## A Rigid Borescopic Fringe Projection System for 3D Measurement

Published: Thursday, August 13, 2015 - 15:17

The inspection and characterization of machine parts is still subject to research. Although optical means of measurement are very promising in respect to speed and precision, they are still not capable of measuring difficult-to-reach inner geometries. In this article, we will demonstrate our development of a borescope-based fringe-projection system that is capable of fast and precise geometry acquisition. The system features a small measurement head that is capable of measuring areas with limited available space.

When measuring an object with a conventional fringe-projection system it is usually not possible to cover all areas with the desired measurement sensitivity. Depending on the geometry some areas might be measured in a large angle that affects the quality of the acquired data. The measurement head has a diameter of less than a centimeter and the length of the borescope may be adjusted according to the measurement task. It is possible to position the measurement system according to the surface measured. The angle between the area of measurement and the measured objects surface can therefore be optimized. Additionally, the measurement head is small enough to be positioned in-between parts of an object with a complex geometry, e.g., between the blades of a blisk (a blade-integrated disk).

Although the principles of fringe projection have not changed, calibrating this system represented a new challenge. The borescope has relatively strong lens distortion which adds up to the distortion of other optical systems—the projector light engine and a c-mount adapter used for projecting the images. As a consequence, algorithms had to be developed to improve the state-of-the-art calibration techniques and develop new models for the lens distortion.

The setup of the measurement system, algorithms, and measurement results will be presented in this article.

### Introduction

The shape of an object gives information as to the functionality, wear, and variances in the production process. For this reason, measuring the shape is an important task in engineering. Having complete and precise information about a part's actual shape leads to new possibilities in the field of regeneration of capital goods. Although capital goods usually have complex shapes (in order to integrate more functionality into one part) the measurement of such shapes is usually quite complicated. For example, the integration of a compressor disk and its blades into a blisk (as seen in figure 1) improves the weight-to-load ratio, but measuring the complete shape is obviously more complex.

**Figure 1:**

Optical means of measurement are capable of fast and precise data measurements. Fringe projection in particular^{1} is often used for shape measurements. To make complete, fast, and precise shape measurements, a rigid borescopic fringe-projection system was developed. It uses one borescope to project fringe patterns onto hard-to-reach surfaces. For image acquisition a chip-on-the-tip camera is used.

### Measurement system

The measurement system consists of a projector and a microcamera. The Texas Instruments LightCrafter 4500 combines a compact light engine with a programmable image memory, HDMI input, and input/output triggers for synchronized image acquisition. The projector is combined with a STORZ rigid borescope with a diameter of 7 mm and a length of 200 mm. Borescopes with different diameters and length may be used, although a smaller diameter would decrease the image quality. To couple the light from the projector into the borescope, a C-MOUNT adapter from STORZ was used.

As a chip-on-the-tip camera, a very small 5 megapixel CMOS camera was used. The model is similar to those employed in modern smartphones. A RaspberryPi single board computer controls the camera and transports the images to the measurement computer. The housing with the lens has a size of approximately 7 x 7 x 5 mm and is therefore approximately as big as the tip of the borescope.

Figure 2 illustrates how the borescopes and the camera are aligned. The fields of view of both, the projector and the camera, may be altered using borescopes with different lengths and angles of view. The clip holding the microcamera may also be altered. The triangulation base can therefore be adjusted according to the measurement task. Currently a triangulation angle of about 30 degree is used.

**Figure 2:**

The projector is used to project sinusoidal patterns with different frequencies and phases to reconstruct the shape of the measured objects. Similar to the approach of Peng,^{2} patterns with one, six, and 36 periods are used. Each pattern is projected with 4 different phases, at 0, 90, 180 and 270 degrees. Therefore, 12 patterns are projected for one measurement. For the reconstruction of the shape, the internal and external parameters of the camera and the projector have to be known.

### Calibration

A pinhole camera is used to model the camera as well as the projector. The model was than extended to take account of the lens distortion.^{3} Modern cameras and their sensors have very little geometric manufacturing variances, so it is usually sufficient to model only radial lens distortion. The projector along with the borescope consists of multiple optical systems, which are not aligned perfectly. Figure 3 shows a measurement of a 20-cent euro coin. It illustrates how complex lens distortion makes the use of a more sophisticated model necessary. Extending the radial distortion polynomial to a rational polynomial and adding a term for tangential distortion improved the results.

