Image labeling is an important task for training computer vision models. In specialized domains, such as healthcare, it is expensive and challenging to recruit specialists for image labeling. We propose HEPHA, a mixed-initiative image labeling tool that elicits human expertise via inductive logic learning to infer and refine labeling rules. Each rule comprises visual predicates that describe the image. HEPHA enables users to iteratively refine the rules by either direct manipulation through a visual programming interface or by labeling more images. To facilitate rule refinement, HEPHA recommends which rule to edit and which predicate to update. For users unfamiliar with visual programming, HEPHA suggests diverse and informative images to users for further labeling. We conducted a within-subjects user study with 16 participants and compared HEPHA with a variant of HEPHA and a deep learning-based approach. We found that HEPHA outperforms the two baselines in both specialized-domain and general-domain image labeling tasks. Our code is available at https://github.com/Neural-Symbolic-Image-Labeling/NSILWeb.