A morphing-based scheme for large deformation analysis with stereo-DIC

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Optics and Lasers in Engineering journal homepage: www.elsevier.com/locate/optlaseng

A morphing-based scheme for large deformation analysis with stereo-DIC Katia Genovese∗, Donato Sorgente School of Engineering, University of Basilicata, Potenza, Italy

a r t i c l e

i n f o

keywords: Large deformation Stereo Digital Image Correlation Image morphing Shape flattening Superplastic sheet metal forming

a b s t r a c t A key step in the DIC-based image registration process is the definition of the initial guess for the non-linear optimization routine aimed at finding the parameters describing the pixel subset transformation. This initialization may result very challenging and possibly fail when dealing with pairs of largely deformed images such those obtained from two angled-views of not-flat objects or from the temporal undersampling of rapidly evolving phenomena. To address this problem, we developed a procedure that generates a sequence of intermediate synthetic images for gradually tracking the pixel subset transformation between the two extreme configurations. To this scope, a proper image warping function is defined over the entire image domain through the adoption of a robust feature-based algorithm followed by a NURBS-based interpolation scheme. This allows a fast and reliable estimation of the initial guess of the deformation parameters for the subsequent refinement stage of the DIC analysis. The proposed method is described step-by-step by illustrating the measurement of the large and heterogeneous deformation of a circular silicone membrane undergoing axisymmetric indentation. A comparative analysis of the results is carried out by taking as a benchmark a standard reference-updating approach. Finally, the morphing scheme is extended to the most general case of the correspondence search between two largely deformed textured 3D geometries. The feasibility of this latter approach is demonstrated on a very challenging case: the full-surface measurement of the severe deformation (> 150% strain) suffered by an aluminum sheet blank subjected to a pneumatic bulge test. © 2017 Elsevier Ltd. All rights reserved.

1. Introduction Since its early development, the Digital Image Correlation (DIC) method [1] has undergone tremendous progress in terms of robustness, efficiency and accuracy of the matching algorithm, as well as regards the entire measurement procedure, including calibration, good-practices and data error analysis. DIC method also extended over new fields of application, now including high-speed, volumetric, microand large-scale measurements [2,3]. Recently, a growing interest has been addressed to large deformations analysis [4–9]. Largely deformed pairs of images may result e.g. from stereo-views of non-flat objects and/or from a motion/deformation not sufficiently sampled over time. Extracting information from such images is particularly challenging because the success of a DIC matching is guaranteed only when a sufficient similarity exists between the images to be correlated. The DIC algorithm, in fact, maps a given point in the reference image to its corresponding point in the target image by comparing the gray intensity distribution over a pixel subset centered in the point of interest on the basis of a given correlation coefficient [10]. In particular, once the form of the shape function potentially describing the subset deformation

∗

is selected, an iterative non-linear optimization procedure is used to maximize (or minimize, depending on the formulation, [1,10]) the correlation coefficient as a function of the subset transformation parameters. The definition of the initial guess for the deformation parameters strongly affects the convergence speed and the solution accuracy of this optimization process. In particular, when the level of deformation surpasses a certain threshold, the automatic initialization of the DIC analysis with conventional approaches (e.g. through an integer pixel search) may fail or yield to erroneous results. This problem was studied in the case of 2D temporal matching of images undergoing large rigid body motions (20° rotation in [9]) and deformation (36% compression in [7] ). Both of these works start the analysis with an automated seed-point search performed with a feature-based matching algorithm widely used in Computer Vision, the Scale Invariant Feature Transform (SIFT) [11]. Hence, the deformation parameters of the area around the seed-point is firstly refined by iterative optimization, then propagated as initial guess to the neighborhood by adopting a reliability guided strategy [4]. In this work we extended the large deformation analysis to the 3D case, by developing a morphing- based method able to perform both spatial and temporal matching of stereo images with a multi-seed point

Corresponding author. E-mail address: [email protected] (K. Genovese).

http://dx.doi.org/10.1016/j.optlaseng.2017.06.020 Received 30 April 2017; Received in revised form 12 June 2017; Accepted 26 June 2017 Available online xxx 0143-8166/© 2017 Elsevier Ltd. All rights reserved.

