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In recent years, Denoising Diffusion Probabilistic Models (DDPM) have caught significant attention. By composing a Markovian process that starts in the data domain and then gradually adds noise until reaching pure white noise, they achieve superior performance in learning data distributions. Yet, these models require a large number of diffusion steps to produce aesthetically pleasing samples, which is inefficient. In addition, unlike common generative adversarial networks, the latent space of diffusion models is not interpretable. In this work, we propose to generalize the denoising diffusion process into an Upsampling Diffusion Probabilistic Model (UDPM), in which we reduce the latent variable dimension in addition to the traditional noise level addition. As a result, we are able to sample images of size $256\times 256$ with only 7 diffusion steps, which is less than two orders of magnitude compared to standard DDPMs. We formally develop the Markovian diffusion processes of the UDPM, and demonstrate its generation capabilities on the popular FFHQ, LSUN horses, ImageNet, and AFHQv2 datasets. Another favorable property of UDPM is that it is very easy to interpolate its latent space, which is not the case with standard diffusion models. Our code is available online \url{//github.com/shadyabh/UDPM}

Low-dose computed tomography (CT) image denoising is crucial in medical image computing. Recent years have been remarkable improvement in deep learning-based methods for this task. However, training deep denoising neural networks requires low-dose and normal-dose CT image pairs, which are difficult to obtain in the clinic settings. To address this challenge, we propose a novel fully unsupervised method for low-dose CT image denoising, which is based on denoising diffusion probabilistic model -- a powerful generative model. First, we train an unconditional denoising diffusion probabilistic model capable of generating high-quality normal-dose CT images from random noise. Subsequently, the probabilistic priors of the pre-trained diffusion model are incorporated into a Maximum A Posteriori (MAP) estimation framework for iteratively solving the image denoising problem. Our method ensures the diffusion model produces high-quality normal-dose CT images while keeping the image content consistent with the input low-dose CT images. We evaluate our method on a widely used low-dose CT image denoising benchmark, and it outperforms several supervised low-dose CT image denoising methods in terms of both quantitative and visual performance.

Diffusion models, as a kind of powerful generative model, have given impressive results on image super-resolution (SR) tasks. However, due to the randomness introduced in the reverse process of diffusion models, the performances of diffusion-based SR models are fluctuating at every time of sampling, especially for samplers with few resampled steps. This inherent randomness of diffusion models results in ineffectiveness and instability, making it challenging for users to guarantee the quality of SR results. However, our work takes this randomness as an opportunity: fully analyzing and leveraging it leads to the construction of an effective plug-and-play sampling method that owns the potential to benefit a series of diffusion-based SR methods. More in detail, we propose to steadily sample high-quality SR images from pretrained diffusion-based SR models by solving diffusion ordinary differential equations (diffusion ODEs) with optimal boundary conditions (BCs) and analyze the characteristics between the choices of BCs and their corresponding SR results. Our analysis shows the route to obtain an approximately optimal BC via an efficient exploration in the whole space. The quality of SR results sampled by the proposed method with fewer steps outperforms the quality of results sampled by current methods with randomness from the same pretrained diffusion-based SR model, which means that our sampling method ``boosts'' current diffusion-based SR models without any additional training.

We develop a class of data-driven generative models that approximate the solution operator for parameter-dependent partial differential equations (PDE). We propose a novel probabilistic formulation of the operator learning problem based on recently developed generative denoising diffusion probabilistic models (DDPM) in order to learn the input-to-output mapping between problem parameters and solutions of the PDE. To achieve this goal we modify DDPM to supervised learning in which the solution operator for the PDE is represented by a class of conditional distributions. The probabilistic formulation combined with DDPM allows for an automatic quantification of confidence intervals for the learned solutions. Furthermore, the framework is directly applicable for learning from a noisy data set. We compare computational performance of the developed method with the Fourier Network Operators (FNO). Our results show that our method achieves comparable accuracy and recovers the noise magnitude when applied to data sets with outputs corrupted by additive noise.

In recent years, many positivity-preserving schemes for initial value problems have been constructed by modifying a Runge--Kutta (RK) method by weighting the right-hand side of the system of differential equations with solution-dependent factors. These include the classes of modified Patankar--Runge--Kutta (MPRK) and Geometric Conservative (GeCo) methods. Compared to traditional RK methods, the analysis of accuracy and stability of these methods is more complicated. In this work, we provide a comprehensive and unifying theory of order conditions for such RK-like methods, which differ from original RK schemes in that their coefficients are solution-dependent. The resulting order conditions are themselves solution-dependent and obtained using the theory of NB-series, and thus, can easily be read off from labeled N-trees. We present for the first time order conditions for MPRK and GeCo schemes of arbitrary order; For MPRK schemes, the order conditions are given implicitly in terms of the stages. From these results, we recover as particular cases all known order conditions from the literature for first- and second-order GeCo as well as first-, second- and third-order MPRK methods. Additionally, we derive sufficient and necessary conditions in an explicit form for 3rd and 4th order GeCo schemes as well as 4th order MPRK methods.

