In this work, the amplification-superposition principle of high-order moiré imaging is analyzed. Based on the first-order geometric-transformation-moiré method, an improved design and analysis method for moiré magnifiers by distinguishing the unit amplification and periodic amplification from the moiré imaging is proposed. Our method is used to simulate high-order moiré imaging in MATLAB. By designing and fabricating various high-order moiré magnifiers, an experimental analysis of high-order moiré imaging is also carried out. The results embody the amplification-composition effect of high-order moiré imaging, thereby validating our method.