Image compression has been an important research topic for many decades. Recently, deep learning has achieved great success in many computer vision tasks, and its use in image compression has gradually been increasing. In this paper, we present an energy compaction-based image compression architecture using a convolutional autoencoder (CAE) to achieve high coding efficiency. Our main contributions include three aspects: 1) we propose a CAE architecture for image compression by decomposing it into several down(up)sampling operations; 2) for our CAE architecture, we offer a mathematical analysis on the energy compaction property and we are the first work to propose a normalized coding gain metric in neural networks, which can act as a measurement of compression capability; 3) based on the coding gain metric, we propose an energy compaction-based bit allocation method, which adds a regularizer to the loss function during the training stage to help the CAE maximize the coding gain and achieve high compression efficiency. The experimental results demonstrate our proposed method outperforms BPG (HEVC-intra), in terms of the MS-SSIM quality metric. Additionally, we achieve better performance in comparison with existing bit allocation methods, and provide higher coding efficiency compared with state-of-the-art learning compression methods at high bit rates.