Abstract: The periodicity of Fibonacci-Q transformation and its inverse one is discussed．The periodic relation of Fibonacci-Q and Arnold transformation is proved．Using the intrinsic relation between Fibonacci-Q and Arnold transformation，a new nonlinear transformation，named A-F transformation，is defined．The periodicity of A-F transformation and its inverse one is studied．A simple algorithm to work out their periodicity is presented．The periodicity of A-F and Fibonacci-Q is quite different from that of Arnold，which shows better security in scrambling digital image．