass balance equation per analyte, including an axial dispersion term, and assumes the analyte concentrations in the mobile and stationary phases to be in local equilibrium. To account for the finite mass transfer rate between the phases, the model employs an apparent dispersion coefficient. Two expressions are available for this coefficient, one being used much more frequently than the other. In this paper, we aimed to clarify which one should be favoured. A desirable feature of simple models is that they can be derived from more general ones with appropriate physical assumptions and rigorous mathematical methods. Thus, to answer our research question, we derived the EDM from the more general pore diffusion model (POR), using an asymptotic method. The expression obtained for the apparent dispersion coefficient does agree with one of the two reported in the literature – the less frequently used. To test the validity of this expression, we simulated elution profiles using the two versions of the EDM and compared the results against those from the POR model. The simulations were conducted in the range where the POR and EDM models should be essentially equivalent, their results confirming the outcome of the asymptotic analysis. This work offers a solid theoretical grounding for the EDM, clarifies which formulation of the model is correct, and provides usable applicability conditions for the model.

Dissemination of knowledge potential in the e-learning process

The key terms in the process of knowledge management and knowledge potential are analyzed. Groups of internal and external factors affecting knowledge potential are indicated. The factors of influence on the choice of electronic educational resources are highlighted. The interaction of participants in the educational process is depicted schematically, particularly in communities of the electronic educational environment. The list of probabilistic selection rules for choosing a source of knowledge and learning is given. The model of dynamics of dissemination of knowledge potential, taking into account the flow of knowledge from source to agent, is indicated. Modeling is described in the form of a generalized diffusion model of processes of redistribution of knowledge potential during e-learning, taking into account the replenishment of the source of knowledge. The influence of electronic educational resources on the replenishment of the teacher's knowledge, which transfers knowledge to students within a certain community, is given. The general structure of the process of formation of knowledge potential during e-learning, indicating sources of knowledge, factors of influence on participants of the educational process is shown, the processes of replenishment, transfer, and redistribution of knowledge are indicated.

Turing‐Turing and Turing‐Hopf bifurcations in a general diffusive Brusselator model

AbstractA general reaction‐diffusion Brusselator model subject to homogeneous Neumann boundary conditions is investigated in this paper. First, the stability of the unique positive equilibrium is studied, and we identify the existence of the Hopf bifurcation. Then, occurrence conditions of the Turing instability, the Turing‐Turing, and the Turing‐Hopf bifurcations are established. To explore the spatiotemporal solutions resulting from the bifurcation, the amplitude equations of the Turing‐Turing and the Turing‐Hopf bifurcations are established via the method of multiple time scale. It is found that the model admits the nonconstant steady state, the mixed nonconstant steady state, the spatially homogeneous periodic solution, and the spatially nonhomogeneous periodic solution. As such, the nonhomogeneous spatiotemporal solutions appear in the model. In the end, numerical simulations verify the validity of the theoretical results.

Weak equilibria for time‐inconsistent control: With applications to investment‐withdrawal decisions

AbstractThis paper considers time‐inconsistent problems when control and stopping strategies are required to be made simultaneously (called stopping control problems by us). We first formulate the time‐inconsistent stopping control problems under general multidimensional controlled diffusion model and propose a formal definition of their equilibria. We show that an admissible pair of control‐stopping policy is equilibrium if and only if the auxiliary function associated with it solves the extended HJB system, providing a methodology to verify or exclude equilibrium solutions. We provide several examples to illustrate applications to mathematical finance and control theory. For a problem whose reward function endogenously depends on the current wealth, the equilibrium is explicitly obtained. For another model with a nonexponential discount, we prove that any constant proportion strategy can not be equilibrium. We further show that general nonconstant equilibrium exists and is described by singular boundary value problems. This example shows that considering our combined problems is essentially different from investigating them separately. In the end, we also provide a two‐dimensional example with a hyperbolic discount.

Thermodynamically consistent phase-field modelling of activated solute transport in binary solvent fluids

Active diffusion of substances in binary immiscible and incompressible fluids with different densities occurs universally in nature and industry, but relevant mathematical models and numerical simulation have been studied scarcely so far. In this paper, a thermodynamically consistent phase-field model is established to describe the activated solute transport in binary fluids with different densities. A mixed free-energy function for multiple solutes is proposed, which leads to different solute chemical potentials in binary solvent fluids, thus it has the ability to characterize the solubility difference of the solutes in two solvents. The two-phase flow is governed by a general hydrodynamic phase-field model that can account for general average velocity and different densities. The proposed model is derived rigorously using the second law of thermodynamics. Moreover, a general multi-component solute diffusion model is established using the Maxwell–Stefan approach, which involves the crossing influences between different solutes. To solve the model effectively, an efficient, linearized and decoupled numerical method is proposed for the model as well. The proposed numerical method can preserve the thermodynamical consistency, i.e. obeying an energy dissipation law at the discrete level, as well as guarantee the mass conservation law for the solutes and solvent fluids. Numerical experiments are carried out to show that the proposed model and numerical method can simulate various processes of the solute active diffusion in two-phase solvent fluids.

Adapting network theory for spatial network externalities in agriculture: A case study on hemp cross‐pollination

AbstractGrowers have increasingly expressed frustration over the negative externalities created by their neighbor's production practices. These spatial agricultural network problems include issues such as cross‐pollination and herbicide drift. We develop novel methods for estimating parameters that allow us to adapt and apply general network diffusion models to these spatial agricultural network problems. Doing so allows us to calculate externality damage within a region and calculate cost‐effective policies for alleviating that externality. We empirically illustrate, motivate, and test this approach by applying it to hemp. We find that network structure is an important factor in externality size and cost‐effective policy response for spatial agricultural network problems. We also find that policies that are implemented early and proactively are more likely to be successful and cost effective than policies implemented retroactively. Finally, we find that in our application of limiting the cross‐pollination damage experienced by growers of feminized hemp from non‐feminized hemp growers, the most cost‐effective policy is to establish a regional quota on non‐feminized production combined with intertemporal cultivar spacing. This policy response will likely change across time and region as economic and network variables evolve.

