Application and Optimisation of the Spatial Phase Shifting ...

3.2 Phase-shifting ESPI 53
known since decades [Bru74] and has recently been referred to as DFT (digital Fourier transform) formula
[Sur96]:
ϕ
O
N −1
− ∑ In
sinαn
π
mod 2π
n 0
2
= arctan
=
with α
N
n = n⋅
.
−1
(3.14)
N
∑ In
cosαn
n=
0
With this choice of the a n and b n , numerator/denominator represent the digital implementation of a
Fourier sine/cosine transform [Bra87, p.17], where α(x,y,t) has an angular frequency of 2π/(N samples)
and the sample interval is in time or space units; the Fourier aspect of phase sampling will be treated in
greater detail in 3.2.2. The signs of numerator and denominator are used to generate a 0-2π arctan, in
contrast to its mathematical definition used in Chapter 2, where it ranges from –π/2 to π/2. This is more
convenient when converting the phases to grey levels.
For 3-step formulae, one can also choose n ∈{-1, 0, 1}, thus assume phase shifts of {-α, 0, α} and write
down the generally valid expression [Cre88, Schwi90, Gre92]
ϕ
O
⎛ 1−
cosα
I−1 − I1
⎞ ⎛ ⎛ α ⎞ I−1 − I1
mod 2 π = arctan⎜
⎟ = arctan⎜
tan⎜
⎟
⎝ sinα
2I − I − I ⎠ ⎝ ⎝ 2 ⎠ 2I − I − I
0 −1 1
0 −1 1
⎞
⎟ . (3.15)
⎠
Much work has been done to improve these simple approaches to very sophisticated sampling schemes,
frequently at the expense of increased N . These are often called algorithms, although their flow diagrams
are trivial; to distinguish them from another class of phase-retrieval methods that are truly algorithms
[Ger72, Fie82, Rav99], I will avoid the term "algorithm" henceforth. Today, there are not only tailored
formulae with excellent rejection of various errors [Schwi83, Har87, Lar92b, Sur93, Schwi93, dGro95,
Hib95, MYo95, Schmi95a, dGro97, Hib97, Küch97, Ser97b, Sto97, Zha99], but also, the properties of
phase-shifting formulae are by now so well understood [Fre90a, Lar92a, Rat95, Sur96, Phi97, Sur97b,
Sur98c, Dor99] that for many purposes phase-extraction schemes can be tailored to adapt to the particular
task. Good measurements reach an accuracy of about λ/100 [Schwi83, Har87].
But the basic approaches with N=3 to 5 have survived in ESPI because superb theoretical accuracy would
remain theoretical where speckle noise and decorrelation set the limits. Also, since ESPI is obviously not
concerned with precision surfaces, the requirements are often lower.
Moreover, a small N helps to determine phases very quickly: since the most time-consuming step in phase
calculation is the arctangent operation, it is advantageous to map all possible values of numerator and
denominator in two-dimensional look-up tables (LUTs).
The size of these LUTs depends on the digital resolution as well as on the respective number of samples
involved. In the case of (3.15) with 8-bit digitisation, the LUT would have 5111021 entries, because the
numerator can range from –255 to 255 and the denominator from –510 to 510. These integers then serve
as matrix indices to retrieve the associated phase value, which is often represented by an 8-bit integer as