Adaptive search is one of the fastest fractal compression algorithms and has gained great success in many industrial applications. By substituting the luminance offset by the range block mean, the authors create a completely new version for both the encoding and decoding algorithms. In this paper, theoretically, they prove that the proposed decoding algorithm converges at least as fast as the existing decoding algorithms using the luminance offset. In addition, they prove that the attractor of the decoding algorithm can be represented by a linear combination of range-averaged images. These theorems are very important contributions to the theory and applications of fractal image compression. As a result, the decoding image can be represented as the sum of the DC and AC component images, which is similar with discrete cosine transform or wavelet transform. To further speed up this algorithm and reduce the complexity of range and domain blocks matching, they propose two improvements in this paper, that is, employing the post-quantisation and geometric neighbouring local search to replace the currently used pre-quantisation and the global search, respectively. The corresponding experimental results show the proposed encoding and decoding algorithms can provide a better performance compared with the existing algorithms.