We present a specialized form of error diffusion (ED) that addresses certain long-standing problems associated with operating on images possessing halftone structure and other local high-contrast structures. For instance, image-quality defects often occur when rendering a scanned halftone image to printable form via quantization reduction. Rendering the scanned halftone via conventional ED produces fragmented dots, which can appear grainy and be unstable in printed density. Rendering by thresholding or rehalftoning often produces moiré, and descreening blurs the image. Another difficulty arises in printers that utilize a binary image path, where an image is rasterized directly to halftone form. In that form it is difficult to perform basic image processing operations such as applying a digital tone reproduction curve. Rank-order error diffusion (ROED) has been developed to address these problems. ROED utilizes brightness ranking of pixels within a diffusion mask to diffuse quantization error at a pixel. This approach results in an image-structure-adaptive quantization with useful properties. We describe the basic methodology of ROED as well as several applications in processing halftone images.