Application and Optimisation of the Spatial Phase Shifting ...

3.2 Phase-shifting ESPI 61
These formulae help to save processing time, since (i) no intermediate images are formed, and (ii) only
one arctangent calculation per pixel is required. But due to the involved multiplications, this method can
be accelerated by LUTs only if enormous storage space or substantial data reduction in the LUT are
acceptable.
As the complex-division method is mathematically equivalent to the difference-of-phases approach, the
performance in terms of σ ∆ϕ is exactly the same for both of them. It has however been demonstrated in
[Vik93] that phase differences can be determined from six intensity samples even with an unknown phase
shift.
3.2.2 Spectral transfer properties of few-sample phase shifting formulae
In our context of spatial phase shifting, the number of phase samples must be as small as possible, e.g.
three or four; at the same time, the phase extraction method should possess the best possible tolerance of
speckle intensity and phase gradients. The latter cause deviations of the phase shift from its nominal
value. A valuable tool to investigate the behaviour of phase-sampling formulae under linear phase-shift
miscalibrations (also called "detuning") is the so-called "Fourier description" of phase-shifting formulae.
It was begun in [Ohy86, Ohy88], developed to its full potential in [Fre90a] and is nowadays a common
tool to assess the performance of phase-sampling formulae [Lar92a, Hib95, MYo95, Schmi95a, Hib97,
Zha99, Mal00]. We will restrict the discussion to linear miscalibration sensitivity here, for which the
Fourier description is particularly suitable. Moreover, it will provide a means to quantify how the signal
sidebands in the frequency spectra of SPS interferograms (cf. Fig. 3.29) will be used and/or altered by the
phase calculation.
To understand the behaviour of some few-sample methods in the frequency domain, we will briefly review
the underlying principles. Some emphasis is put on the spatial version of phase extraction; but the phaseshift
parameter x, denoting one spatial co-ordinate, can be replaced by t as well. As (3.14) indicates, the
general task in phase determination is to generate signals that are proportional to sine and cosine of the phase
of an unknown signal, say, I(x), and then extract its phase ϕ by an arctangent operation. We start with the
continuous (analogue) description of the process, which will help to clarify the properties of the discrete
(digital) version. An extensive overview of the formalism, and also of the spectral characteristics of many
phase-shifting formulae besides the ones that we will examine here, can be found in [Mal98, pp. 113-245].
3.2.2.1 Analogue synchronous detection
When I(x) is modulated with a so-called carrier frequency ν x , we can write
I ( x) = Ib ( x) + M I ( x) ⋅ cos( ϕ( x) + 2 πνx
x)
(3.30)
and use the well-known method of "synchronous detection" to extract the phase ϕ(x) in (3.30). An early
application of this method to spatial fringe analysis has been given in [Ich72]; moreover, it is the principle
upon which lock-in amplifiers are based. The first step of synchronous detection is to multiply the input