Local X-ray Transform on Asymptotically Hyperbolic Manifolds via Projective Compactification

We prove local injectivity near a boundary point for the geodesic X-ray transform for an asymptotically hyperbolic metric even mod $O(\rho^5)$ in dimensions three and higher.

Real-time detection of particleboard surface defects based on improved YOLOV5 target detection

Particleboard surface defect detection technology is of great significance to the automation of particleboard detection, but the current detection technology has disadvantages such as low accuracy and poor real-time performance. Therefore, this paper proposes an improved lightweight detection method of You Only Live Once v5 (YOLOv5), namely PB-YOLOv5 (Particle Board-YOLOv5). Firstly, the gamma-ray transform method and the image difference method are combined to deal with the uneven illumination of the acquired images, so that the uneven illumination is well corrected. Secondly, Ghost Bottleneck lightweight deep convolution module is added to Backbone module and Neck module of YOLOv5 detection algorithm to reduce model volume. Thirdly, the SELayer module of attention mechanism is added into Backbone module. Finally, replace Conv in Neck module with depthwise convolution (DWConv) to compress network parameters. The experimental results show that the PB-YOLOv5 model proposed in this paper can accurately identify five types of defects on the particleboard surface: Bigshavings, SandLeakage, GlueSpot, Soft and OliPollution, and meet the real-time requirements. Specifically, recall, F1 score, [email protected], [email protected]:.95 values of pB-Yolov5s model were 91.22%, 94.5%, 92.1%, 92.8% and 67.8%, respectively. The results of Soft defects were 92.8%, 97.9%, 95.3%, 99.0% and 81.7%, respectively. The detection of single image time of the model is only 0.031 s, and the weight size of the model is only 5.4 MB. Compared with the original YOLOv5s, YOLOv4, YOLOv3 and Faster RCNN, the PB-Yolov5s model has the fastest Detection of single image time. The Detection of single image time was accelerated by 34.0%, 55.1%, 64.4% and 87.9%, and the weight size of the model is compressed by 62.5%, 97.7%, 97.8% and 98.9%, respectively. The mAP value increased by 2.3%, 4.69%, 7.98% and 13.05%, respectively. The results show that the PB-YOLOV5 model proposed in this paper can realize the rapid and accurate detection of particleboard surface defects, and fully meet the requirements of lightweight embedded model.

Stability of the Non-abelian X-ray Transform in Dimension $$\ge 3$$

Non-abelian X-ray tomography seeks to recover a matrix potential $$\Phi :M\rightarrow {\mathbb {C}}^{m\times m}$$ Φ : M → C m × m in a domain M from measurements of its so-called scattering data $$C_\Phi $$ C Φ at $$\partial M$$ ∂ M . For $$\dim M\ge 3$$ dim M ≥ 3 (and under appropriate convexity and regularity conditions), injectivity of the forward map $$\Phi \mapsto C_\Phi $$ Φ ↦ C Φ was established in (Paternain et al. in Am J Math 141(6):1707–1750, 2019). The present article extends this result by proving a Hölder-type stability estimate. As an application, a statistical consistency result for $$\dim M =2$$ dim M = 2 (Monard et al. in Commun Pure Appl Math, 2019) is generalised to higher dimensions. The injectivity proof in (Paternain et al. in Am J Math 141(6):1707–1750, 2019) relies on a novel method by Uhlmann and Vasy (Invent Math 205(1):83–120, 2016), which first establishes injectivity in a shallow layer below $$\partial M$$ ∂ M and then globalises this by a layer stripping argument. The main technical contribution of this paper is a more quantitative version of these arguments, in particular, proving uniform bounds on layer depth and stability constants.