NOISY PHASE UNWRAP FOR FRINGE TECHNIQUES: ADAPTIVE LOCAL POLYNOMIAL APPROXIMATIONS Vladimir Katkovnik, Jaakko Astola, Karen Egiazarian Signal Processing Institute, University of Technology of Tampere, P. O. Box 553, Tampere, Finland. E-mail: katkov@cs.tut.fi. ABSTRACT Many imaging systems deal with phase measurements using coherent radiation in order to illuminate objects. The reflected scattered return carries information on physical and geomet- rical properties of illuminated objects. It can be information on shape, deformation, movement and structure of the ob- ject’s surface. We propose a novel phase unwrap filtering for noisy holographic data. Based on the window size adaptive local polynomial approximation it allows dramatically reduce level of noise and in the same time preserve phase variation features. Simulation shows that the technique enables an ad- vanced accuracy for phase reconstruction from wrapped noisy observations. Index Terms— Fringe techniques, hologram, local poly- nomial approximation, phase unwrap, speckle interferometry. 1. INTRODUCTION A variety of imaging systems deal with phase measurements using coherent radiation in order to illuminate objects. The reflected scattered return carries information on physical and geometrical properties of the illuminated objects. It can be information on shape, deformation, movement and structure of the object’s surface. Common to these applications is that the observations are periodical functions of the phase which can be interpreted as the principal phase value defined on the interval [−π, π). Ac- cordingly, it is impossible to unambiguously reconstruct the original, nonrestricted values, hereafter referred to as the true phase, unless additional assumptions are introduced. If a true phase value is outside the principal interval [−π, π), the ob- served value is wrapped into this interval, corresponding to an addition or subtraction of an integer number of 2π. Many approaches start from estimation of the phase for the princi- pal interval and further extend these estimates to nonrestricted values. This last procedure is known as phase unwrapping. What make this problem even more dif ficult is that the mea- sured values are usually corrupted by noise. The standard formulation of the phase unwrapping starts from the observation model in the form z φ = W (ϕ + n ϕ ), where ϕ is the true phase, n ϕ is a random noise additive to the phase ϕ, z φ is the observed phase. Here W is a wrapping operator transforming the noisy phase to the basic phase in- terval [−π, π). There is an obvious link between the wrapped φ and non-wrapped true phase ϕ ϕ = φ +2πk, φ ∈ [−π,π), (1) where k is an integer. The wrapping operator is equivalent to division by module 2π, φ = mod{ϕ + π, 2π} − π, which sep- arate ϕ on two parts: fractional and integer parts. The latter is defined as 2πk with an integer k. The basic unwrapping problem is to reconstruct ϕ(x, y), x, y ∈ X, from the ob- servations z φ (x, y), x, y ∈ X. In our simulation we assume that X is an integer 2D grid, X = {x, y : x =1, 2, ...N 1 , y =1, 2, ...N 2 }. The equation (1) shows that there is no one- to-one relation between the wrapped and unwrapped phase. Surprisingly, differentiation of the observations z ϕ can re- solve this ambiguity or at least to reduce it dramatically. The techniques developed for phase unwrapping can be roughly separated in two large classes. The algorithms of the first class use a two stage approach with estimation of the gradient at the first stage and the following integration of this gradient at the second stage. There are two main dif ficulties in this approach. First, the non-alising sampling conditions sufficient for perfect reconstruction of the phase from the gra- dient: −π ≤ ∆ x ϕ(x, y) <π, − π ≤ ∆ y ϕ(x, y) < π, (2) ∆ x ϕ(x, y)= ϕ(x, y) − ϕ(x − 1,y), ∆ y ϕ(x, y)= ϕ(x, y) − ϕ(x, y − 1), often are not fulfilled for noisy phase φ. Then smoothness assumptions imposed on the true phase are used for regular- ization of the problem in order to avoid the aliasing affects. Second, numerical differentiation as well as numerical inte- gration are not trivial operation for noisy data as the noise suppression should be incorporated. In particular, the differ- entiation results in increasing the noise level and the integra- tion is an inverse ill-conditined operation. The algorithms of the second class are based on direct approximation of the reconstructed phase. In this way the dif- ferentiation as well as the integration are completely avoided.