Finding influential edges in multilayer networks: Perspective from multilayer diffusion model.

With the popularization of social network analysis, information diffusion models have a wide range of applications, such as viral marketing, publishing predictions, and social recommendations. The emergence of multiplex social networks has greatly enriched our daily life; meanwhile, identifying influential edges remains a significant challenge. The key problem lies that the edges of the same nodes are heterogeneous at different layers of the network. To solve this problem, we first develop a general information diffusion model based on the adjacency tensor for the multiplex network and show that the n-mode singular value can control the level of information diffusion. Then, to explain the suppression of information diffusion through edge deletion, efficient edge eigenvector centrality is proposed to identify the influence of heterogeneous edges. The numerical results from synthetic networks and real-world multiplex networks show that the proposed strategy outperforms some existing edge centrality measures. We devise an experimental strategy to demonstrate that influential heterogeneous edges can be successfully identified by considering the network layer centrality, and the deletion of top edges can significantly reduce the diffusion range of information across multiplex networks.

Diffusive host-pathogen model revisited: Nonlocal infections, incubation period and spatial heterogeneity

We formulate and analyze a general diffusive host-pathogen model with an incubation period and nonlocal infections in a spatially heterogeneous environment. The model system is partially degenerate and the solution map is not compact. We first prove that the solutions of the model exist globally and are ultimately bounded. Next, we define the basic reproduction number R0 as the spectral radius of the sum of two next generation operators corresponding to direct and indirect infection modes, and prove that R0 is decreasing with respect to the incubation period and the diffusion coefficient of infectious hosts under some conditions. Finally, we demonstrate that the model system possesses a global attractor, and explore the global dynamics of the system. Especially, we show that the infection-free steady state is globally asymptotically stable if R0≤1. On the other hand, if R0>1, then the infection will persist and the model system admits at least one positive steady state. For spatial homogeneous system, global asymptotic stability of the positive steady state is proved via Lyapunov functional technique. Numerical simulation is conducted to explore singular perturbation phenomenon when the diffusion coefficients approach zero. We observe that the diffusion will not only spread the infection to the low-risk region, but it may also increase the infection level in the high-risk region.

Adaptive Greedy versus Non-adaptive Greedy for Influence Maximization

We consider the adaptive influence maximization problem: given a network and a budget k, iteratively select k seeds in the network to maximize the expected number of adopters. In the full-adoption feedback model, after selecting each seed, the seed-picker observes all the resulting adoptions. In the myopic feedback model, the seed-picker only observes whether each neighbor of the chosen seed adopts. Motivated by the extreme success of greedy-based algorithms/heuristics for influence maximization, we propose the concept of greedy adaptivity gap, which compares the performance of the adaptive greedy algorithm to its non-adaptive counterpart. Our first result shows that, for submodular influence maximization, the adaptive greedy algorithm can perform up to a (1 − 1/e)-fraction worse than the non-adaptive greedy algorithm, and that this ratio is tight. More specifically, on one side we provide examples where the performance of the adaptive greedy algorithm is only a (1−1/e) fraction of the performance of the non-adaptive greedy algorithm in four settings: for both feedback models and both the independent cascade model and the linear threshold model. On the other side, we prove that in any submodular cascade, the adaptive greedy algorithm always outputs a (1 − 1/e)-approximation to the expected number of adoptions in the optimal non-adaptive seed choice. Our second result shows that, for the general submodular diffusion model with full-adoption feedback, the adaptive greedy algorithm can outperform the non-adaptive greedy algorithm by an unbounded factor. Finally, we propose a risk-free variant of the adaptive greedy algorithm that always performs no worse than the non-adaptive greedy algorithm.

Estimating Life Cycle Sales of Technology Products with Frequent Repeat Purchases: A Fractional Calculus-Based Approach

Accurately predicting the sales trajectory of a product is critically important for firms’ medium- and long-term planning. However, reliable sales prediction models are very difficult to find when repeat purchases or subscription renewals account for a large proportion of product sales, which is often the case for technological products. This study introduces a new sales growth model, the generalized diffusion model with repeat purchases (GDMR), to address this problem. The GDMR draws upon a branch of mathematics called fractional calculus and formulates a product’s sales growth rate using a novel noninteger-order integral equation. Compared with benchmark methods, the GDMR is simple and easy to implement, is suitable for a wide variety of products, and predicts better than benchmark models such as time series and machine learning models. Furthermore, the GDMR can reliably recover a product’s progress of adoptions even when only sales data are available. Because of these important advantages, the GDMR can help firms better understand their products’ market positions and, subsequently, make more informed decisions in production and inventory planning, transportation and logistics, and sales and marketing, thus improving the effectiveness and efficiency of their business operations.

On the modelling of spatially heterogeneous nonlocal diffusion: deciding factors and preferential position of individuals.

We develop general heterogeneous nonlocal diffusion models and investigate their connection to local diffusion models by taking a singular limit of focusing kernels. We reveal the link between the two groups of diffusion equations which include both spatial heterogeneity and anisotropy. In particular, we introduce the notion of deciding factors which single out a nonlocal diffusion model and typically consist of the total jump rate and the average jump length. In this framework, we also discuss the dependence of the profile of the steady state solutions on these deciding factors, thus shedding light on the preferential position of individuals.