Current state-of-the-art segmentation models encode entire images before focusing on specific objects. As a result, they waste computational resources - particularly when small objects are to be segmented in high-resolution scenes. We introduce FLIP (Fovea-Like Input Patching), a parameter-efficient vision model that realizes object segmentation through biologically-inspired top-down attention. FLIP selectively samples multi-resolution patches centered on objects of interest from the input. As a result, it allocates high-resolution processing to object centers while maintaining coarser peripheral context. This off-grid, scale-invariant design enables FLIP to outperform META's Segment Anything models (SAM) by large margins: With more than 1000x fewer parameters, FLIP-Tiny (0.51M parameters) reaches a mean IoU of 78.24% while SAM-H reaches 75.41% IoU (641.1M parameters). FLIP-Large even achieves 80.33% mean IoU (96.6M parameters), still running about 6$\times$ faster than SAM-H. We evaluate on six benchmarks in total. In five established benchmarks (Hypersim, KITTI-360, OpenImages, COCO, LVIS) FLIP consistently outperforms SAM and various variants of it. In our novel ObjaScale dataset, which stress-tests scale invariance with objects ranging from 0.0001% up-to 25% of the image area, we show that FLIP segments even very small objects accurately, where existing models fail severely. FLIP opens new possibilities for real-time, object-centric vision applications and offers much higher energy efficiency. We believe that FLIP can act as a powerful foundation model, as it is very well-suited to track objects over time, for example, when being integrated into slot-based scene segmentation architectures.