This paper proposes a new family of 4-dimensional chaotic cat maps. This family is then used in the design of a novel block-based image encryption scheme. This scheme is composed of two independent phases, a robust light shuffling phase and a masking phase which operate on image-blocks. It utilizes measures of central tendency to mix blocks of the image at hand to enhance security against a number of cryptanalytic attacks. The mixing is designed so that while encryption is highly sensitive to the secret key and the input image, decryption is robust against noise and cropping of the cipher-image. Empirical results show high performance of the suggested scheme and its robustness against well-known cryptanalytic attacks. Furthermore, comparisons with existing image encryption methods are presented which demonstrate the superiority of the proposed scheme.