## 1 Introduction  
Repeated patterns can be found everywhere, not only in everyday environments, such as on textiles, packaging, buildings, and architectural structures, but also within molecular structures and biological processes that are invisible to the naked eye. A repeated pattern is defined as the minimum representative unit (or tile) of an image, which can replicate the image's original appearance when tiled [1, 2]. Detection of repeated patterns from input images is a fundamental computer vision problem, which serves as a key enabler for diverse real-world applications. For example, in the textile industry, repeated pattern detection automates quality control, defect detection, and assists design element extraction for resource preparation, significantly reducing the reliance on manual labor. Similarly, in architecture, detecting repeating patterns in buildings supports place recognition, urban planning, and 3D modeling. Repeated patterns differ considerably in terms of _image content_ and _topological regularity_. Figure 1 shows examples of repeated patterns with distinct content and topological regularity. Specifically for textile images, they often exhibit regular or near-regular topological structures but feature highly diverse visual content, making them a unique and challenging domain for repeated pattern detection. Recognizing recurring patterns is practically instinctive for humans, but it is difficult for computers due to the wide variety of image types and the complexities introduced by perspective distortions, lighting variations, occlusions, and so forth. Over the years, many methods or algorithms have been proposed for repeated pattern detection. Earlier approaches, such as template matching and signal processing methods, were computationally efficient and effective for regular and noise-free patterns [3; 4]. However, these methods often fail when faced with structural variations caused by occlusion, projection distortions, or deformations. To overcome these limitations, feature-based methods were developed, which abstract input images into a set of local features (e.g., keypoints or descriptors) and detect repeated patterns by identifying repetition rules among these features [5; 6; 7; 8]. While feature-based methods offer improved adaptability to structural variations, they rely heavily on handcrafted features, such as Scale-Invariant Feature Transform (SIFT) [7] or a Histogram of Oriented Gradients (HOGs) [8], which have limited capacity to handle diverse content and are prone to failure in dynamic or irregular scenarios. In recent decades, Convolutional Neural Networks (CNNs) have revolutionized computer vision by enabling robust feature extraction at multiple levels of abstraction. CNNs have shown great performance in tasks such as image classification [9], signal processing [10], and object detection [11]. For repeated pattern detection, CNNs have been successfully applied to extract hierarchical features for pattern size prediction and position estimation [2; 12]. Unlike traditional handcrafted descriptors, pre-trained CNNs can generalize effectively to new tasks without requiring large task-specific datasets, which makes them particularly attractive for repeated pattern detection. However, CNN-based methods still face three critical challenges. First, the computational cost of CNNs scales rapidly with image size, making them unsuitable for high-resolution or complex images. Second, although CNNs improve robustness to diverse content, achieving an optimal trade-off between accuracy and efficiency remains an open question. Third, CNN-based methods often lack the ability to refine predictions dynamically to adapt to irregular patterns or noisy data, limiting their applicability in real-world scenarios. Regardless of the approaches, repeated pattern detection methods usually cover two key components: (1) feature extraction and processing and (2) prediction model optimization.