This letter introduces a two-dimensional bilinear mapping operator referred to as the cubic phase (CP) function. For first-, second-, or third-order polynomial phase signals, the energy of the CP function is concentrated along the frequency rate law of the signal. The function, thus, has an interpretation as a time-frequency rate representation. The peaks of the CP function yield unbiased estimates of the instantaneous (angular) frequency rate (IFR) and, hence, can be used as the basis for an IFR estimation algorithm. The letter defines an IFR estimation algorithm and theoretically analyzes it. The estimation is seen to be asymptotically optimal at the center of the data record for high signal-to-noise ratios. Simulations are provided to verify the theoretical claims.

A new technique for instantaneous frequency rate estimation

Phase unwrapping is an ill-posed classical problem in many practical applications of significance such as 3D profiling through fringe projection, synthetic aperture radar and magnetic resonance imaging. Conventional phase unwrapping techniques estimate the phase either by integrating through the confined path (referred to as path-following methods) or by minimizing the energy function between the wrapped phase and the approximated true phase (referred to as minimum-norm approaches). However, these conventional methods have some critical challenges like error accumulation and high computational time and often fail under low SNR conditions. To address these problems, this paper proposes a novel deep learning framework for unwrapping the phase and is referred to as “PhaseNet 2.0”. The phase unwrapping problem is formulated as a dense classification problem and a fully convolutional DenseNet based neural network is trained to predict the wrap-count at each pixel from the wrapped phase maps. To train this network, we simulate arbitrary shapes and propose new loss function that integrates the residues by minimizing the difference of gradients and also uses $L_{1}$ loss to overcome class imbalance problem. The proposed method, unlike our previous approach PhaseNet, does not require post-processing, highly robust to noise, accurately unwraps the phase even at the severe noise level of −5 dB, and can unwrap the phase maps even at relatively high dynamic ranges. Simulation results from the proposed framework are compared with different classes of existing phase unwrapping methods for varying SNR values and discontinuity, and these evaluations demonstrate the advantages of the proposed framework. We also demonstrate the generality of the proposed method on 3D reconstruction of synthetic CAD models that have diverse structures and finer geometric variations. Finally, the proposed method is applied to real-data for 3D profiling of objects using fringe projection technique and digital holographic interferometry. The proposed framework achieves significant improvements over existing methods while being highly efficient with interactive frame-rates on modern GPUs.

PhaseNet 2.0: Phase Unwrapping of Noisy Data Based on Deep Learning Approach

A fresh phase unwrapping algorithm based on iterated unscented Kalman filter is proposed to estimate unambiguous unwrapped phase of interferometric fringes. This method is the result of combining an iterated unscented Kalman filter with a robust phase gradient estimator based on amended matrix pencil model, and an efficient quality-guided strategy based on heap sort. The iterated unscented Kalman filter that is one of the most robust methods under the Bayesian theorem frame in non-linear signal processing so far, is applied to perform simultaneously noise suppression and phase unwrapping of interferometric fringes for the first time, which can simplify the complexity and the difficulty of pre-filtering procedure followed by phase unwrapping procedure, and even can remove the pre-filtering procedure. The robust phase gradient estimator is used to efficiently and accurately obtain phase gradient information from interferometric fringes, which is needed for the iterated unscented Kalman filtering phase unwrapping model. The efficient quality-guided strategy is able to ensure that the proposed method fast unwraps wrapped pixels along the path from the high-quality area to the low-quality area of wrapped phase images, which can greatly improve the efficiency of phase unwrapping. Results obtained from synthetic data and real data show that the proposed method can obtain better solutions with an acceptable time consumption, with respect to some of the most used algorithms.

Iterated unscented Kalman filter for phase unwrapping of interferometric fringes.

