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INTERFEROMETRIC TECHNIQUES
1. Moiré • shadow• projection
2. Digital fringe analysis• intensity methods• temporal phase measurement• spatial phase measurement• phase unwrapping
1. Moiré
Moiré patterns result from the interference (superposition) of high frequency gratings or interference of electromagnetic waves :
Superposition of gratings is equivalent to their product, and this is in turn the sameas sampling : white strips separation is the horizontal sampling period, while the verticalperiod is 1 (no sampling).
Moiré patterns are just a type of aliasing , that is, sampling a grating of period p at afrequency less than 2/p (Nyquist limit).
sinusoidal grating128 (1+sin(2x /p +))
gratingrotated
Moiré interference :low frequency pattern of black and white stripes
1. Moiré
The Moiré effect can be achieved through gratings, white light projected fringes orinterference of coherent light waves : the difference is the wavelength and consequentlythe period of fringes.
Often gratings are transparences with transmittances given by a squarewave functioninstead of a sinusoidal one. The result is similar : all types of periodic gratings can bedescomposed as a sum of sinusoidal gratings.
1. Moiré
Projection Moiré
Fringes are projected on the surface and the image is interfered digitally in the computer with a stored grating, or the images the same grating projected on two surfaces are interfered.
The camera sees a grating of period
)cos(0
p
p
surface
x
payxt
2cos1),(1
1. Moiré
Should the angle between lighting and plane normal or the viewing direction change, theobserved period p would be different but constant : fringes would be equally spaced.
Now substitute the flat surface for a curved one :
)( tan),(),( yxhyxu
ubut now
p
yxuyx
),(),(
surface height with respect to the reference plane : the plane thatwould produce equally spacedfringes with period p : t1 , the grating used to demodulate (x,y)
),(2cos1),(2 yx
p
xayxt
1. Moiré
),(1 yxt ),(2 yxt
)),(2cos1(128 yx),(),( 21 yxtyxt
1. Moiré
level curves of the face of twogood sinks : lines are mostly parallel
level curves of a good surface (red) and one with an slight bump (green) :lines cross ones to the others much more
1. Moiré
Let be t1 a sinusoidal grating of constant frequency and vertically oriented :
amplitude 0
period grating
2cos1),(1
a
p
xp
ayxt
The principle behind measurement methods through Moiré interference is that somehowthe quantity to measure (x,y) 3D shape ,
1) gets codified into the horizontal displacement u(x,y) of the grating lines from its original position divided by the grating period p :
2) mathematically : it becomes phase modulated :
),(2cos1),(2 yx
p
xayxt
p
yxuyx
),(),(
a = 128p = 40 pixelsx = 0 ... 511
1. Moiré
When the two gratings are superposed (interfere) the resulting transmittance t is :
),(2cos2
1
),(2
2cos2
1
),(2cos2
cos1),(),(),( 221
yx
yxp
x
yxp
xx
payxtyxtyxt
original gratings
second grating with doubled frequency
Moiré pattern
t(x, y) has a maximum (centers of brigh fringes) whenever (x,y) = n, for n = 0, 1, 2, ...and minima (centers of black fringes) whenever(x,y) = n + 1/2, for n = 0, 1, 2, ... the Moiré pattern forms level curves of (x,y)
1. Moiré
),( yxu
),(),(),( 21 yxtyxtyxt
),(1 yxt ),(2 yxt
)),(2cos1(128 yx
right part very damaged part
Backlight + Moiré
Deformation detection in TV screen grids
Backlight + Moiré
Backlight setting
test grid
right referencegrid
Moiré interference
camera
Backlight + Moiré
a simple thresholding outlines the main bumps
1. Moiré
Shadow Moiré
The Moiré pattern is formed by the interference of a grating and the shadow it castsover a surface
lightingdirection
viewingdirection
Point P0 is projected to a point P1 on the surface which, by viewing, is projected to thepoint P2 again on the grating. Thus, the displacement of the grating relative to itsshadow is
)tan(tan ),( 2121 yxhuuu
)tan(tan ),(),(
),( 21 p
yxh
p
yxuyx
Therefore
1. Moiré
(x,y) = n + 1/2, for n = 0, 1, 2, ...
A bright fringe is obtained whenever
(x,y) = n, for n = 0, 1, 2, ...
tantan
),(21
np
yxh
and
tantan
)21(),(
21
pn
yxh
In this way a topographic map is formedover the surface. However, we have assumedpoint source and camera at infinite distancesso that angles don’t change among points
Shadow Moire videos :
Video 1 : deforming wet A4 paper
Video 2 : two overlaping A4 papers
Shadow Moiré
Temporal Phase Modulation Interferometry (TPMI)
)],(cos[),(),(),( yxyxbyxayxI
)/()(tan
]),(cos[),(
]),(cos[),(
]),(cos[),(
12321
33
22
11
IIII
yxbayxI
yxbayxI
yxbayxI

2. Digital fringe analysis
if 1= /4, 2= 3/4 and 3= 5/4,
Example : FringePro
phase unwrapped phase plot