Image compression has always been an important topic in the last decades due to the explosive increase of images. The popular image compression formats are based on different transforms which convert images from the spatial domain into compact frequency domain to remove the spatial correlation. In this paper, we focus on the exploration of data-driven transform, Karhunen-Loéve transform (KLT), the kernels of which are derived from specific images via Principal Component Analysis (PCA), and design a high efficient KLT based image compression algorithm with variable transform sizes. To explore the optimal compression performance, the multiple transform sizes and categories are utilized and determined adaptively according to their rate-distortion (RD) costs. Moreover, comprehensive analyses on the transform coefficients are provided and a band-adaptive quantization scheme is proposed based on the coefficient RD performance. Extensive experiments are performed on several class-specific images as well as general images, and the proposed method achieves significant coding gain over the popular image compression standards including JPEG, JPEG 2000, and the state-of-the-art dictionary learning based methods.