In this paper, image compression utilizing visual redundancy is investigated. Inspired by recent advancements in image inpainting techniques, we propose an image compression framework towards visual quality rather than pixel-wise fidelity. In this framework, an original image is analyzed at the encoder side so that portions of the image are intentionally and automatically skipped. Instead, some information is extracted from these skipped regions and delivered to the decoder as assistant information in the compressed fashion. The delivered assistant information plays a key role in the proposed framework because it guides image inpainting to accurately restore these regions at the decoder side. Moreover, to fully take advantage of the assistant information, a compression-oriented edge-based inpainting algorithm is proposed for image restoration, integrating pixel-wise structure propagation and patch-wise texture synthesis. We also construct a practical system to verify the effectiveness of the compression approach in which edge map serves as assistant information and the edge extraction and region removal approaches are developed accordingly. Evaluations have been made in comparison with baseline JPEG and standard MPEG-4 AVC/H.264 intra-picture coding. Experimental results show that our system achieves up to 44% and 33% bits-savings, respectively, at similar visual quality levels. Our proposed framework is a promising exploration towards future image and video compression.