Estimation of surface curvature from full-field shape data using principal component analysis

Three-dimensional digital image correlation (3D-DIC) is a popular image-based experimental technique for estimating surface shape, displacements and strains of deforming objects. In this technique, a calibrated stereo rig is used to obtain and stereo-match pairs of images of the object of interest from which the shapes of the imaged surface are then computed using the calibration parameters of the rig. Displacements are obtained by performing an additional temporal correlation of the shapes obtained at various stages of deformation and strains by smoothing and numerically differentiating the displacement data. Since strains are of primary importance in solid mechanics, significant efforts have been put into computation of strains from the measured displacement fields; however, much less attention has been paid to date to computation of curvature from the measured 3D surfaces. In this work, we address this gap by proposing a new method of computing curvature from full-field shape measurements using principal component analysis (PCA) along the lines of a similar work recently proposed to measure strains (Grama and Subramanian 2014 Exp. Mech. 54 913-33). PCA is a multivariate analysis tool that is widely used to reveal relationships between a large number of variables, reduce dimensionality and achieve significant denoising. This technique is applied here to identify dominant principal components in the shape fields measured by 3D-DIC and these principal components are then differentiated systematically to obtain the first and second fundamental forms used in the curvature calculation. The proposed method is first verified using synthetically generated noisy surfaces and then validated experimentally on some real world objects with known ground-truth curvatures.