Zhijiang Zhang

Zhijiang Zhang, Weihua Deng, George Em Karniadakis

The fractional Laplacian $Δ^{β/2}$ is the generator of $β$-stable Lévy process, which is the scaling limit of the Lévy fight. Due to the divergence of the second moment of the jump length of the Lévy fight it is not appropriate as a physical model in many practical applications. However, using a parameter $λ$ to exponentially temper the isotropic power law measure of the jump length leads to the tempered Lévy fight, which has finite second moment. For short time the tempered Lévy fight exhibits the dynamics of Lévy fight while after sufficiently long time it turns to normal diffusion. The generator of tempered $β$-stable Lévy process is the tempered fractional Laplacian $(Δ+λ)^{β/2}$ [W.H. Deng, B.Y. Li, W.Y. Tian, and P.W. Zhang, Multiscale Model. Simul., in press, 2017]. In the current work, we present new computational methods for the tempered fractional Laplacian equation, including the cases with the homogeneous and nonhomogeneous generalized Dirichlet type boundary conditions. We prove the well-posedness of the Galerkin weak formulation and provide convergence analysis of the single scaling B-spline and multiscale Riesz bases finite element methods. We propose a technique for efficiently generating the entries of the dense stiffness matrix and for solving the resulting algebraic equation by preconditioning. We also present several numerical experiments to verify the theoretical results.

Wenhui Chen, Zhijiang Zhang, Liang Yu et al.

Airport runway segmentation can effectively reduce the accident rate during the landing phase, which has the largest risk of flight accidents. With the rapid development of deep learning (DL), related methods achieve good performance on segmentation tasks and can be well adapted to complex scenes. However, the lack of large-scale, publicly available datasets in this field makes the development of methods based on DL difficult. Therefore, we propose a benchmark for airport runway segmentation, named BARS. Additionally, a semiautomatic annotation pipeline is designed to reduce the annotation workload. BARS has the largest dataset with the richest categories and the only instance annotation in the field. The dataset, which was collected using the X-Plane simulation platform, contains 10,256 images and 30,201 instances with three categories. We evaluate eleven representative instance segmentation methods on BARS and analyze their performance. Based on the characteristic of an airport runway with a regular shape, we propose a plug-and-play smoothing postprocessing module (SPM) and a contour point constraint loss (CPCL) function to smooth segmentation results for mask-based and contour-based methods, respectively. Furthermore, a novel evaluation metric named average smoothness (AS) is developed to measure smoothness. The experiments show that existing instance segmentation methods can achieve prediction results with good performance on BARS. SPM and CPCL can effectively enhance the AS metric while modestly improving accuracy. Our work will be available at https://github.com/c-wenhui/BARS.

Weihua Deng, Zhijiang Zhang

Because of the finiteness of the life span and boundedness of the physical space, the more reasonable or physical choice is the tempered power-law instead of pure power-law for the CTRW model in characterizing the waiting time and jump length of the motion of particles. This paper focuses on providing the variational formulation and efficient implementation for solving the corresponding deterministic/macroscopic models, including the space tempered fractional equation and time tempered fractional equation. The convergence, numerical stability, and a series of variational equalities are theoretically proved. And the theoretical results are confirmed by numerical experiments.

Weihua Deng, Zhijiang Zhang

This paper focuses on providing the computation methods for the backward time tempered fractional Feynman-Kac equation, being one of the models recently proposed in [Wu, Deng, and Barkai, Phys. Rev. E, 84 (2016) 032151]. The discretization for the tempered fractional substantial derivative is derived, and the corresponding finite difference and finite element schemes are designed with well established stability and convergence. The performed numerical experiments show the effectiveness of the presented schemes.

Zhijiang Zhang, Weihua Deng

For describing the probability distribution of the positions and times of particles performing anomalous motion, fractional PDEs are derived from the continuous time random walk models with waiting time distribution having divergent first order moment and/or jump length distribution which has divergent second order moment. It can be noted that the fractional PDEs are essentially dealing with the multiscale issues. Generally the regularity of the solutions for fractional PDEs is weak at the areas close to boundary and initial time. This paper focuses on developing the applications of wavelet bases to numerically solving fractional PDEs and digging out the potential benefits of wavelet methods comparing with other numerical methods, especially in the aspects of realizing preconditioning, adaptivity, and keeping the Toeplitz structure. More specifically, the contributions of this paper are as follows: 1. the techniques of efficiently generating stiffness matrix with computational cost $\mathcal{O}(2^J)$ are provided for first, second, and any order bases; 2. theoretically and numerically discuss the effective preconditioner for time-independent equation and multigrid method for time-dependent equation, respectively; 3. the wavelet adaptivity is detailedly discussed and numerically applied to solving the time-dependent (independent) equations. In fact, having reliable, simple, and local regularity indicators is the striking benefit of the wavelet in adaptively solving fractional PDEs (it seems hard to give a local posteriori error estimate for the adaptive finite element method because of the global property of the operator).

Jianlin Chen, Gongyang Li, Zhijiang Zhang et al.

