Simple and low-cost interferogram projection system for small-size automotive parts profilometry: benchmarking with a commercial apparatus

Three-dimensional (3D) contouring has become very important in industry and in many other production systems. The optical techniques provide many attractive properties for such measurement due to their precision, reliability, accuracy and ability to measure small and fragile objects. In this work we report the study, the development and the performance of a low-cost optical device based on interferogram projection in order to measure the submillimetric relief of polymeric plates containing biomimetic textures used in the automotive industry. The interferogram was generated by a Twyman-Green interferometer illuminated by a 532-nm green laser. The measurement was performed by means of phase-shifting and phase-unwrapping procedures and the results were benchmarked with the ones obtained by a commercial device.


Introduction
Optical methods for 3D shape measurement have gained increasing relevance and have been applied in several areas of science and technology. Due to their precision and sensitivity, those techniques find applications in electrical engineering (He et al, 2006) and precision mechanics (Son et al, 2002). Since those techniques provide non-contact testing, they are very suitable for measuring delicate and soft textures, like e.g. in quality control of vegetable shape and aging in agricultural engineering (Cardoso et al, 2014) or in evaluation of the retina in ophthalmology procedures (Drexler et al, 2001). Low-coherence speckle metrology enabled also good results in lens characterization (Barbosa et al, 2013) and laser scanning provided applications in historical restoration (Liang et al, 2016). Techniques based on moiré effect, like dual-phase shifting moiré (Chen et al, 2011), projection moiré (Suzuki et al, 1988, and shadow moiré (Du et al, 2018) found many applications for surface contouring.
Among the optical techniques, the structured light projection became possibly the one with most commercial relevance with a very large amount of industrial and technological applications, due to its simplicity, reliability and precision. Structured light consists of a geometrically known and spatially periodic light pattern. When this pattern is projected onto a surface with oblique incidence and observed by a proper angle, the resulting light undergoes deformations according to the surface shape (Angelo et al, 2019;Jeng, 2011). By evaluating the deformed light pattern one obtains the contour map of the studied surface.
The most common and well-established geometry of structured light is formed by parallel and straight bars or stripes. This geometry can be usually achieved by means of light interference. In this work we propose the use of a Twyman-Green (TG) interferometer illuminated by a solid state laser to generate and to project an interference patternthe interferogramformed by straight and parallel fringes to perform the 3D reconstruction of biomimetic micro-textures (patterns inspired in Nature models) for polymeric automotive parts.
In order to evaluate the projected fringe pattern was sequentially phase-shifted with 4 steps of one quarter of a fringe ( 2  rad) and stored, the so-called four stepping procedure.
The four acquired frames are then combined to get the phase map of the studied object. This resulting wrapped phase map is then deconvoluted by means of phase-unwrapping procedures to provide the reconstructed surface. In our work we performed the measurement of biomimetic inspired polymeric structures used in panels for the automotive industry. Those structures were manufactured whether by etching (chemical process) or by laser ablation. We compare the obtained contour maps with the ones obtained with a commercial device.

Materials and Methods
In this section the theoretical basis of fringe projection profilometry is briefly discussed and the experimental setup is described.
Our methodological approach consisted firstly of a bibliographic research for a further theoretical and conceptual analysis of the principles and phenomena that support our experimental work. Hence, we opted for a quantitative laboratorial approach (Pereira et al, 2018).
From the theoretical study of light wavefronts, interference and triangulation we built an interferometer to generate the fringe pattern and developed an optical setup in order to achieve the best measurement sensitivity. This optical setup was adjusted in order to confirm by means of the experiments our theoretical prediction. The experiments and all measurements were performed at the Laboratório de Óptica Aplicada at Faculdade de Tecnologia de São Paulo (Fatec-SP). Research, Society and Development, v. 9, n. 9, e486997499, 2020 (CC BY 4.0) | ISSN 2525-3409 | DOI: http://dx.doi.org/10.33448/rsd-v9i9.7499 5 2.1 Theory of shape measurement by interference fringe projection Figure 1 shows the oblique incidence of a uniform interferogram (a regularly spaced fringe pattern) onto the studied surface on xy-plane. Consider that a CCD collects the image of the surface covered by the resulting contour fringes along the z-axis (the "observer" in Figure 1).
If the illuminated surface is planar, the contour pattern is formed by straight and parallel stripes which comprise the interferogram. Otherwise, if the surface is not perfectly flat, the projected interferogram will appear deformed to the observed according to the surface relief. After acquiring four phase-shifted interferograms with intensities 0 I , 1 I , 2 I and 3 I the containing the coded surface contour is determined according to Kreath where (x,y) is a point of the illuminated surface. By performing the phase unwrapping procedure (Ghiglia et al, 1987) the phase is deconvoluted. In 8-bit imaging systems like the Research, Society and Development, v. 9, n. 9, e486997499, 2020 (CC BY 4.0) | ISSN 2525-3409 | DOI: http://dx.doi.org/10.33448/rsd-v9i9.7499 6 one used in this work the phase is displayed in 256 gray levels, ranging from 0 (black, lowest z-level) to 255 (white, highest z-level).
The phase map appears pseudo-tilted by an angle  relatively to the z-axis of Figure   1.
By applying the coordinate transformation of equation (2) the reconstructed surface is positioned in the correct position with respect to the observer.

