This paper proposes a block-based reversible data embedding method applied to grayscale cover images. The image is first divided into two areas: data embedding area and auxiliary information embedding area. The data embedding area is further partitioned into 3∗3 blocks. According to two predefined thresholds, all the blocks are divided into three categories: smooth, normal or complex. The complex blocks do not embed any data. A smooth block will embed twice the data of a normal block. After predicting the edge direction in a smooth or normal block, Alattar's difference expansion of a generalized integer transform method is exploited to embed data into each vector in the block. Then, the location map and other auxiliary information are embedded in the auxiliary information embedding area using a novel least significant bit (LSB) substitution method. Our method not only improves the embedded image quality but also provides a higher payload capacity compared to Alattar's method. For some smoother images, the embedding capacity of the proposed method is higher than 1 bit/pixel in a single pass and also derives acceptable embedded image quality (PSNR⩾30). To the best of our knowledge, the proposed method is the first to do this.