**Figure 3:**

The basic model for pinhole cameras is:

with *u* and *v* being the pixel position where the object points (*x y z*) ^{t} have been projected. *S* is a scaling factor to normalize the z component to one, *f _{u}* and

*f*resemble the focal length and

_{v}*u*and

_{o}*u*the point where the optical axis meets the sensor.

_{v}*R*is the rotational matrix and

*t*the translation vector which define the position of the optical system in the reference coordinate system. Before applying the projection into pixel space, which corresponds to multiplication with the camera matrix, the distortion formula is used on the coordinates to take account for the properties of the lenses.

A rational polynomial is used together with an additive term that models the tangential distortion of the lenses,

with the corrected coordinates *x'* and *y'* and the distance to the optical axis *r*. The same model is used for the camera as well as the projector. To simplify the calibration, the camera is located in the origin of the reference coordinate system and the optical axis matches the z-axis.

One advantage of this gray-box model is the option to calculate the inverse; black-box models usually do not provide an easy way to achieve this. An inverse of the model may be used for advanced triangulation algorithms like inverse fringe projection.^{4}

### Results

To characterize the abilities of the measurement system, simple geometric objects were measured. Figure 4 shows the measurement of a sand blasted aluminum plane. The area of measurement is approximately 30 times 20 mm. Most of the measurement points are within 30 *µ*m of the fitted reference plane. Figure 5 illustrates the measurement uncertainties of a sphere with a diameter of 30 mm.

**Figure 4:**

Again, most of the data points have a distance less than 30 *µ*m to the fitted reference sphere. As both measurements show, the system achieves quite good measurement results, although there is still some kind of systematic error left. The edges of the measurement of the sand blasted plane are bent towards the camera, which indicates that the model of the lens distortion of the measurement system is still not sufficient. The measurement of the sphere has some sinusoidal structures. These structures appear in an area with glare due to the glossy surface of the measured sphere. Additionally some dust particles which have not been removed prior to the measurement can be seen.

### Conclusion

As seen in this article, we developed a very compact and mobile measurement system. Although an extended model for lens distortion was able to compensate for the complex lens distortion, more effort is necessary to further improve the results. Using splines to model the lens distortion^{5} will probably lead to better measurement results. Combining another borescope with an industrial camera may improve the image quality and the measurement uncertainties. Nonetheless, the measurement system is already capable of measuring in very limited space. It might improve in both the capability to measure complex shaped parts as well as measuring the geometry of parts inside a machine.

**Figure 5:**

Figure 6 shows a possible use case, where a blisk will be measured semiautomatically. The measurement system will be positioned so that all areas are measured with an ideal angle. Gravitational effects on the borescope will have to be analyzed and modeled.

**Figure 6:**

**Figure 7:**

### Acknowledgements

We would like to thank the German Research Foundation (DFG) for funding this project within the Collaborate Research Center (SFB) 871 regeneration of complex capital goods (http://www.sfb871.de).

### References

^{1} Kästner, M., "Optische Geometrieüberprüfung präzisionsgeschmiedeter Hochleistungsbauteile," Dissertation, Leibniz Universität, Hannover, 2008.^{2} Peng, T., "Algorithms and Models for 3-D Shape Measurement Using Digital Fringe Projections," Dissertation, University of Maryland, 2007.^{3} Zhang, Z., "A Flexible New Technique for Camera Calibration," *IEEE Transactions on Pattern Analysis and Machine Intelligence*, 2000.^{4} Pösch, A., Vynnyk, T., and Reithmeier, E., "Fast Detection of Geometry Defects on Free Form Surfaces Using Inverse Fringe Projection," *The Journal of the CMSC* , Vol. 8, No. 1, pp. 10–13, 2013.^{5} Schmid, S., Jiang, X., and Schäfers, K., "High-Precision Lens Distortion Correction Using Smoothed Thin Plate Splines," *Computer Analysis of Images and Patterns*, pp. 432–439, 2013.