Moiré artifacts are common in digital photography, resulting from the interference between high-frequency scene content and the color filter array of the camera. Existing deep learning-based demoiréing methods trained on large scale datasets are limited in handling various complex moiré patterns, and mainly focus on demoiréing of photos taken of digital displays. Moreover, obtaining moiré-free ground-truth in natural scenes is difficult but needed for training. In this paper, we propose a self-adaptive learning method for demoiréing a high-frequency image, with the help of an additional defocused moiré-free blur image. Given an image degraded with moiré artifacts and a moiré-free blur image, our network predicts a moiré-free clean image and a blur kernel with a self-adaptive strategy that does not require an explicit training stage, instead performing test-time adaptation. Our model has two sub-networks and works iteratively. During each iteration, one sub-network takes the moiré image as input, removing moiré patterns and restoring image details, and the other sub-network estimates the blur kernel from the blur image. The two sub-networks are jointly optimized. Extensive experiments demonstrate that our method outperforms state-of-the-art methods and can produce high-quality demoiréd results. It can generalize well to the task of removing moiré artifacts caused by display screens. In addition, we build a new moiré dataset, including images with screen and texture moiré artifacts. As far as we know, this is the first dataset with real texture moiré patterns.