Application and Optimisation of the Spatial Phase Shifting ...

6.7 Extensions of SPS by temporal unwrapping 165
Dj
(
x, y) mod 2p
6p
F (
x, y)
4p
2p
2p n(x, y)
n=1 n=2 n=3
x
j
O ( x,y, t) mod 2p
6
p
F (
x, y, t)
4p
2p
2
pn(x, y, t)
n=1 n=2 n=3
t
Fig. 6.23: Principle sketch of spatial (top row) and temporal (bottom row) phase unwrapping; for details, see text.
In spatial unwrapping, a sawtooth image (top, left), representing a temporal phase history ∆ϕ(x, y) mod 2π
= (ϕ O (x, y, t f ) mod 2π–ϕ O (x, y, t i ) mod 2π) mod 2π, with i and f referring to undeformed and deformed
object state, is converted to a continuous displacement phase Φ(x, y) (top, right) by appropriate additions
of 2π, i.e. by finding the correct step function 2πn(x,y), n∈. This is done by a simple criterion: when
the data satisfy the sampling theorem spatially, there will be no phase changes >π from pixel to pixel. If
such a transition is detected nevertheless, it must then be a 02π jump that is wrapped back onto [0, π)
by in- or decrementing n. This procedure along the x direction at an image row y is sketched in Fig. 6.23
in the centre of the upper row.
Temporal unwrapping starts from an empty displacement map (bottom, left) and tracks the phase history
of every pixel (x,y) in time by comparing it with an initial phase map ϕ O (x, y, t i ). The unwrapping criterion
is applied temporally, as shown in the centre of the bottom row for some pixel (x, y); the temporal
sampling rate must be high enough to keep differences of ϕ O (x, y, t) from frame to frame smaller than π on
each pixel, this is, the sampling theorem must be fulfilled temporally. The phase differences are
unwrapped by addition of 2πn(x, y, t) as required and used to continuously update Φ(x, y, t), which may
conveniently be represented by grey levels as well, as on the right in the bottom row. The advantage of
this method is that errors due to faulty – mostly badly modulated – pixels will not spread across the image
as this may be the case for spatial unwrapping.
In longer monitoring sequences however, the advantage of TPU can become a disadvantage: it
accumulates data, including errors, and severely corrupted unwrapped phase maps Φ(x, y) cannot be
restored a posteriori. It is therefore favourable to store both the temporally unwrapped data and several
phase maps ϕ O (x, y, t). If the former are doubtful, the latter may yield conventional sawtooth images
∆ϕ (x, y) that can easily be unwrapped spatially when they contain few fringes.