Application and Optimisation of the Spatial Phase Shifting ...

160 Improvements on SPS
uncertain, frequently occur at locations that are different in the v and h fields. This gets clear when we
express the probability of finding a "bad" pixel (denoted by subscript b) at (x,y) in either speckle pattern
by P vb (x,y) and P hb (x,y) and that of finding a bad pixel in both speckle patterns by P bb (x,y). Then we have
Pbb ( x, y) ≅ Pvb ( x, y) Phb
( x, y)
, (6.24)
which is exact when c=0. Provided both P vb (x,y) and P hb (x,y) are distinctly smaller than unity, this means
that it is possible to replace most of the bad pixels from one speckle pattern by valid pixels from the other
one. There are of course always several points (even for c = 0) where bad pixels in both speckle fields
coincide. But in any case, the number of bad points in the phase map can be minimised by suitable
merging of ϕ O,v (x,y) and ϕ O,h (x,y).
The merging process is carried out by analysing M I (x,y) in the interferograms between the reference wave
(ideally linearly polarised at 45°) and the vertically or horizontally polarised object wave [Cre88],
2
I , vi 1vi 3vi 2vi 1vi 3vi
M ( x, y) = 3( I − I ) + ( 2I − I − I )
2
2
(6.25)
M ( x, y) = 3( I − I ) + ( 2I − I − I ) ,
I , hi 1hi 3hi 2hi 1hi 3hi
where α=120°/sample and the index i refers to the initial object state, for reasons to become clear shortly.
It should be emphasised that these M I are derived from the "sine" and "cosine" terms of (3.17), which
must then be used for the subsequent phase determination; if other α, or phase-calculation formulae, are
used, the respective "sine" and "cosine" terms have to be inserted under the square root. Due to the low
correlation between the v and h speckle patterns, the two maps of M I,vi (x,y) and M I,hi (x,y) will also be
different, and the higher of the two values ought to indicate the prospect of a more accurate phase
measurement. Admittedly, (6.25) is not as reliable in SPS as in TPS [Su 94] due to the underlying speckle
structure that may yield bogus modulation when the pixel triplet crosses speckle "boundaries"; but as far
as a comparison of M I,vi (x,y) and M I,hi (x,y) is concerned, this approach still works rather well, as we shall
see.
The interferograms are recorded by a CCD camera behind a polariser in the vertical or horizontal position;
setting the plane of polarisation of the reference wave to ideally 45° assures P vb (x,y)P hb (x,y). For each
point (x,y) in both interferograms we determine M I,vi (x,y) and M I,hi (x,y) and the phase distributions
ϕ O,vi (x,y) and ϕ O,hi (x,y). Then, starting with ϕ O,vi (x,y), we replace all phase values in this map by those
from ϕ O,hi (x,y) at all the locations where M I,vi (x,y) < M I,hi (x,y). Thus, a pixel is considered "bad" in the
sense of (6.24) when a better measurement is available. The locations of replaced pixels are stored in a
binary mask B i (x,y).
Repeating this modulation analysis for the final object state would lead to a slightly different map B f (x,y)
due to speckle decorrelation by the object deformation and statistical temporal fluctuations like camera
noise. Therefore, B i (x,y) is used for the final object state too. That is, the phase values in the map
ϕ O,vf (x,y) are replaced by those of the map ϕ O,hf (x,y) at the same locations where the replacement is done
2