Experimental setup
The optical arrangement is shown in Figure 2.  Research, Society and Development, v. 9, n. 9, e486997499, 2020 (CC BY 4.0) | ISSN 2525-3409 | DOI: http://dx.doi.org/10.33448/rsd-v9i9.7499 7 The imaging system shown in Figure 2 is comprised by a 640-lines monochrome CCD camera (pixel size 15 m) and a computer for image acquiring and further processing. One of the mirrors of the interferometer (PZT-M in Figure 2) was attached and supported by a piezoelectric transducer in order to phase shift the interferograms in the phase stepping procedures.
For the sake of comparison, the patterns were also scanned by a high-resolution commercial device, the optical contact profilometer model 3D UHD Scanner from the AKK® company. The resolution of this system is 3600-dpi with a scanning area of 20 x 20 mm 2 .

Results and Discussion
We measured biomimetic patterns produced by etching or by laser ablation inspired in corn grains and in seaweeds. The results below show the 3D reconstructions of the patterns obtained by fringe projection, four-stepping phase mapping and phase unwrapping performed by the branch-cut method. After each measurement, we obtained the height profile of the pattern and compared it with the correspondent measurement performed by the 3D UHD Scanner.

Corn grains
One of the four phase-shifted frames of the 6.0 x 4.5-mm 2 (x  y) surface area of the corn grain pattern made by laser ablation is shown Figure 3a. Notice that the surface is covered by the interference fringes which became deformed due to the surface relief. Figure   3b shows the phase map after combining the four frames in equation (1). The unwrapped phase of the sample from the phase map is shown in Figure 3c.
This figure mistakenly suggests that the object plane is tilted with respect to the z-axis of Figure 1, with the left side (dark) of the frame having lower z-values and the right side (bright) of the frame having higher z-values.
In order to compensate this pseudo-inclination, equations (2a) and (2b) were applied to provide the 3D reconstructed surface shown in Figure 3d, with z given in arbitrary units (a.u.). Research, Society and Development, v. 9, n. 9, e486997499, 2020 (CC BY 4.0) | ISSN 2525-3409 | DOI: http://dx.doi.org/10.33448/rsd-v9i9.7499  Figure 4a shows the profile of the corn grain pattern measured by our method, while Figure 4b shows the same corresponding profile obtained by the commercial 3D UHD scanner. Both curves were obtained along line AB shown in Figure 3A. At first glance a good similarity between both measurements can be observed, but there are also some discrepancies, however. In the central "corn grain", the distance between the superior edges (points P and Q in both graphs) is ~ 1,7 mm in the fringe projection measurement, while it is ~2,1 mm obtained by the 3D UHD device. It is due to the fact that the sides of the "grains" have different inclinations in each measurement, which in turn can be mainly attributed to a higher resolution of the commercial device, if compared to our home-made, fringe projection profilometer. The "grain" heights obtained by both measurement are different also, ~140 m for the fringe projection system and ~166 m for the 3D UHD scanner.  Figure 5 shows analogous results obtained for the biomimetic seaweed polymeric sample after applying the same procedures described in section 2 and applied in section 3.1.

Seaweed
The etched seaweed pattern illuminated by the interference fringes is shown in Figure 5a while Figure 5b shows the resulting 3D reconstruction. Research, Society andDevelopment, v. 9, n. 9, e486997499, 2020 (CC BY 4.0) | ISSN 2525-3409 | DOI: http://dx.doi.org/10.33448/rsd-v9i9.7499    (a) Research, Society and Development, v. 9, n. 9, e486997499, 2020 (CC BY 4. Another noticeable difference between the graphs 6a and 6b is the symmetry of the elevations. While the benchmark measurement of Figure 6b has symmetric elevations with respect to the z-axis, the fringe projection elevations shown in Figure 6a are remarkably asymmetric. This difference may be mainly attributed to the fact the sample was illuminated from one side only, as shown in Figure 1.
A possible solution for this error is using two illuminating light patterns, each one coming from the opposite sides with respect to the z-axis. Hence, two 3D reconstructions could be obtained and averaged in order to provide the resulting object contouring.

Conclusion
A simple and reliable optical device for surface contouring was developed and its effectiveness was demonstrated. The simple, low-cost and easily available components of the optical setup allowed for the construction of a cost-effective equipment which points out for applications in industrial plants and other production systems.
Biomimetic polymeric samples with details and structures of the order of 150 m were successfully reconstructed and resolved. Our device was benchmarked with a high-resolution commercial apparatus and the results showed compatible performances concerning the shape and the dimensions of the structures. Some improvements and further development must be implemented, nevertheless.
Fringe evaluation by Fourier transform would be very suitable for such measurement and could provide faster testing without the need of a transducer and a moveable mirror to perform phase stepping.
As discussed in the previous section, the use of two illuminating light patterns could enhance the measurement accuracy and without turning the process to be significantly more complex. The use of higher resolution -and widely available -cameras would undoubtedly allow for more precise results either without significantly rising the cost of the device.