INTERFEROMETRIC TECHNIQUES 1. Moiré shadow projection 2. Digital fringe analysis intensity methods temporal phase measurement spatial phase measurement.

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 square-wave 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