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Graph Neural Network (GNN) with its ability to integrate graph information has been widely used for data analyses. However, the expressive power of GNN has only been studied for graph-level tasks but not for node-level tasks, such as node classification, where one tries to interpolate missing nodal labels from the observed ones. In this paper, we study the expressive power of GNN for the said classification task, which is in essence a function interpolation problem. Explicitly, we derive the number of weights and layers needed for a GNN to interpolate a band-limited function in $\mathbb{R}^d$. Our result shows that, the number of weights needed to $\epsilon$-approximate a bandlimited function using the GNN architecture is much fewer than the best known one using a fully connected neural network (NN) - in particular, one only needs $O((\log \epsilon^{-1})^{d})$ weights using a GNN trained by $O((\log \epsilon^{-1})^{d})$ samples to $\epsilon$-approximate a discretized bandlimited signal in $\mathbb{R}^d$. The result is obtained by drawing a connection between the GNN structure and the classical sampling theorems, making our work the first attempt in this direction.

We develop a method that recovers the surface, materials, and illumination of a scene from its posed multi-view images. In contrast to prior work, it does not require any additional data and can handle glossy objects or bright lighting. It is a progressive inverse rendering approach, which consists of three stages. First, we reconstruct the scene radiance and signed distance function (SDF) with our novel regularization strategy for specular reflections. Our approach considers both the diffuse and specular colors, which allows for handling complex view-dependent lighting effects for surface reconstruction. Second, we distill light visibility and indirect illumination from the learned SDF and radiance field using learnable mapping functions. Third, we design a method for estimating the ratio of incoming direct light represented via Spherical Gaussians reflected in a specular manner and then reconstruct the materials and direct illumination of the scene. Experimental results demonstrate that the proposed method outperforms the current state-of-the-art in recovering surfaces, materials, and lighting without relying on any additional data.

Neural radiance fields (NeRFs) are able to synthesize realistic novel views from multi-view images captured from distinct positions and perspectives. In NeRF's rendering pipeline, neural networks are used to represent a scene independently or transform queried learnable feature vector of a point to the expected color or density. With the aid of geometry guides either in occupancy grids or proposal networks, the number of neural network evaluations can be reduced from hundreds to dozens in the standard volume rendering framework. Instead of rendering yielded color after neural network evaluation, we propose to render the queried feature vectors of a ray first and then transform the rendered feature vector to the final pixel color by a neural network. This fundamental change to the standard volume rendering framework requires only one single neural network evaluation to render a pixel, which substantially lowers the high computational complexity of the rendering framework attributed to a large number of neural network evaluations. Consequently, we can use a comparably larger neural network to achieve a better rendering quality while maintaining the same training and rendering time costs. Our model achieves the state-of-the-art rendering quality on both synthetic and real-world datasets while requiring a training time of several minutes.

Despite the promising results of multi-view reconstruction, the recent neural rendering-based methods, such as implicit surface rendering (IDR) and volume rendering (NeuS), not only incur a heavy computational burden on training but also have the difficulties in disentangling the geometric and appearance. Although having achieved faster training speed than implicit representation and hash coding, the explicit voxel-based method obtains the inferior results on recovering surface. To address these challenges, we propose an effective mesh-based neural rendering approach, named FastMESH, which only samples at the intersection of ray and mesh. A coarse-to-fine scheme is introduced to efficiently extract the initial mesh by space carving. More importantly, we suggest a hexagonal mesh model to preserve surface regularity by constraining the second-order derivatives of vertices, where only low level of positional encoding is engaged for neural rendering. The experiments demonstrate that our approach achieves the state-of-the-art results on both reconstruction and novel view synthesis. Besides, we obtain 10-fold acceleration on training comparing to the implicit representation-based methods.

In this paper, we present and analyze a linear fully discrete second order scheme with variable time steps for the phase field crystal equation. More precisely, we construct a linear adaptive time stepping scheme based on the second order backward differentiation formulation (BDF2) and use the Fourier spectral method for the spatial discretization. The scalar auxiliary variable approach is employed to deal with the nonlinear term, in which we only adopt a first order method to approximate the auxiliary variable. This treatment is extremely important in the derivation of the unconditional energy stability of the proposed adaptive BDF2 scheme. However, we find for the first time that this strategy will not affect the second order accuracy of the unknown phase function $\phi^{n}$ by setting the positive constant $C_{0}$ large enough such that $C_{0}\geq 1/\Dt.$ The energy stability of the adaptive BDF2 scheme is established with a mild constraint on the adjacent time step radio $\gamma_{n+1}:=\Dt_{n+1}/\Dt_{n}\leq 4.8645$. Furthermore, a rigorous error estimate of the second order accuracy of $\phi^{n}$ is derived for the proposed scheme on the nonuniform mesh by using the uniform $H^{2}$ bound of the numerical solutions. Finally, some numerical experiments are carried out to validate the theoretical results and demonstrate the efficiency of the fully discrete adaptive BDF2 scheme.