In this paper, we propose a particle-filter-based technique for the analysis of a reconstructed interference field. The particle filter and its variants are well proven as tracking filters in non-Gaussian and nonlinear situations. We propose to apply the particle filter for direct estimation of phase and its derivatives from digital holographic interferometric fringes via a signal-tracking approach on a Taylor series expanded state model and a polar-to-Cartesian-conversion-based measurement model. Computation of sample weights through non-Gaussian likelihood forms the major contribution of the proposed particle-filter-based approach compared to the existing unscented-Kalman-filter-based approach. It is observed that the proposed approach is highly robust to noise and outperforms the state-of-the-art especially at very low signal-to-noise ratios (i.e., especially in the range of -5 to 20 dB). The proposed approach, to the best of our knowledge, is the only method available for phase estimation from severely noisy fringe patterns even when the underlying phase pattern is rapidly varying and has a larger dynamic range. Simulation results and experimental data demonstrate the fact that the proposed approach is a better choice for direct phase estimation.

Particle-filter-based phase estimation in digital holographic interferometry

This paper proposes a technique for the simultaneous estimation of interference phase derivative and phase from a complex interferogram recorded in an optical interferometric setup. The complex interferogram is represented as a spatially varying autoregressive process in a given row or column at a time. The phase derivative is estimated from the poles of the transfer function representation of the autoregressive process. The poles are computed using the spatially varying autoregressive coefficients which are estimated by a computationally efficient Rauch-Tung-Striebel smoothing algorithm. The estimated phase derivative is used as a control input to a state space model designed for the phase estimation at each pixel. The unscented Kalman filter is utilized to deal with the nonlinear measurement process for the accurate estimation of the unwrapped phase. Numerical and experimental results substantiate the ability of the proposed method in handling noisy phase fringe patterns.

https://iopscience.iop.org/article/10.1088/0957-0233/27/6/065203/pdf

Simultaneous estimation of phase derivative and phase using parallel Kalman filter implementation

In this research work, we introduce a novel approach for phase estimation from noisy reconstructed interference fields in digital holographic interferometry using an unscented Kalman filter. Unlike conventionally used unwrapping algorithms and piecewise polynomial approximation approaches, this paper proposes, for the first time to the best of our knowledge, a signal tracking approach for phase estimation. The state space model derived in this approach is inspired from the Taylor series expansion of the phase function as the process model, and polar to Cartesian conversion as the measurement model. We have characterized our approach by simulations and validated the performance on experimental data (holograms) recorded under various practical conditions. Our study reveals that the proposed approach, when compared with various phase estimation methods available in the literature, outperforms at lower SNR values (i.e., especially in the range 0-20 dB). It is demonstrated with experimental data as well that the proposed approach is a better choice for estimating rapidly varying phase with high dynamic range and noise. (C) 2014 Optical Society of America

Signal tracking approach for phase estimation in digital holographic interferometry

Phase estimation plays an important role in various signal processing areas like Radar, Sonar, power systems, speech analysis, communications and many others. The phase of the analytic form of the non stationary signals can be used to find instantaneous frequency. This paper addresses the problem of simultaneous phase and instantaneous frequency estimation from polynomial phase signals embedded in Gaussian noise. Here we have introduced the modified signal tracking approach which is then realized using unscented Kalman filter. The state space model is derived using Taylor series expansion of the phase of polynomial phase signal as process model while Polar to Cartesian conversion as measurement model. Proposed method, compared with state-of-the-art, performs better for signals with higher order polynomial phase variations at lower Signal-to-Noise-Ratio (0-5dB). We also present the simulation results for phase estimation.

Signal tracking approach for simultaneous estimation of phase and instantaneous frequency

Simultaneous measurement of multi-dimensional deformations can be done in digital holographic Moire by illuminating the object with multiple beams from different angles and recording the interference pattern due to multiple beams. This technique involves reliable estimation of resultant multiple phase maps from a one record of the reconstructed interference moire field. In this paper, we introduce a novel method, based on Unscented Kalman Filter (UKF), for the estimation of multiple phase maps from a complex multi-component signal encountered in digital holographic moire. The State space model used in this method is determined by Taylor series expansion of phase component as process model and polar to Cartesian conversion as measurement model.

G. R. K. Sai Subrahmanyam

Papers

Signal tracking approach for phase estimation in digital holographic interferometry

Signal tracking approach for simultaneous estimation of phase and instantaneous frequency

UKF based multi-component phase estimation in digital holographic Moiré