Please cite this article as: K. Genovese, D. Sorgente, Optics and Lasers in Engineering (2017), http://dx.doi.org/10.1016/j.optlaseng.2017.06.020

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scheme. The goal of the proposed method is to allow the adoption of an incremental calculation approach [12,4,5] through the generation of a series of synthetic images that simulate a gradual rotation (in the case of spatial matching, [13]) or a gradual deformation (in the case of temporal matching, [4,5]) between two extremes represented by the original largely deformed images. The graduality of the transformation throughout the image series is achieved by defining a suitable warping function to interpolate the pixels position in the in-between sequence, using a Computer Vision technique known as image morphing [14]. Symmetrically deformed images have been already used in DIC to iteratively enhance the cross-correlation matching between two consecutive images of a sequence [8]. In our case DIC fails when used straight to match the two only images available, hence a different strategy needs to be adopted to define the warping function that maps one image to the other. In particular, a first crude information on the disparity between the two images is calculated in correspondence of a large number of keypoints matched via SIFT. Then, under the deformation continuity assumption, the sparse and partially incorrect information obtained from SIFT is fitted with non-uniform rational B-splines (NURBS) [15] and expanded over the entire region of interest (ROI). The morphing function so obtained is hence used to generate a suitable number of intermediate synthetic images. Finally, a reference updating scheme [12] is used to propagate the correlation analysis through the images sequence yielding to find a pair of corresponding point grids in the two original images to use in the subsequent refinement stage of the DIC analysis. This morphing scheme was extended also to the case in which two different stereo-systems are employed to capture the images relative to the undeformed and deformed configurations, i.e. when the input to the correlation analysis is not images but textured geometries. Two sets of experiments were planned for this research. A first experiment test aimed to collect the set of images needed to develop and validate the method in a well-known case of large and heterogeneous deformation: the axisymmetric indentation of a circular membrane [16]. For this first test-case, the load was gradually applied to the sample and the corresponding frame sequence was captured and processed through a standard reference-updating scheme [12,4,5]. The obtained results have been thus used as a quantitative benchmark for our developed ‘one-step’ procedure that used only the first (undeformed) and last (deformed) images of the recorded sequence. A second experiment aimed to apply the so validated procedure to a very challenging test case: the one-step full-surface shape and deformation measurement on a bulge test sample obtained with superplastic forming. This latter investigation was motivated by a recent growing interest in hot sheet metal forming processes due to their unique capabilities in manufacturing complex-shaped components in both steel [17] and light alloys [18] for a large variety of industrial applications. Given the large (> 200%) and heterogeneous strain field generated by superplastic forming, it is undoubted that both material characterization and process design could benefit from the full-surface information obtainable with DIC. However, due to the extreme process conditions in which a hot forming operation is carried out (i.e. elevated temperature and solid lubricants related issues), a DIC in-situ data acquisition such that performed during cold forming processes [19,20] is still considered a challenging task [21]. Hence, the one-step large deformation measurement developed in this work could represent an optimal practical choice to obtain valuable DIC full-surface information from the sole images of the sample captured before and after the forming process.