Transformer-based methods for RGB-D Salient Object Detection (SOD) have gained significant interest, owing to the transformer's exceptional capacity to capture long-range pixel dependencies. Nevertheless, current RGB-D SOD methods face challenges, such as the quadratic complexity of the attention mechanism and the limited local detail extraction. To overcome these limitations, we propose a novel Superpixel Token Enhancing Network (STENet), which introduces superpixels into cross-modal interaction. STENet follows the two-stream encoder-decoder structure. Its cores are two tailored superpixel-driven cross-modal interaction modules, responsible for global and local feature enhancement. Specifically, we update the superpixel generation method by expanding the neighborhood range of each superpixel, allowing for flexible transformation between pixels and superpixels. With the updated superpixel generation method, we first propose the Superpixel Attention Global Enhancing Module to model the global pixel-to-superpixel relationship rather than the traditional global pixel-to-pixel relationship, which can capture region-level information and reduce computational complexity. We also propose the Superpixel Attention Local Refining Module, which leverages pixel similarity within superpixels to filter out a subset of pixels (i.e., local pixels) and then performs feature enhancement on these local pixels, thereby capturing concerned local details. Furthermore, we fuse the globally and locally enhanced features along with the cross-scale features to achieve comprehensive feature representation. Experiments on seven RGB-D SOD datasets reveal that our STENet achieves competitive performance compared to state-of-the-art methods. The code and results of our method are available at https://github.com/Mark9010/STENet.

Yichun Tai, Kun Yang, Tao Peng et al.

The task of steel surface defect recognition is an industrial problem with great industry values. The data insufficiency is the major challenge in training a robust defect recognition network. Existing methods have investigated to enlarge the dataset by generating samples with generative models. However, their generation quality is still limited by the insufficiency of defect image samples. To this end, we propose Stable Surface Defect Generation (StableSDG), which transfers the vast generation distribution embedded in Stable Diffusion model for steel surface defect image generation. To tackle with the distinctive distribution gap between steel surface images and generated images of the diffusion model, we propose two processes. First, we align the distribution by adapting parameters of the diffusion model, adopted both in the token embedding space and network parameter space. Besides, in the generation process, we propose image-oriented generation rather than from pure Gaussian noises. We conduct extensive experiments on steel surface defect dataset, demonstrating state-of-the-art performance on generating high-quality samples and training recognition models, and both designed processes are significant for the performance.

Yichun Tai, Zhenzhen Huang, Tao Peng et al.

Current saliency-based defect detection methods show promise in industrial settings, but the unpredictability of defects in steel production environments complicates dataset creation, hampering model performance. Existing data augmentation approaches using generative models often require pixel-level annotations, which are time-consuming and resource-intensive. To address this, we introduce DefFiller, a mask-conditioned defect generation method that leverages a layout-to-image diffusion model. DefFiller generates defect samples paired with mask conditions, eliminating the need for pixel-level annotations and enabling direct use in model training. We also develop an evaluation framework to assess the quality of generated samples and their impact on detection performance. Experimental results on the SD-Saliency-900 dataset demonstrate that DefFiller produces high-quality defect images that accurately match the provided mask conditions, significantly enhancing the performance of saliency-based defect detection models trained on the augmented dataset.

Wei Shen, Kai Zhao, Yuan Jiang et al.

Object skeleton is a useful cue for object detection, complementary to the object contour, as it provides a structural representation to describe the relationship among object parts. While object skeleton extraction in natural images is a very challenging problem, as it requires the extractor to be able to capture both local and global image context to determine the intrinsic scale of each skeleton pixel. Existing methods rely on per-pixel based multi-scale feature computation, which results in difficult modeling and high time consumption. In this paper, we present a fully convolutional network with multiple scale-associated side outputs to address this problem. By observing the relationship between the receptive field sizes of the sequential stages in the network and the skeleton scales they can capture, we introduce a scale-associated side output to each stage. We impose supervision to different stages by guiding the scale-associated side outputs toward groundtruth skeletons of different scales. The responses of the multiple scale-associated side outputs are then fused in a scale-specific way to localize skeleton pixels with multiple scales effectively. Our method achieves promising results on two skeleton extraction datasets, and significantly outperforms other competitors.

Zhijiang Zhang, Weihua Deng

The functional distributions of particle trajectories have wide applications, including the occupation time in half-space, the first passage time, and the maximal displacement, etc. The models discussed in this paper are for characterizing the distribution of the functionals of the paths of anomalous diffusion described by time-space fractional diffusion equation. This paper focuses on providing effective computation methods for the models. Two kinds of time stepping schemes are proposed for the fractional substantial derivative. The multiresolution Galerkin method with wavelet B-spline is used for space approximation. Compared with the finite element or spectral polynomial bases, the wavelet B-spline bases have the advantage of keeping the Toeplitz structure of the stiffness matrix, and being easy to generate the matrix elements and to perform preconditioning. The unconditional stability and convergence of the provided schemes are theoretically proved and numerically verified. Finally, we also discuss the efficient implementations and some extensions of the schemes, such as the wavelet preconditioning and the non-uniform time discretization.