Deep learning-based approaches have achieved remarkable performance in single-image denoising. However, training denoising models typically requires a large amount of data, which can be difficult to obtain in real-world scenarios. Furthermore, synthetic noise used in the past has often produced significant differences compared to real-world noise due to the complexity of the latter and the poor modeling ability of noise distributions of Generative Adversarial Network (GAN) models, resulting in residual noise and artifacts within denoising models. To address these challenges, we propose a novel method for synthesizing realistic noise using diffusion models. This approach enables us to generate large amounts of high-quality data for training denoising models by controlling camera settings to simulate different environmental conditions and employing guided multi-scale content information to ensure that our method is more capable of generating real noise with multi-frequency spatial correlations. In particular, we design an inversion mechanism for the setting, which extends our method to more public datasets without setting information. Based on the noise dataset we synthesized, we have conducted sufficient experiments on multiple benchmarks, and experimental results demonstrate that our method outperforms state-of-the-art methods on multiple benchmarks and metrics, demonstrating its effectiveness in synthesizing realistic noise for training denoising models.

Denoising diffusion models represent a recent emerging topic in computer vision, demonstrating remarkable results in the area of generative modeling. A diffusion model is a deep generative model that is based on two stages, a forward diffusion stage and a reverse diffusion stage. In the forward diffusion stage, the input data is gradually perturbed over several steps by adding Gaussian noise. In the reverse stage, a model is tasked at recovering the original input data by learning to gradually reverse the diffusion process, step by step. Diffusion models are widely appreciated for the quality and diversity of the generated samples, despite their known computational burdens, i.e. low speeds due to the high number of steps involved during sampling. In this survey, we provide a comprehensive review of articles on denoising diffusion models applied in vision, comprising both theoretical and practical contributions in the field. First, we identify and present three generic diffusion modeling frameworks, which are based on denoising diffusion probabilistic models, noise conditioned score networks, and stochastic differential equations. We further discuss the relations between diffusion models and other deep generative models, including variational auto-encoders, generative adversarial networks, energy-based models, autoregressive models and normalizing flows. Then, we introduce a multi-perspective categorization of diffusion models applied in computer vision. Finally, we illustrate the current limitations of diffusion models and envision some interesting directions for future research.

Diffusion models are a class of deep generative models that have shown impressive results on various tasks with dense theoretical founding. Although diffusion models have achieved impressive quality and diversity of sample synthesis than other state-of-the-art models, they still suffer from costly sampling procedure and sub-optimal likelihood estimation. Recent studies have shown great enthusiasm on improving the performance of diffusion model. In this article, we present a first comprehensive review of existing variants of the diffusion models. Specifically, we provide a first taxonomy of diffusion models and categorize them variants to three types, namely sampling-acceleration enhancement, likelihood-maximization enhancement and data-generalization enhancement. We also introduce in detail other five generative models (i.e., variational autoencoders, generative adversarial networks, normalizing flow, autoregressive models, and energy-based models), and clarify the connections between diffusion models and these generative models. Then we make a thorough investigation into the applications of diffusion models, including computer vision, natural language processing, waveform signal processing, multi-modal modeling, molecular graph generation, time series modeling, and adversarial purification. Furthermore, we propose new perspectives pertaining to the development of this generative model.

Forecasting has always been at the forefront of decision making and planning. The uncertainty that surrounds the future is both exciting and challenging, with individuals and organisations seeking to minimise risks and maximise utilities. The large number of forecasting applications calls for a diverse set of forecasting methods to tackle real-life challenges. This article provides a non-systematic review of the theory and the practice of forecasting. We provide an overview of a wide range of theoretical, state-of-the-art models, methods, principles, and approaches to prepare, produce, organise, and evaluate forecasts. We then demonstrate how such theoretical concepts are applied in a variety of real-life contexts. We do not claim that this review is an exhaustive list of methods and applications. However, we wish that our encyclopedic presentation will offer a point of reference for the rich work that has been undertaken over the last decades, with some key insights for the future of forecasting theory and practice. Given its encyclopedic nature, the intended mode of reading is non-linear. We offer cross-references to allow the readers to navigate through the various topics. We complement the theoretical concepts and applications covered by large lists of free or open-source software implementations and publicly-available databases.

High spectral dimensionality and the shortage of annotations make hyperspectral image (HSI) classification a challenging problem. Recent studies suggest that convolutional neural networks can learn discriminative spatial features, which play a paramount role in HSI interpretation. However, most of these methods ignore the distinctive spectral-spatial characteristic of hyperspectral data. In addition, a large amount of unlabeled data remains an unexploited gold mine for efficient data use. Therefore, we proposed an integration of generative adversarial networks (GANs) and probabilistic graphical models for HSI classification. Specifically, we used a spectral-spatial generator and a discriminator to identify land cover categories of hyperspectral cubes. Moreover, to take advantage of a large amount of unlabeled data, we adopted a conditional random field to refine the preliminary classification results generated by GANs. Experimental results obtained using two commonly studied datasets demonstrate that the proposed framework achieved encouraging classification accuracy using a small number of data for training.