Phase Unwrapping with Kalman Filter based Denoising in Digital Holographic Interferometry P. Ram Sukumar ∗ , Rahul G. Waghmare ∗ , Rakesh Kumar Singh † , G. R. K. S. Subrahmanyam ‡ , Deepak Mishra ∗ ∗ Dept. of Avionics, Indian Institute of Space Science and Technology, Thiruvananthapuram, India 695547 deepak.mishra@iist.ac.in † Dept. of Physics, Indian Institute of Space Science and Technology, Thiruvananthapuram, India 695547 ‡ Dept. of Earth and Space Sciences, Indian Institute of Space Science and Technology, Thiruvananthapuram, India 695547 Abstract—Phase information recovered through interferomet- ric techniques is mathematically wrapped in the interval (-π,π]. Obtaining the original unwrapped phase is very important in numerous number of applications. This paper discusses a Fourier transform based phase unwrapping method. Kalman filter is proposed for denoising in post processing step to restore the unwrapped phase without any noise. The proposed method is highly robust to noise and performs better even at lower SNR values (5-10dB) with a very less value of RMS error. Also, the time taken for execution is very less compared to the many available methods in the literature. Keywords—Phase unwrapping, Kalman filter, Fast phase un- wrapping, Digital holographic interferometry I. I NTRODUCTION Extracting the phase information encoded in the interfer- ence pattern is vital in most of the interferometric techniques. Phase information relates to a distinct physical quantity in various interferometric techniques such as surface topography in SAR Interferometry, mapping the internal structures of human body in MRI imaging, Non-destructive testing and deformation assessment in Digital holographic interferometry etc. Underlying phase of the signal can be extracted using arctan function and retrieval of the phase in this method often limits the phase to lie in its principal range (-π,π] and is known as wrapped phase [1]. However true phase may vary over any range greater than 2π. True phase can be expressed in terms of addition of wrapped phase φ w (r) and integer multiples of 2π as φ(r)= φ w (r)+2πn(r) (1) where r refers to the pixel location and n(r) is the integer value at each pixel. Wrapping operation introduces artificial discontinuities in the retrieved phase pattern. Unwrapping operation is addition or subtraction of integer multiples of 2π at each pixel value to obtain the unwrapped phase map without any induced discontinuities. Most of the algorithms for phase unwrapping present in the literature can be broadly classified in to two class of approaches [2] as path following algorithms and least- squares approach. In path following algorithms, the path for unwrapping is chosen such that first areas of lesser incon- sistencies are processed and then further progressed to areas of higher inconsistencies. Quality maps [3], [4], [5] are used as metric to determine the quality of pixels and assess the consistent and inconsistent areas in the phase map. However, unique solution cannot be obtained for unwrapping by these approaches and depends on the path chosen for processing. Some of the methods which employ path following strategies for unwrapping are branch-cut algorithm proposed by Huntley [6], Flynn’s mask cut Algorithm [7], tree branch cut technique proposed by Goldstein et. al [8]. Some of the methods which employ Least squares approach are non-weighted least squares approach proposed by Hunt [9], Weighted least squares ap- proach proposed by Flynn in [10] and L p norm criteria pro- posed by Ghiglia in [11]. Though the approach for unwrapping is unique in each algorithm, the basic concept of unwrapping remains the same. All the unwrapping methods available in the literature don’t consider the effect of noise in the wrapped phase and result in noisy phase maps and frequently employ ad-hoc heuristic techniques such as median filtering to remove noise. The piecewise polynomial approximation approach [12] and signal tracking approach [15], [16] provides unwrapped phase directly, but the non-linear measurement model limits the performance of those methods. In this paper we propose a mathematically sound and more reliable restoration approach for the estimation of accurate phase map by exploiting the properties of Taylor series expansion with Kalman filter frame- work. For the comparison purpose, we are using the outcomes of FFT based phase unwrapping method proposed in [13]. For the sake of completeness, we explain the FFT based method in section-II. It is observed that, the denoised estimates obtained by the proposed method are comparatively more accurate than the state-of-the-art approaches. Our work is validated by experimental measurements taken from Digital holographic Interferometry. The rest of this paper is organized as follows. Section-II discusses the FFT method for phase unwrapping, Section-III discusses the proposed Kalman filter approach for denoising the unwrapped estimate and section-IV discusses the performance of proposed algorithm when tested on simulated and real holographic data and section-V concludes the paper. II. FFT METHOD FOR PHASE UNWRAPPING This section discusses briefly about the FFT method for phase unwrapping discussed in [13]. True phase can be ex- pressed in terms of wrapped phase as shown by equation 1. The aim is to determine the n(r) to obtain the unwrapped phase. This can be accomplished by formulating the problem initially in terms of Laplacian and further expressing Laplacian in terms of Fourier transform. 2256 978-1-4799-8792-4/15/$31.00 c 2015 IEEE