Multi-frequency Phase Unwrapping from Noisy Data: Adaptive Local Maximum Likelihood Approach

Abstract

Abstract. The paper introduces a new approach to absolute phase estimation from frequency diverse wrapped observations. We adopt a discontinuity preserving nonparametric regression technique, where the phase is reconstructed based on a local maximum likelihood criterion. It is shown that this criterion, applied to the multifrequency data, besides filtering the noise, yields a 2πQ-periodic solution, where Q > 1 is an integer. The filtering algorithm is based on local polynomial (LPA) approximation for the design of nonlinear filters (estimators) and the adaptation of these filters to the unknown spatially smoothness of the absolute phase. Depending on the value of Q and of the original phase range, we may obtain complete or partial phase unwrapping. In the latter case, we apply the recently introduced robust (in the sense of discontinuity preserving) PUMA unwrapping algorithm [1]. Simulations give evidence that the proposed method yields state-of-the-art performance, enabling phase unwrapping in extraordinary difficult situations when all other algorithms fail.

A Closed-Form Robust Chinese Remainder Theorem and Its Performance Analysis

“…With the erroneous remainders in (10), (1) becomes (15) When the folding integers in (15) are accurately solved, the unknown parameter can be estimated as (16) and thus , i.e., is a robust estimate of . We now show how to accurately determine the folding integers as in [8].…”

“…The above sufficient condition is the same as that in [8] for the searching based robust CRT as we described in Section II. Similar to the searching based robust CRT, after every for is uniquely determined, the unknown parameter can be estimated as (16) and the estimate error of is thus upper bounder by . Therefore, the above reconstruction algorithm is robust similar to the searching based robust CRT obtained in [8].…”

Multi-frequency Phase Unwrap from Noisy Data: Adaptive Least Squares Approach

“…LPA is exploited with the uniform square windows w h defined on the integer symmetric grid {(x, y) : |x|, |y| ≤ h}; thus, the number of pixels of w h is (2h + 1). The ICI parameter was set to Γ = 2.0 and the window sizes to H ∈ {1, 2, 3, 4} and H = H − 1 for ML-MF-PEARLS (see [10] for details) and LS-MF-PEARLS algorithms, respectively. The frequencies (ĉ 2,s ,ĉ 3,s ) defined in (12) were computed via FFT zero-padded to the size 64 × 64.…”

“…Tables I and II show root mean square errors (RMSE) for the following algorithms: PEARLS, introduced in [9], LS-MF-PEARLS (the proposed algorithm), and ML-MF-PEARLS, introduced in [10]). LS-MF-PEARLS and ML-MF-PEARLS shows systematically better accuracy and manage to unwrap the phase when single frequency algorithms fail.…”

“…In this paper, we use the wrapped phase prefiltering technique developed in [9] for a single frequency phase unwrapping. The results presented in this paper is a further elaboration of the maximum likelihood approach proposed and analyzed in [10].…”