We propose a deep learning method for 3D volumetric reconstruction in low-dose helical cone-beam computed tomography. Prior machine learning approaches require reference reconstructions computed by another algorithm for training. In contrast, we train our model in a fully self-supervised manner using only noisy 2D X-ray data. This is enabled by incorporating a fast differentiable CT simulator in the training loop. As we do not rely on reference reconstructions, the fidelity of our results is not limited by their potential shortcomings. We evaluate our method on real helical cone-beam projections and simulated phantoms. Our results show significantly higher visual fidelity and better PSNR over techniques that rely on existing reconstructions. When applied to full-dose data, our method produces high-quality results orders of magnitude faster than iterative techniques.

Denoising Diffusion Probabilistic Models (DDPM) have shown remarkable efficacy in the synthesis of high-quality images. However, their inference process characteristically requires numerous, potentially hundreds, of iterative steps, which could lead to the problem of exposure bias due to the accumulation of prediction errors over iterations. Previous work has attempted to mitigate this issue by perturbing inputs during training, which consequently mandates the retraining of the DDPM. In this work, we conduct a systematic study of exposure bias in diffusion models and, intriguingly, we find that the exposure bias could be alleviated with a new sampling method, without retraining the model. We empirically and theoretically show that, during inference, for each backward time step $t$ and corresponding state $\hat{x}_t$, there might exist another time step $t_s$ which exhibits superior coupling with $\hat{x}_t$. Based on this finding, we introduce an inference method named Time-Shift Sampler. Our framework can be seamlessly integrated with existing sampling algorithms, such as DDIM or DDPM, inducing merely minimal additional computations. Experimental results show that our proposed framework can effectively enhance the quality of images generated by existing sampling algorithms.

Estimating human pose and shape from monocular images is a long-standing problem in computer vision. Since the release of statistical body models, 3D human mesh recovery has been drawing broader attention. With the same goal of obtaining well-aligned and physically plausible mesh results, two paradigms have been developed to overcome challenges in the 2D-to-3D lifting process: i) an optimization-based paradigm, where different data terms and regularization terms are exploited as optimization objectives; and ii) a regression-based paradigm, where deep learning techniques are embraced to solve the problem in an end-to-end fashion. Meanwhile, continuous efforts are devoted to improving the quality of 3D mesh labels for a wide range of datasets. Though remarkable progress has been achieved in the past decade, the task is still challenging due to flexible body motions, diverse appearances, complex environments, and insufficient in-the-wild annotations. To the best of our knowledge, this is the first survey to focus on the task of monocular 3D human mesh recovery. We start with the introduction of body models and then elaborate recovery frameworks and training objectives by providing in-depth analyses of their strengths and weaknesses. We also summarize datasets, evaluation metrics, and benchmark results. Open issues and future directions are discussed in the end, hoping to motivate researchers and facilitate their research in this area. A regularly updated project page can be found at //github.com/tinatiansjz/hmr-survey.

We consider the problem of discovering $K$ related Gaussian directed acyclic graphs (DAGs), where the involved graph structures share a consistent causal order and sparse unions of supports. Under the multi-task learning setting, we propose a $l_1/l_2$-regularized maximum likelihood estimator (MLE) for learning $K$ linear structural equation models. We theoretically show that the joint estimator, by leveraging data across related tasks, can achieve a better sample complexity for recovering the causal order (or topological order) than separate estimations. Moreover, the joint estimator is able to recover non-identifiable DAGs, by estimating them together with some identifiable DAGs. Lastly, our analysis also shows the consistency of union support recovery of the structures. To allow practical implementation, we design a continuous optimization problem whose optimizer is the same as the joint estimator and can be approximated efficiently by an iterative algorithm. We validate the theoretical analysis and the effectiveness of the joint estimator in experiments.

Semantic reconstruction of indoor scenes refers to both scene understanding and object reconstruction. Existing works either address one part of this problem or focus on independent objects. In this paper, we bridge the gap between understanding and reconstruction, and propose an end-to-end solution to jointly reconstruct room layout, object bounding boxes and meshes from a single image. Instead of separately resolving scene understanding and object reconstruction, our method builds upon a holistic scene context and proposes a coarse-to-fine hierarchy with three components: 1. room layout with camera pose; 2. 3D object bounding boxes; 3. object meshes. We argue that understanding the context of each component can assist the task of parsing the others, which enables joint understanding and reconstruction. The experiments on the SUN RGB-D and Pix3D datasets demonstrate that our method consistently outperforms existing methods in indoor layout estimation, 3D object detection and mesh reconstruction.