Reproducibility of two calibration procedures for phase-measuring deflectometry

Abstract

Phase-measuring deflectometry is an optical inspection technique for reflective surfaces. It enables absolute, quantitative surface measurements, given a calibrated measurement setup. Two general calibration approaches can be found in literature: First, the stepwise approach uses a calibration pattern and determines internal camera parameters and external geometrical parameters in separate, consecutive steps. Second, the holistic approach optimizes all parameters collectively, based on deflectometric measurements of a calibration mirror.Whereas both approaches have been compared regarding the accuracy of subsequent surface measurements, the present contribution focuses on experimental examination of their reproducibility. In experiment E1, we assess the parameter variability by repeating both calibration procedures ten times. In an additional experiment E2, we repeat all calibration measurements related to a mirror/pattern position ten times in a row before rearranging the mirror/pattern, in order to examine the purely noise-related parameter variability. Finally, we calculate the coordinate variability of a set of world points projected onto the image planes of the calibrated cameras.The measured variability is consistently higher in E1 than in E2 (average ratio: 3.2). Unexpectedly, in both experiments, the external parameter variability also turns out to be higher for the holistic approach compared to stepwise calibration (average ratio: 2.3). This is of importance, since the holistic approach is known from literature to be more accurate than the stepwise approach, regarding their respective application to surface measurements. The image coordinate variability is comparable for both calibration approaches with an average of 0.84 and 0.21 camera pixels for E1 and E2, respectively.

The effects of pattern screen surface deformation on deflectometric measurements: a simulation study

“…It may further be of interest to examine the effects of a non-planar calibration mirror on the holistic calibration process, given that the current implementations assume the mirror surface to be a perfect plane. 20 Finally, future developments could extend the simulation framework with features such as alternative camera models or depthof-field effects.…”

“…They follow the scheme we use for real calibrations: one measurement with the mirror lying on the base table; three with the mirror parallel to the table but elevated by different amounts; and a total of twelve measurements with the mirror rotated around the x-or y-axis by different angles (three positive and three negative rotation angles per axis). 20 For ∆h = 0 mm, the calibration procedure yields the correct internal and external camera parameters, i.e. they are identical with the parameters used for the simulated measurements.…”

“…1). Each coordinates transformation is defined by six parameters: three for the translation vector and three rotation components (we use three Givens rotation angles for that purpose 19,20 ). The transformation matrices are inverted and combined to provide the transformation matrix from any origin coordinate system (index o) to any destination coordinate system (index d) by…”

“…In short, the projection of a point p (c) = (x c , y c , z y ) T (c) given in the camera coordinate system onto the corresponding image pixel coordinates p (i) = (x i , y i ) T (i) of the resulting image is modeled as a three-step process. 20 Note that throughout the text, subscripts in parentheses are meant to denote the coordinate system in which a vector is given; (c) and (i) denote the camera and image coordinate systems, respectively. The first step projects p (c) onto the image plane (i.e.…”

Deflectometry for specular surfaces: an overview

“…An alternative approach that potentially may reduce the need for a precision calibration object (or eliminate it entirely) is to include these six parameters in the "global optimization loop" so that the object reconstruction and the setup calibration happen in parallel within some "holistic" or probabilistic framework. As the cost function to minimize, one often chooses ray re-projection errors or similar metrics [3,264,[267][268][269][270][271]. As a price, however, one may need to record several object poses per measurement.…”