baseline and a 20° stereo-angle. A fairly uniform illumination of the sample was provided by two symmetrically placed LED panels (Yongnuo YN600L, 3200–5500 K adjustable color temperature, 4800 LM). At the selected magnification (allowing the full framing of all the samples tested in this study), an image spatial resolution of ∼ 5 ⋅ 10−2 mm/pixel was obtained. Stereo-system was calibrated twice, before and after each test, in order to exclude any possible system drift. Calibration was performed by using a three-dimensional target provided with a regular pattern of dots whose 3D position in the world reference system is known with reasonable accuracy. In a first step, the calibration procedure allowed to find the intrinsic and extrinsic parameters of the cameras by using a closed-form solution based on the distortion-free pin-hole camera model [22]. The so obtained set of parameters were then used as initial guess values for the iterative non-linear optimization procedure aimed to find a larger set of parameters describing a more complex camera model including a 3rd order radial lens distortion. All codes used in this work have been written in MATLAB (Math-Works, Natick, MA). 2.2. Axisymmetric indentation of a silicone circular membrane To perform the first test, aimed to collect the set of images needed to develop and validate the proposed procedure, a circular silicone elastomeric membrane of 1 mm thickness and 76 mm diameter was cut out from a swim cup, glued along its outer border to a ring gasket and then secured in place within a 3″ lens holder with a retaining ring. Before testing, a coarse speckle pattern was manually applied to the membrane surface by using a black tempera paint. The membrane was hence fixed to the optical bench and then subjected to a central transverse load applied through a 27 mm diameter nearly hemispherical punch mounted on a three axes translation stage (see [23] for a picture of the sample and loading fixture). In particular, the indenter was firstly fairly centered to the membrane by acting on the X and Y stage axes, then it was moved through sixteen Z positions at 2 mm step increments for a total out-of-plane (applied) displacement of about 32 mm (at the start, the punch travelled idle for a short while before touching the membrane). A pair of stereo images were captured at each loading position for subsequent data analysis (see [23] for a video of the stereo-frame sequence). In particular, three different approaches were used to retrieve the 3D deformed shape and the full-surface deformation of the membrane: (i) the conventional DIC reference updating scheme that uses the entire sequence of captured images (hereafter shortly named DIC-incremental method), and two morphing-based methods, namely (ii) the image-morphing method and (iii) the shape-morphing method, specially designed for one-step large deformation analysis and requiring only the two most extreme configurations of the image series. 2.2.1. DIC incremental method Following this well-known approach (see e.g. [4,5]), the displacement of a given set of evenly spaced points within the region of interest (ROI) on the object surface (the control point grid) is tracked through the captured sequence of N images starting from a given image selected as a reference. After the correspondence is found for the first couple of images Ir and It (hereafter generally indicated as the reference -superscript r- and the target -superscript t-), the current target image is used as an updated reference for the next image and the correlation analysis similarly proceeds up to the last image of the series. The overall large deformation is hence obtained by cumulating the so calculated 𝑁 − 1 incremental small deformation fields. To ensure a reliable matching and limit the propagation of the error, the deformation between two adjacent images should set to be reasonably small. Fig. 1(a) and (b) reports the left ILUND and right IR images of the undeformed configuraUND tion of the indented membrane and the two corresponding images ILDEF and IR of the last deformed configuration (the entire frame sequence DEF is reported in the supplementary material). As expected, the angled view has a little effect for a flat surface (the undeformed membrane, compare Fig. 1(a) and (b)) while it yields to largely dissimilar images

2. Materials and methods 2.1. Experimental set-up To perform the 3D measurements reported in this work, a standard stereo-DIC system was set-up featuring two scientific graded cameras (Dalsa Falcon 4M30, 2352 × 1728 pixels CMOS sensor, 8 bit) equipped with 28–105 mm Nikkor zoom lenses (at 70 mm and f/8) with a 225 mm 2

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Fig. 1. The original stereo-images of the undeformed (a)-(b) and deformed (c)-(d) configurations for a silicone rubber membrane under axisymmetric indentation with superimposed the DIC correlated (coarse) mesh grids.

in the case of a more complex geometry (the deformed membrane, Fig. 1(c) and (d)). A single image sequence was used here to perform both the spatial and temporal matching of the undeformed and deformed configurations. In particular, a regular point grid defined on the ILDEF image was taken as a reference and tracked through the backward temporal sequence ILDEF → ILUND of the left camera. Then, a straightforward correspondence search between the two very similar left and right images of the flat undeformed configuration allowed the spatial matching between the two stereo views. Finally, a forward temporal matching for the right camera was performed through the image sequence IR → IR . A total number of 34 images was hence processed UND DEF to retrieve the full shape and deformation of the indented membrane. The DIC codes used in this work for image matching implement the Normalized Cross Correlation coefficient (NCCC) [24] with a 21 × 21 pixels subset size, a 41 × 41 pixels search window size, a 8 pixels point spacing. Two different levels of matching accuracy and resolution were implemented. A fast crude estimation of the matched positions of corresponding points in the two images to be correlated can be achieved with an integer pixel-search scheme with a zero-order shape function [1] (the coarse-DIC code). A zero-order shape function ( ) ( ) 𝑥𝑟1 − 𝑥𝑟𝐶 𝑦𝑟1 − 𝑦𝑟𝐶 ⎛ 𝑥𝑡1 − 𝑥𝑟1 ⎞ ⎡1 0 ⎜ 𝑦𝑡1 − 𝑦𝑟1 ⎟ ⎢0 1 0 0 ⎜ ⎟ ⎢ … … ⎜ ⎟=⎢ ⎜ … ⎟ ⎢ … ( ) ( ) ⎜𝑥 − 𝑥 ⎟ ⎢1 0 − 𝑥 𝑥 𝑦𝑟𝑚 − 𝑦𝑟𝐶 𝑟𝑚 ⎟ 𝑟𝑚 𝑟𝐶 ⎜ 𝑡𝑚 ⎢ ⎝ 𝑦𝑡𝑚 − 𝑦𝑟𝑚 ⎠ ⎣0 1 0 0

