Frequency Decomposition and Spatial Dependency Mathematical Modeling for Small-Scale Open-World Object Detection

Intelligent transportation and aerial remote sensing scenes suffer from complex scene variations, abundant miniature targets and unpredictable out-of-distribution obstacles, which brings tough mathematical challenges to open-world detection tasks. Conventional detection algorithms lack rigorous frequency-domain separation and spatial constraint mathematical formulations, resulting in severe tiny-object feature attenuation, inefficient multimodal feature matching and catastrophic forgetting during incremental category iteration. To solve these mathematical bottlenecks, this paper constructs the TPCA-Net model built upon frequency decomposition and spatial dependency mathematical modelling. The entire framework consists of four fixed core modules: High-Frequency-Aware Multi-Scale Feature Enhancement (HSE), Reparameterized Adaptive Text–Visual Alignment (RTA), Double Wildcard Spatial Dependency Fusion (WSF), and Incremental Forgetting-Free Dual-Path Detection (DPD). From the mathematical perspective, the HSE module adopts discrete cosine transform-based filtering equations to split high-frequency object details from low-frequency background signals and establishes cross-attention spatial constraint formulas to make up for missing contextual information of small targets. The RTA module introduces low-rank decomposition mathematical optimization and reparameterized tensor fusion rules to realize domain-adaptive text embedding calibration and zero-cost cross-modal mapping at the inference stage. The WSF module constructs dual-wildcard self-supervised mathematical loss to finish unsupervised unknown-object identification and builds decoupled semantic–spatial fusion equations to improve the positioning precision of novel targets. The DPD module designs two sets of independent optimization objective functions and category-freezing incremental mathematical constraints to avoid conflicting parameter updates and eliminate forgetting defects in new-class expansion. Validated on COCO, DOTA and AI-TOD datasets, TPCA-Net achieves 56.0% AP on COCO, 79.30% mAP on DOTA, and 40.5% overall AP with 28.7% small-object AP on AI-TOD while delivering an inference throughput of 101.2 FPS on the Tesla T4 edge GPU. The proposed method outperforms existing mainstream open-world detection algorithms in tiny-object and rare-category recognition while maintaining efficient inference speed.