APPENDIX

Principle of fringe projection

In a fringe projection profilometry system (FPP), a series of straight, vertical or horizontal and equally spaced fringes are generated by computer and projected onto the surface of an object using a digital video projector. At the same time, a CCD (charged-coupled device) camera captures the intensity of the fringes from the surface of the object for later processing. By using the phase shifting (PS) method, “N” fringe patterns of frames that were captured by the camera, each nth intensity pattern I n( x , y ) can be commonly described as:

I n ( x , y ) = a ( x , y ) + b ( x , y ) cos [ (2 π / p ) x + φ ( x , y ) + θ ∙ n ]; n=1...N

where N is the total number of images acquired; n is the number of the phase-shifting steps, ( x , y ) denotes the coordinates of an arbitrary point in the object being analyzed; p is the period of the equally spaced fringes on the reference plane; φ ( x , y ) is the phase map related to the object profile, also called wrapped phase ; and θ is the assigned phase shift value that is usually equal to 2π/N (Ma et al., 2012).

The numerical process to recover the wrapped phase from the set of captured images is performed with a phase shift algorithm (PSA). The correct design of this algorithm, in accordance with a specific setup, determines the complexity and resolution of the results that, briefly, uses the intensity values obtained by shifting the fringes on the object to calculate the phase (Rathjen, 1995). Next, an unwrapping procedure is needed to make this phase map continuous, which is done by removing the artificial discontinuities added by the FPP technique. This phase’s unwrapping is a complex process and the techniques used to perform it assume that the neighboring pixel differences of the unwrapped phase can be estimated by adding an integral multiple of 2π, when these differences are less than π. Using some mathematical methods it is possible to obtain a cloud of data points that is proportional to the analyzed object (Ghiglia and Pritt, 1998). After this, the temporal carrier is calculated and subtracted (Li et al., 1998) and the topography or profile is calculated by:

h ( x, y ) = l 0 ∙ φ ( x, y )/[ 2 π f 0 d + φ ( x, y ) ]

where l 0 is the distance between the CCD camera and the reference plane, and d is the distance between the camera and the projector (Takeda and Mutoh, 1983).