Projection moir$\acute{e}$ topography is useful tool for measuring 3-D form of large object. It provides interferometric moir$\acute{e}$ fringe of which phase is function of height in object space. By adapting phase-shifting technique in moir$\acute{e}$ fringe analysis, 3-D information can be easily obtained. But the phase of moir$\acute{e}$ fringe differs from that of traditional interferogram. For common interferometry, the height difference between consecutive fringes is determined by wavelength of illumination source and is constant over the entire measuring range. In a projection moir$\acute{e}$ fringe, fringe distance which is called $\lambda_{eq}$ changes with height in the measuring range. And for many reason like lens aberration, optical axis misalignment and phase-shifting error, phase of moir$\acute{e}$ fringe is not only function of height but also lateral coordinates in the measuring range. These imperfections of measurement system and variation of $\lambda_{eq}$ cause measurement error if fringe is analyzed as a conventional phase-shifting interferogram.
This paper discusses such measurement errors, and proposes N-plane 3-D calibration method to enhance measurement uncertainty in projection moir$\acute{e}$. With this method, distorted absolute moir$\acute{e}$ fringe phase distribution and 2-D camera calibration matrix are obtained by two-frequency projection moir$\acute{e}$ and by N-plane 2-D camera calibration method.
The advantage of this method is that phase to height calibration and 2-D calibration are performed at the same time. And by adapting two-frequency projection moir$\acute{e}$, 2π ambiguity problem which is inherent limitation of phase measuring interferometer can be solved.
The effectiveness of this method is demonstrated by measuring step and radius of cylinder type step specimen. Calibration results prove that the measurement accuracy of calibrated system is less than 0.18% in the ratio of measurement error/calibrated height.
