Noise robust linear dynamic system for phase unwrapping and smoothing

Julio C Estrada,Juan A Quiroga,Manuel Servin

doi:10.1364/oe.19.005126

Abstract

Phase unwrapping techniques remove the modulus ambiguities of wrapped phase maps. The present work shows a first-order feedback system for phase unwrapping and smoothing. This system is a fast sequential unwrapping system which also allows filtering some noise because in deed it is an Infinite Impulse Response (IIR) low-pass filter. In other words, our system is capable of low-pass filtering the wrapped phase as the unwrapping process proceeds. We demonstrate the temporal stability of this unwrapping feedback system, as well as its low-pass filtering capabilities. Our system even outperforms the most common and used unwrapping methods that we tested, such as the Flynn's method, the Goldstain's method, and the Ghiglia least-squares method (weighted or unweighted). The comparisons with these methods shows that our system filters-out some noise while preserving the dynamic range of the phase-data. Its application areas may cover: optical metrology, synthetic aperture radar systems, magnetic resonance, and those imaging systems where information is obtained as a demodulated wrapped phase map.

Highlights

There are several applications where information of interest is phase modulated within the received signals and recovered as a wrapped phase
In optical metrology the information of testing events is phase modulated within the interferograms by the use of optical interferometers [3]
Other examples where information is phase modulated occur in synthetic aperture radars, magnetic resonance images, acoustic imaging, and X-ray crystallography

Summary

Introduction

There are several applications where information of interest is phase modulated within the received signals and recovered as a wrapped phase. The function W [·] is the modulus 2π operator; the main use of this operator in the phase derivative (or differences in a discreet domain) is to wrap or remove the outliers of the difference operator generated by the 2π phase jumps of the wrapped phase This 1D line integration may be stated as the solution of the following continuous system dφ(t) = W dφw(t) ,. The linear system of Eq (4) can be rewritten as φ(n) − (1 − τ)φ(n − 1) − τφw(n) = 0 To solve this difference equation we can use the z-transform, and obtain that the response of our dynamic phase unwrapping (4) for any arbitrary input φw(n) is: n φ(n) = (1 − τ)n+1φ(−1) + τ ∑ (1 − τ)kφw(k). The frequency response of the system behaves like a low-pass filter and its bandwidth is controlled by the parameter τ

Two-dimensional extension

Tests and results

Conclusions