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# An Optimized Measurement Process of 3D Freeform Objects and Discussion of the Object Dependent Uncertainties

## Dipl.-Ing. Dr. Veronika Putz

 Supervisory committee: Univ.-Prof. Dipl.-Ing. Dr. Bernhard Zagar a.Univ.-Prof. Dipl.-Ing. Dr. Josef Scharinger Final exam: September 21, 2010

Within an industrial environment, the measurement of complex shapes and surfaces is used for quality control, e.g. to detect shape or surface defects. Being contact-free and thus non-destructive, optical methods which capture the object under test using digital foto- or video-cameras are highly beneficial: Even low-cost CMOS- and CCD-sensors allow to capture several millions of data points on the tested surface at one shot.

A frequently used method to measure 3D objects with diffuse reflection behavior which requires at least three images of the surface under test, is known as phase shifting profilometry. Figure 1 (left) shows a basic measurement setup, which consists of an LCD-projector and a camera. It is used to measure a hemispherical object in front of a planar background. A series of sinusoidal fringe patterns is projected onto the object under test, whereas the phase offset varies across the images. Figure 1 (right) shows four simulated images with offsets of 0°, 90°, 180° and 270°. From the four images, a geometry-dependent phase distortion can be found, which is independent from variations of background illumination.

Figure 1: left: a basic measurement setup for phase shifting profilometry, right: captured series of phase-shifted images (obtained from simulation)

Figure 2: left: dynamic speaker, diameter approx. 18 cm, right: measured phase image showing principal values of the phase (black: -180°, white: 180°)

Figure 3: left: measured phase distortion related to the xy-plane (in °), right: reconstructed shape of the speaker

If the imaging properties of both camera and projector with regard to an outer coordinate base (indicated by the vectors x , y and z , see Figure 1 left) are known, from the measured phase distortion the shape of the object can be obtained. For this purpose, to each camera- and projectorpixel a line of sight is assigned, which can be described as straight line in 3D. The parameters of each straight line can be measured using an appropriate measurement procedure (bundle adjustment for the camera, line-of-sight-calibration for the projector). With a modified triangulation procedure, the shape of the measured object is reconstructed.

The uncertainty of the measured shape depends upon the uncertainty of the measured lines of sight of both camera and projector and furthermore upon the shape of the object itself. With a combination of analytical and numerical methods, which is introduced in the thesis, the propagation of the individual errors can be estimated, and an area of uncertainty enclosing the reconstructed shape is obtained. With a known probability, the true shape of the object under test lies within this area. Figure 4 shows the area of uncertainty for the measured loudspeaker.

Figure 4: Calculated area of uncertainty of the reconstructed shape shown in Figure 3 right (admeasurements in mm): With a probability of 68.3 %, the true shape of the object lies within this area of uncertainty.

Keywords: photogrammetry, phase shift grating projection, fringe projection, non-destructive testing