# Rebar Recognition Using Multi-Hyperbolic Attention in Faster R-CNN
## 2 Materials and Methods  
In ground-penetrating radar (GPR) images, subsurface targets like rebar typically manifest as hyperbolic shapes. This occurs due to the electromagnetic waves emitted by the GPR, which reflect back to the receiver at varying times upon encountering underground objects. The propagation time of electromagnetic waves emitted by the GPR is related to the distance between the GPR antenna and the target object, and this relationship can be expressed as  
\[t=\frac{2d}{v} \tag{1}\]  
where \(v\) denotes the propagation speed of electromagnetic waves within the medium. As the GPR antenna moves, the reflection points at different positions vary over time, generating a hyperbolic trajectory. Specifically, if the radar antenna emits electromagnetic waves and receives a reflected signal at position \(x\), the hyperbolic trajectory of the target in the image can be expressed as  
\[t(x)=\frac{2\sqrt{\left(x-x_{0}\right)^{2}+z_{0}^{2}}}{v} \tag{2}\]  
In this equation, \((x_{0},z_{0})\) represents the coordinates of the target object. Figure 1 illustrates the appearance of rebar as a hyperbolic curve in the radar image. Multiple reflections of radar waves from the rebar and its surrounding environment result in the appearance of hyperbolic curves in the radar image. These curves represent the reflection signals of radar waves traveling along different paths. A single hyperbolic curve is typically associated with the reflection from a single rebar, where the radar waves are reflected at various angles and times depending on the rebar's position relative to the radar transmitter and receiver. In the case of multiple rebars, each rebar produces a distinct hyperbolic curve, leading to the presence of several overlapping or separate hyperbolic patterns in the radar image. This phenomenon occurs due to the varying reflection paths and times of the radar waves as they interact with multiple rebars located at different positions within the structure. A Convolutional Neural Network (CNN) is highly effective in image processing, particularly in extracting local features. In radar images, the hyperbolic patterns representing rebar can be detected through CNN operations, allowing for the extraction of local features such as edges and corner points. In GPR images, the hyperbolic features of rebar are represented as changes in local characteristics, specifically as variations in amplitude. This study integrates A DAS that utilizes Deformable Convolution (DC) and Depthwise Separable Convolution (DSC), enabling the model to better adapt to the diversity and complexity of  
Figure 1: Principle of hyperbolic imaging. rebar reflection features in radar images. The DAS [17] mechanism dynamically adjusts the attention area, focusing on capturing prominent hyperbolic reflection information. This significantly enhances the recognition capability for multiple hyperbolic reflections of rebar. Building upon the work in [18] and the concept of self-attention mechanisms, this paper introduces a hyperbolic attention mechanism designed to enhance the model's ability to focus specifically on hyperbolic shapes in radar images, such as those formed by rebar. In the Faster R-CNN model based on multi-hyperbolic attention, the Convolutional Neural Network (CNN) extracts deep features from the image through multiple layers of convolution and pooling operations, particularly capturing the hyperbolic features exhibited by rebar. The extraction of these features provides abundant information for the subsequent Region Proposal Network (RPN). The RPN slides a window over the feature map to generate candidate regions, classifying them and performing bounding box regression to rapidly screen areas likely to contain rebar. Finally, the classifier evaluates each candidate region to determine if it contains rebar, also performing bounding box regression to precisely adjust the position and size of the candidate boxes, thus improving detection accuracy and spatial Precision.