the search can be then done by using the information obtained in the previous step as the initial guess for an iterative non-linear optimization routine aimed to find a more accurate estimate of the position of Pt in the target image with a 0.1 sub-pixel resolution (achieved from a 2nd order polynomial fitting of the NCCC over the search window). In this second code (hereafter designated as the fine-DIC code), a more complex model was implemented for describing the deformation of the pixel subset. In particular, without loss of generality, we assumed that the reference subset warps to the target subset by following an affine transformation (first-order shape function) to give: ( ) ( ) { 𝑥𝑡 = 𝑥𝑟 + 𝑢 + 𝑢𝑥 𝑥𝑟 − 𝑥𝑟𝐶 + 𝑢𝑦 𝑦𝑟 − 𝑦𝑟𝐶 ( ) ( ) (1) 𝑦𝑡 = 𝑦𝑟 + 𝑣 + 𝑣𝑥 𝑥𝑟 − 𝑥𝑟𝐶 + 𝑣𝑦 𝑦𝑟 − 𝑦𝑟𝐶

0 ( ) 𝑥𝑟𝑚 − 𝑥𝑟𝐶

assumes that the pixel subset centered in the point of interest Pr (xrC , yrC ) in the reference image undergoes a simple rigid translation to the corresponding Pt (xtC , ytC ) position in the target image. A refinement of

built upon the coordinates of m (m ≥ 3) neighborhoods to the points pair of interest Pr /Pt previously matched with the coarse-DIC code. For this first correlation analysis, a control grid with > 5200 points evenly sampling the ROI was tracked through the entire sequence of

where xrC and yrC are the coordinates of the reference subset center, u and v are the translations along the x and y direction, and ux , uy , vx , vy are the components of the first-order deformation gradients. For each point of the control grid, the optimization process can be initialized by using as initial guess for the unknown parameters, the vector 𝒑0 = (𝑢0 , 𝑣0 , 𝑢𝑥0 , 𝑢𝑦0 , 𝑣𝑥0 , 𝑣𝑦0 ) obtained by solving in a least square sense the following system of linear equations: 0 ( ) 𝑥𝑟1 − 𝑥𝑟𝐶

3

0 𝑢 ( ) ⎤⎛ 0 ⎞ 𝑦𝑟1 − 𝑦𝑟𝐶 ⎥⎜ 𝑣0 ⎟ ⎥⎜ ⎟ ⎥⎜𝑢𝑥0 ⎟ ⎥⎜ 𝑢𝑦0 ⎟ ⎥⎜𝑣 ⎟ 0 ( )⎥⎜ 𝑥0 ⎟ 𝑦𝑟𝑚 − 𝑦𝑟𝐶 ⎦⎝𝑣𝑦0 ⎠

(2)

