Phase-measuring deflectometry (PMD) with active display registration (ADR) is a ray-optics-based technique for the shape measurement of specular surfaces. To obtain quantitative results, the relative position of the cameras of the PMD–ADR setup needs to be determined by geometric calibration. Geometric calibration can be performed by inserting a planar mirror into the setup that brings all camera fields of view to overlap on an active pattern display. The mirror is tilted to multiple positions and each time the cameras capture the displayed images, which yields sufficient data to obtain the relative camera positions and the positions of the mirror. In this article, we give a more detailed description of PMD–ADR and its calibration. We also implement a laser-tracker-based reference method to measure the mirror positions and use its result to expose systematic errors in the geometric calibration.

Phase-measuring deflectometry (PMD) is a geometric-optics-based measurement technique for the robust noncontact full-field measurement of specular surfaces

Additionally, the display surface deviates from often assumed planarity, which leads to shape measurement errors.

As a method of regularization and a solution to the nonideal behavior of the display,

Figure

Schematic of PMD–ADR measurement setup. To measure the SUT shape, the display is moved multiple distances. Using correspondences of phase values, multiple 3D points can be calculated and assigned to each camera 1 pixel. Intersecting the line fit through the points with the corresponding ray of vision yields a 3D point of the SUT.

Since the 3D points of the display surface are determined directly by cameras 2 and 3, PMD–ADR does not require calibration of the display position and shape. Therefore, only the cameras' rays of vision and their relative position need to be calibrated. We use the generic camera model and vision ray calibration

To obtain the transformation

The cost function is a sum of squares of the minimal distances between each point and the corresponding reflected ray. Only a fraction of the data of all camera 1 pixels is used in the numeric calculation to reduce the computational effort.

A laser tracker is a device that tracks a reference target and measures its 3D coordinates

During geometric calibration, the tracker measures the coordinates of the nests for each mirror position. To obtain the mirror vectors, we fit a plane through the coordinates of the nests. Although the nests generally do not lie on a plane, a rigid transformation of the nest coordinates will lead to the same transformation of the fit plane coordinates. Since the nests and therefore the fit plane are rigidly connected to the mirror, there exists a coordinate system

To compare the mirror positions obtained from the tracker with those from the geometric calibration, they need to be transformed to a common coordinate system. Figure

Transformation of mirror vector from system

Figure

Experimental setup. Camera 1 observes the display via reflection on the SUT. Cameras 2 and 3 observe the display directly.

As SUTs we use a planar mirror and two convex spherical mirrors with radii of curvature of approximately 200 and 100 mm. All samples have a diameter of 50 mm, of which the clear aperture of 45 mm diameter is used for evaluation. Each SUT is measured individually by placing it in the setup. To obtain data for shape reconstruction, the display is moved to five distances and each time a phase measurement is performed. For the 100 mm spherical mirror, multiple measurements are being recorded for each distance. This is necessary to cover the camera 1 field of view, which is enlarged by reflection on the high-curvature SUT.

Prior to SUT measurements, data for geometric calibration are gathered using the setup depicted in Fig.

Calibration setup. Camera 1 observes the display via reflection on a large planar calibration mirror. Cameras 2 and 3 observe the display directly. The tracker records the coordinates of the SMR which can be placed at four nests at the rear of the mirror holder.

Rear of the calibration mirror holder with four nests and the SMR attached to one of the nests.

The determination of nest coordinates by the tracker might be affected by multiple error sources that do not originate from the tracker itself, such as thermal expansion of the calibration mirror hold or an imperfect snapping of the SMR to a nest due to dust or scratches. We repeat for each mirror position the measurement of the four nest coordinates four times to check repeatability. For all measurements and all nests, the deviation of coordinates of a nest from their mean is less than 5

As described in Sect.

The mirror vectors of the tracker are given in some coordinate frame

We use the geometric calibration result to reconstruct the shape of measured SUTs. A performance criterion for shape reconstruction is the deviation of the reconstructed shape from reference. The reference radii of the spherical mirrors were determined at Physikalisch Technische Bundesanstalt (PTB), Braunschweig, Germany, to be

Shape residuals.

Shape residuals' peak-to-valley (PV) results for multiple SUTs. Right column: maximum difference of the residuals to the residuals obtained with the merged geometric calibration. For an explanation of merged geometric calibration see Sect.

The residuals of the flat mirror are small. It is therefore not suitable to expose calibration errors, or the general shape reconstruction accuracy, of the PMD–ADR method. Still, the good reconstruction of the flat mirror shape indicates consistency of the recorded phase data with the camera calibration. This means that camera 1 is probably calibrated well, and cameras 2 and 3 are probably calibrated well in the center of their field of view, where the phase values were recorded that go into shape reconstruction. For nonzero curvature, the deviations are larger. It is of interest if the geometric calibration error that caused the deviation of mirror vectors can cause such shape deviations.

To investigate whether the observed deviation of mirror vectors during geometric calibration can cause the observed shape deviations, we also perform a merged geometric calibration, where the mirror poses determined with the laser tracker are incorporated into the geometric calibration procedure. The optimization of the mirror vectors is thereby replaced by the optimization of

Figure

Multiple sources of error are possible. Refraction at the display cover glass can cause deviations of the measured phase

We have presented phase-measuring deflectometry (PMD) with active display registration (ADR), which is a shape measurement technique for specular surfaces. Geometric calibration is an important step for PMD–ADR. As one result of geometric calibration, one obtains the positions of the calibration mirror. We implemented a laser-tracker-based method to perform a reference measurement of the calibration mirror positions. The reference method yields different mirror positions than the geometric calibration. This shows systematic errors in the geometric calibration. By employing a merged geometric calibration, we were able to show that these errors are sufficient to yield shape reconstruction deviations of similar size as the observed deviations. This result helps us to guide our future efforts to improve modeling towards the calibration stage of PMD–ADR.

Code and data underlying the results presented may be obtained from the authors upon reasonable request.

YS performed the investigation and prepared the manuscript. RBB conceptualized and administered the project, acquired the funding, and supervised this work. YS and RBB reviewed and edited the manuscript.

The contact author has declared that neither of the authors has any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors.

This article is part of the special issue “Sensors and Measurement Science International SMSI 2023”. It is a result of the 2023 Sensor and Measurement Science International (SMSI) Conference, Nuremberg, Germany, 8–11 May 2023.

The authors gratefully acknowledge the radius measurement of the spherical mirrors at Physikalische Technische Bundesanstalt (PTB).

This research has been supported by the Deutsche Forschungsgemeinschaft (grant no. 444018140).

This paper was edited by Andreas Schütze and reviewed by two anonymous referees.