2D Gaussian Splatting has emerged as a novel image representation technique that can support efficient rendering on low-end devices. However, scaling to high-resolution images requires optimizing and storing millions of unstructured Gaussian primitives independently, leading to slow convergence and redundant parameters. To address this, we propose Structured Gaussian Image (SGI), a compact and efficient framework for representing high-resolution images. SGI decomposes a complex image into multi-scale local spaces defined by a set of seeds. Each seed corresponds to a spatially coherent region and, together with lightweight multi-layer perceptrons (MLPs), generates structured implicit 2D neural Gaussians. This seed-based formulation imposes structural regularity on otherwise unstructured Gaussian primitives, which facilitates entropy-based compression at the seed level to reduce the total storage. However, optimizing seed parameters directly on high-resolution images is a challenging and non-trivial task. Therefore, we designed a multi-scale fitting strategy that refines the seed representation in a coarse-to-fine manner, substantially accelerating convergence. Quantitative and qualitative evaluations demonstrate that SGI achieves up to 7.5x compression over prior non-quantized 2D Gaussian methods and 1.6x over quantized ones, while also delivering 1.6x and 6.5x faster optimization, respectively, without degrading, and often improving, image fidelity. Code is available at https://github.com/zx-pan/SGI.