Application and Optimisation of the Spatial Phase Shifting ...

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4 Quantification of displacement-measurement errors
Since a major part of this work deals with the "quality" of displacement phase maps, it is vital to have a
numerical figure of merit at hand that allows to compare measurements accurately enough. While the
human brain's image processing allows to tell a "bad" sawtooth image ∆ϕ(x,y) from a "good" one at just a
glance, it runs into problems when small quality differences have to be found or even quantified.
Therefore, we must find a reliable and standardised method to determine noise levels numerically.
The general problem when determining the noise in measured displacement phase maps ∆ϕ meas (x,y) is:
what does the noise-free reference phase map ∆ϕ ref(x,y) look like, and how can one obtain it? In practice,
unless excellently calibrated displacements are available, one has to fall back upon the actual
measurement. One common approach is to generate ∆ϕ ref(x,y) by spatially smoothing the noisy phase map
∆ϕ meas (x,y) as much as possible and to obtain an average displacement phase-measurement error
δϕ=F∆ϕ meas (x,y)–∆ϕ ref(x,y)F or a so-called root-mean-square (r.m.s.) displacement phasemeasurement
error σ ∆ϕ = ( ∆ϕ
( x, y) − ∆ϕ
( x, y))
meas
ref
2
from a comparison of the "raw" and the
smoothed data. Such approaches are widely used and give reasonable results, but the best way to reduce
the noise in a sawtooth image will most likely depend on the input image; this is, the smoothing filter's
parameters and/or the number of iterations remain a matter of user judgement. Since we intend to
compare TPS and SPS, and to find improved phase-extraction methods for SPS later on, we need
comparable performance data throughout a very wide range of fringe densities and noise levels, so that
smoothing images "by hand" does not seem to be universal and accurate enough. Therefore, to generalise
the process of finding the best-matching ∆ϕ ref(x,y), I felt the need to develop an almost fully automatic
procedure.
4.1 Previous methods
We start with a brief survey of some existing noise reduction methods; while their objective has seldom
been an accurate quantification of experimental errors, their purpose is certainly to improve the reliability
of experimental data, which happens by approximating ∆ϕ ref(x,y), the true phase map, as closely as
possible. Although we are aiming at a method to evaluate sawtooth images, we also include some
achievements of noise handling in secondary interferograms. We will, however, put some emphasis on the
processing of sawtooth images and point out specific difficulties with various filtering schemes.
4.1.1 Processing of correlation fringes
There is a wealth of smoothing and filtering methods to generate clearer fringes from ESPI subtraction
images that can then be used for the phase-of-difference method, or possibly for direct evaluation. It is
important to realise that the design of filters for correlation fringes must take into account that the speckle
noise is multiplicative in secondary interferograms. This is not a generic property of the speckle effect