光
子
学
报
( ESPI) ,is a full-field,non-contact,high-accuracy optical measurement technique for micro /nano-deformation
[1-3]
measurement,which has been widely used in manufacturing,aerospace,biomedicine,and so on
. Precise
measurement of deformation by DSPI depends on quantitative determination of interferometric phase. The obtained
raw phase is wrapped and varies within the range of [0,2 ) because of the use of an inverse tangent calculation,
π
[4-5]
an essential process in phase determination
removed before the calculation of deformation. The process of phase discontinuity removal is called as phase
unwrapping.
. This wrapped form causes phase discontinuities which should be
Traditionally,Spatial Phase Unwrapping ( SPU) is the dominant phase unwrapping algorithm that is used in
DSPI. It demodulates phase along a two-dimensional spatial path which is either a predetermined or varying path.
[6]
Distinguished by strategies of routing,a variety of SPUs have been proposed,including row-by-point algorithm
,
[7]
[8]
[9]
p
[10]
quality-guided algorithm ,least-squares algorithm ,branch-cut algorithm ,minimum L -norm algorithm
,
etc. These SPUs can generate high-quality unwrapped phase maps in most cases,due to the use of modern image
sensor with millions of pixels. The high-density image sensor allows an over sampling of the wrapped phase map,
avoiding fault in phase unwrapping. However,in the presence of object surface discontinuity,abrupt phase change
appears,causing local ultrahigh frequency phase variation which is over the image sensor sampling frequency. The
local ultrahigh frequency blocks the path of phase unwrapping,resulting in the failure of real phase distribution
obtainment. The dependence on continuous phase unwrapping path leads to the failure of using DSPI with SPU to
determine phases of discontinuous surfaces.
An alternate to solve the problem is the utilization of Temporal Phase Unwrapping ( TPU) in DSPI. TPU
demodulates the phase through the time axis,so it can neglect the spatial relation between areas of the object
[11]
surface
. Object discontinuity is not a handicap any longer when phase unwrapping. Another advantage is the
[12]
determination of absolute phase value which is realized by recording the initial phase map before loading
. The
disadvantage of TPU is high demand of camera speed which defines the sampling rate of TPU. Unfortunately,the
sampling rate cannot satisfy the sampling theory in most situations,even if high-speed cameras are used. This makes
DSPI with TPU an unpractical tool in engineering.
Recently, we have proposed a Spatiotemporal Three-dimensional Phase Unwrapping ( STPU ) which
[13]
demodulates the phase not only along the two-dimensional spatial path but also through the time axis . The
combination of phase unwrapping in spatial and temporal domains makes DSPI able to determine the absolute phase
by using an ordinary camera. The ability of the new algorithm in deformation measurement of discontinuous surfaces
is also briefly introduced but has not been demonstrated clearly yet. In this article,the principle and procedure of
using DSPI with STPU to measure deformation of discontinuous surface are demonstrated in detail. The capability of
STPU is further revealed by theoretical explanation as well as experiments.
1 Principle of spatiotemporal three-dimensional phase unwrapping
The original phases obtained by DSPI are wrapped as modulo 2 of the true phases. The relation between the
π
original and true phases is mathematically formulated as
=
=
+
2n
Φ
mod ( ,2 )
Φ π
( 1)
π
where
[0,2 ) is the original phase, is the true phase,and n is an integer.
∈ π Ф
The process of changing the original phase to the real phase is called as phase unwrapping which is
traditionally classified as SPU and TPU,depending on phase unwrapping in spatial or temporal domain respectively.
1.1 Spatial phase unwrapping
SPU compares the phase values at neighboring pixels and adds or subtracts 2 if the phase jumps exist. The
π
unwrapping path is predetermined or varied. Distinguished by strategies of routing,many SPUs have been proposed.
For simplicity,row-by-point algorithm,the simplest SPU,is chosen to explain the basic principle of SPU. Row-by-
point algorithm takes the first pixel,usually locating at the lower left corner of the phase map,as a start point. Then
from this point,it demodulates the phase along the first column. The phase values at the first column are expressed
by
-
m
1
=
+
( i,0)
(0,0)
W[
( i,0) ]
( 2)
Φ
Δ
∑
=
i
1
where (0,0) is the phase at the start point,
Δ
is the difference between two neighboring pixels along the
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