Moire 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 demoireing methods trained on large scale datasets are limited in handling various complex moire patterns, and mainly focus on demoireing of photos taken of digital displays. Moreover, obtaining moire-free ground-truth in natural scenes is difficult but needed for training. In this paper, we propose a self-adaptive learning method for demoireing a high-frequency image, with the help of an additional defocused moire-free blur image. Given an image degraded with moire artifacts and a moire-free blur image, our network predicts a moire-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 moire image as input, removing moire 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 demoired results. It can generalize well to the task of removing moire artifacts caused by display screens. In addition, we build a new moire dataset, including images with screen and texture moire artifacts. As far as we know, this is the first dataset with real texture moire patterns.