Robust phase unwrapping based on non-coprime fringe pattern periods for deflectometry measurements

Phase-measuring deflectometry is a technique for non-contact inspection of reflective surfaces. A camera setup captures the reflection of a sine-modulated fringe pattern shifted across a screen; the location-dependent measured phase effectively encodes the screen coordinates. As the used fringe patterns are much narrower than the screen dimension, the resulting phase maps are wrapped. The number-theoretical solution uses the Chinese remainder theorem to calculate an unwrapped phase map from repeated measurements with coprime fringe widths. The technique is highly susceptible to phase noise, i.e. small deviations of the measured phase values generally lead to unwrapped phase values with large errors. We propose a modification and show how non-coprime period widths make phase unwrapping robust against phase noise. Measurements with two non-coprime fringe period widths introduce the opportunity to discriminate between “legal” measured phase value pairs, that potentially originate from noise-free measurements, and “illegal” phase value pairs, that necessarily result from noise-affected measurements. Arranged as a matrix, the legal measurements lie on distinct diagonals. This insight not only allows to determine the legality of a measurement, but also to provide a correction by looking for the closest legal matrix entry. We present an experimental comparison of the resulting phase maps with reference phase maps. The presented results include descriptive statistics on the average rate of illegal phase measurements as well as on the deviation from the reference. The measured mean absolute deviation decreases from 1.99 pixels before correction to 0.21 pixels after correction, with a remaining maximum absolute deviation of 0.91 pixels

On Solving a Generalized Chinese Remainder Theorem in the Presence of Remainder Errors

In estimating frequencies given that the signal waveforms are undersampled multiple times, Xia et. al. proposed to use a generalized version of Chinese remainder Theorem (CRT), where the moduli are $M_1, M_2, \cdots, M_k$ which are not necessarily pairwise coprime. If the errors of the corrupted remainders are within \tau=\sds \max_{1\le i\le k} \min_{\stackrel{1\le j\le k}{j\neq i}} \frac{\gcd(M_i,M_j)}4, their schemes can be used to construct an approximation of the solution to the generalized CRT with an error smaller than $\tau$. Accurately finding the quotients is a critical ingredient in their approach. In this paper, we shall start with a faithful historical account of the generalized CRT. We then present two treatments of the problem of solving generalized CRT with erroneous remainders. The first treatment follows the route of Wang and Xia to find the quotients, but with a simplified process. The second treatment considers a simplified model of generalized CRT and takes a different approach by working on the corrupted remainders directly. This approach also reveals some useful information about the remainders by inspecting extreme values of the erroneous remainders modulo $4\tau$. Both of our treatments produce efficient algorithms with essentially optimal performance. Finally, this paper constructs a counterexample to prove the sharpness of the error bound $\tau$