INTERFEROMETRIC TECHNIQUES 1. Moiré shadow projection 2. Digital fringe analysis intensity methods temporal phase measurement spatial phase measurement phase unwrapping 3. Light interference 4. Holographic 5. Speckle

1. Moiré Moiré patterns result from the interference (superposition) of high frequency gratings or interference of electromagnetic waves : sinusoidal grating 128 (1+sin(2px /p +f)) grating rotated a Moiré interference : low frequency pattern of black and white stripes Superposition of gratings is equivalent to their product, and this is in turn the same as sampling : white strips separation is the horizontal sampling period, while the vertical period is 1 (no sampling). Moiré patterns are just a type of aliasing , that is, sampling a grating of period p at a frequency less than 2/p (Nyquist limit).

1. Moiré The Moiré effect can be achieved through gratings, white light projected fringes or interference of coherent light waves : the difference is the wavelength and consequently the period of fringes. Often gratings are transparences with transmittances given by a square-wave function instead of a sinusoidal one. The result is similar : all types of periodic gratings can be descomposed 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. surface The camera sees a grating of period

1. Moiré Should the angle between lighting and plane normal or the viewing direction change, the observed period p would be different but constant : fringes would be equally spaced. Now substitute the flat surface for a curved one : u but now surface height with respect to the reference plane : the plane that would produce equally spaced fringes with period p : t1 , the grating used to demodulate y(x,y)

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1. Moiré level curves of the face of two good 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 : a = 128 p = 40 pixels x = 0 ... 511 The principle behind measurement methods through Moiré interference is that somehow the quantity to measure y(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 :

1. Moiré When the two gratings are superposed (interfere) the resulting transmittance t is : original gratings second grating with doubled frequency Moiré pattern t(x, y) has a maximum (centers of brigh fringes) whenever y(x,y) = n, for n = 0,  1,  2, ... and minima (centers of black fringes) whenever y(x,y) = n + 1/2, for n = 0,  1,  2, ...  the Moiré pattern forms level curves of y(x,y)

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1. Moiré If gratings were already modulated by functions y1(x,y) and y2(x,y) then the Moiré pattern would be their difference y1(x,y)  y2(x,y) : we would obtain the level curves of the three-dimensional difference function between two surfaces, for instance for flaw detection or tolerance gauging in surface comparison. We have assumed that both gratings were oriented vertically, except for the phase introduced in t2. Were it not the case, the Moiré effect also occurs but level curves get rotated in the same amount as the angle between the gratings. a p /2sin(a/2)

Deformation detection in TV screen grids Backlight + Moiré Deformation detection in TV screen grids right part very damaged part

Backlight + Moiré camera test grid right reference grid Moiré interference Backlight setting

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 casts over a surface lighting direction viewing Point P0 is projected to a point P1 on the surface which, by viewing, is projected to the point P2 again on the grating. Thus, the displacement of the grating relative to its shadow is Therefore

1. Moiré A bright fringe is obtained whenever y(x,y) = n, for n = 0,  1,  2, ... and y(x,y) = n + 1/2, for n = 0,  1,  2, ... In this way a topographic map is formed over the surface. However, we have assumed point source and camera at infinite distances so that angles don’t change among points

Shadow Moiré Shadow Moire videos : Video 1 : deforming wet A4 paper Video 2 : two overlaping A4 papers

2. Digital fringe analysis Temporal Phase Modulation Interferometry (TPMI) if a1= p/4, a2= 3p/4 and a3= 5p/4,

Example : FringePro phase unwrapped phase plot