THE 14 th INTERNATIONAL SYMPOSIUM OF MEASUREMENT TECHNOLOGY AND INTELLIGENT INSTRUMENTS - September 1 ~ 4, 2019 Modelling Fringe Projection Based on Linear Systems Theory and Geometric Transformation George Gayton*, Rong Su and Richard Leach University of Nottingham, Manufacturing Metrology Team, Advanced Manufacturing Building, Jubilee Campus, Nottingham, NG8 1BB, UK * Corresponding author: george.gayton@nottingham.co.uk Keywords: fringe projection, diffraction, surface topography, metrology, uncertainty Abstract. Fringe projection measurement techniques offer fast, non-contact measurements of the surface form of manufactured parts at relatively low cost. Recent advances in fringe projection have reduced measurement errors from effects such as multiple surface reflections and projector defocus. However, there is no standardised calibration framework for fringe projection systems and an uncertainty estimation of surface measurements is rarely carried out in practice. A calibration framework for estimating spatial frequency-dependent measurement uncertainty built on solid theoretical foundations is required. To move towards traceable surface measurement using fringe projection techniques, we are developing a measurement model to accurately predict the captured image and include all major uncertainty contributors, i.e. a virtual fringe projection system. The first step of the model is to calculate the optical field distribution using the three-dimensional optical transfer function of the projector. Next, a camera image is built up using a ray-tracing model to probe the optical field distribution at the measurement surface boundary. The results are compared to an experimental fringe projection system. The intention is to use this model within a Monte-Carlo framework to move towards estimating the uncertainty at each point-cloud data point. Introduction Fringe projection is a three-dimensional optical measurement technique that measures surface topography and geometrical dimensions of a part and has seen increased use in the aerospace, automotive and medical industries [1, 2]. Fringe projection offers relatively fast measurements in the form of high-density point clouds using relatively cost-effective components: a camera and a projector. A pattern is projected onto a measurement surface. The camera, offset from the projector records the image of the projected pattern, which has become distorted due to the surface geometry. The image is decoded to give correspondence between the reference frame of the camera and the projector, allowing points to be triangulated between the images. Many different fringe projection techniques exist which project different patterns, all optimised for specific measurement scenarios [3, 4]. Unlike conventional contact measurement methods, e.g. coordinate measuring machines (CMMs), there is no standardised calibration framework for fringe projection systems. Uncertainty evaluation of fringe projection surface measurements is rarely carried out in practice, restricting the use of this technique in manufacturing industry. The dependence of fringe projection measurements on surface characteristics, e.g. optical properties and topography, makes current calibration methods given in ISO 15530 part 1-3 [5] (for contact CMMs) unsuitable for fringe projection. Also, it is unclear how to apply the calibration approach in ISO 25178 [6] for areal surface topography measuring instruments. Current work on fringe projection systems focuses on limiting the magnitude of certain artefacts, such as global illumination, multiple surface reflections [7-9]. However, few authors quantify the influence of these artefacts on the measurement outcome. This quantification process is difficult in fringe projection systems; any process that alters the intensity of light scattered/reflected from the measurement surface and recorded by the camera will alter the measurement outcome. A rigorous mathematical model is missing that includes the large number of influence factors that can alter the measurement outcome. The model must also work over a large range of measurement scales, since fringe projection systems operate from millimetre-range surface topography measurements [10, 11] to larger scales of metres and above [12]. Previous attempts to simulate fringe projection have been limited to geometrical optics only, ignoring diffraction and surface effects. Surface optical characteristics strongly influence the