Recent years, image enhancement for single-pixel imaging has developed rapidly and provides an image-free way for extracting image information. However, the conventional image enhancement approaches for single-pixel imaging are still based on the discontinuously adjustable operations such as integer-order derivatives, which are frequently used in edge detection but sensitive to the image noise. Therefore, how to balance between two conflicting demands, i.e. edge detection and noise suppression, is a new challenge. To address this issue, we introduce arbitrary-order fractional operations into single-pixel imaging. In experiment, the proposed technique has the capacity to detect image edges with high quality. Compared with integer-order derivative method which amplifies noise significantly while extracting edges, it offers a nice tradeoff between image SNR and performance of edge enhancement. In addition, it also shows good performance of image smoothing and improvement of image quality, if fractional order is negative. The proposed technique provides the adjustable fractional order as a new degree of freedom for edge extraction and image de-noising and therefore makes up for the shortcomings of traditional method for image enhancement.