Application and Optimisation of the Spatial Phase Shifting ...

6.5 Fourier transform method of phase determination 153
straight lines of pixels anymore, which leads to drawbacks already mentioned; and in the latter case, the
averaging hardly makes sense due to very poor statistical independence of the pixel sets. Accordingly,
these approaches do not deliver any improvement in the whole range of N x values over the results already
shown. Therefore it is also doubtful whether the formally expected increase in spatial resolution would
actually turn up: very fine fringes might just disappear in higher noise.
To continue our quest for maximal accuracy in sawtooth images from spatially phase shifted
interferograms, we will now put aside the phase-shifting methods in favour of the more general concept of
the spatial frequency plane.
6.5 Fourier transform method of phase determination
In the discussion of 3.2.2, we saw that the spectral transfer functions of phase-sampling formulae are
designed to function correctly at their nominal frequency only. While considerable improvements are
possible by simple means, all phase-shifting formulae tend the more to falsify the signal the broader the
sidebands are. So, instead of looking for a phase-evaluation window that delivers low noise while being as
small as possible, one could switch to the other end of the scale and use instead a very large window: the
whole image. Since the signal is encoded in a spectral sideband, it is quite natural – and convenient – to
retrieve its phase from frequency space by a Fourier transform method (henceforth abbreviated by FT or
FTM). It has been applied also to interferograms without a signal carrier [Kre86]; but that approach
requires a-priori knowledge or one temporal phase shift to eliminate the sign ambiguity.
Although it would require sophisticated hard- and software even today to maintain the real-time capability
of an ESPI system with carrier frequency and FT phase calculation, we do investigate the effect of it as a
possible means of a posteriori data processing that still can run entirely automatically. It is intuitively
clear that this approach should offer a distinct advantage over phase sampling: while phase sampling
always works with local information from a very short sequence of samples, the FTM, as a global method,
has access to all the image information simultaneously.
The way to retrieve phase information modulated on a carrier frequency by means of Fourier transforms
has been described in [Tak82, Rod87]. The FTM lends itself to, inter alia, profilometry [Tak83], moiré
[Mor94a], holographic [Qua96] and speckle interferometry [Sal96]. Here, we will of course consider the
method with emphasis on speckle interferometry and also generalise the original 1-D treatment to two
frequency dimensions, as first suggested in [Bon86].
There are numerous analyses as to the attainable accuracy [Mac83, Gre88, Kuj91c, Jo_92], with the main
results that the interferogram should be multiplied by an appropriate window function, or extrapolated, to
minimise edge truncation effects; but as shown in [Koz99], they can also be eliminated exactly. We will
not deal with such refined methods because (i) our digital resolution is rather large (10241024 pixels),
so that the edge effects play a relatively small role, and (ii) the benefit for speckle images would hardly be
significant.