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images with an average value of the NCCC always larger than 0.98 for each correlated image pair (note the smooth variation of deformation and illumination conditions going from the undeformed to the deformed configuration through the frames sequence in [23]). The correlated grids served to reconstruct the undeformed and deformed 3D shapes as well as to evaluate the full-surface three-dimensional displacement field of the indented membrane. Finally, the Green’s surface strains were calculated ‘pointwise’ by assuming a homogeneous deformation over each triangular facet ( ∼ 0.15 mm2 area in the undeformed state) delimited by three closely spaced grid points following a procedure described in detail in [25]. Before any calculations were performed, for each couple of matched reference and target images, the raw disparities along the x and y directions were smoothed by using non-uniform rational B-splines (NURBS) in the Rhinoceros (Robert McNeel & Associates, Seattle, USA) CAD environment. B-splines [15] have been extensively used in the DIC community for interpolating the pattern intensity distribution over the pixel subset [26], for describing the object shape and representing the unknown deformation field before correlation [27,28] as well as for smoothing obtained field data in the post-processing phase of the analysis [29]. When properly applied, the NURBS-based smoothing procedure implemented in this paper and described in detail in previous works (see e.g. [29]) allows to easily remove the possible outliers and to replace them with reliably interpolated data points, while leaving the position of valid matches of the original distribution almost unaltered (the Euclidean distance between the raw and smoothed data points is of the order of 10−2 mm). Despite its not-invasive nature, NURBS-based smoothing of disparities maps demonstrated to have a dramatically beneficial effect on the smoothness of the deformation and, most importantly, of the calculated strain fields. No additional data smoothing was in fact further needed to obtain the results reported in this paper.

large set of keypoints together with a set of descriptors of their appearance. SIFT descriptors are derived from the local intensity gradient orientations and are invariant to the image transformations typically generated by stereo vision and/or by deformation i.e. as translation, rotation and scaling. Once the two sets of features are independently detected in the two images, all the possible matches are compared and sorted on the basis of the similarity (distance) between their descriptors. Depending on the threshold parameter chosen for selecting the ‘true’ matches (defined as the ratio between the descriptors distances of the first two possible matches), a number of incorrect matches may be included in the final set of corresponding features pairs. In this paper, we chose to use 1.5 as the threshold ratio since it yielded to get a sufficiently large number of keypoints distributed across the image while keeping low the percentage of incorrect matches (ranging from 4% to 8% in relation to the degree of similarity between the two images). A Matlab implementation of SIFT is available for download from the VLFeat website [30]. (3) Calculating images disparities. The disparities maps between Ir and It are calculated as follows: 𝑑𝑢 = 𝑥𝑡𝐾 − 𝑥𝑟𝐾 , 𝑑𝑣 = 𝑦𝑡𝐾 − 𝑦𝑟𝐾 .

(4) Smoothing disparities. The four point clouds 𝑊𝑢𝑟 (𝑥𝑟𝐾 , 𝑦𝑟𝐾 , 𝑑𝑢), 𝑊𝑣𝑟 (𝑥𝑟𝐾 , 𝑦𝑟𝐾 , 𝑑𝑣), 𝑊𝑢𝑡 (𝑥𝑡𝐾 , 𝑦𝑡𝐾 , 𝑑𝑢) and 𝑊𝑣𝑡 (𝑥𝑡𝐾 , 𝑦𝑡𝐾 , 𝑑𝑣) are imported in Rhinoceros for data smoothing (only two of them are depicted in the scheme of Fig. 2 for clarity of representation). In particular, a NURBS patch is drawn through each 3D point cloud (see Fig. 2) after setting proper values for the horizontal and vertical spans (i.e. number of control points) and for the stiffness (i.e. the degree of the surface). Since the number of outliers (due to incorrectly matched SIFT points) are usually only a small percentage out of the total number of calculated point pairs, they are automatically cut off from the fitting surface. Whenever present in a large amount, outliers can be easily detected and removed by fitting a plane through the dv disparity distribution that, being relative to the direction perpendicular to the stereo-system parallax, is usually quite flat (see Fig. 2). The distance between each point and its projection onto this plane can be hence used to automatically detect (and discard) the outliers as those points whose distance exceed a fixed threshold. Finally, only the disparities of the inliers are considered for the final NURBS fitting. The four obtained smoothed disparities maps 𝑤𝑟𝑢 ,𝑤𝑟𝑣 ,𝑤𝑡𝑢 and 𝑤𝑡𝑣 represent the approximate warping functions that map the grey level intensity distribution from image Ir to image It and vice versa. (5) Defining the number of intermediate morphed images. Depending on the level of the deformation between the two images to be matched, an even number 𝑁 − 2 of synthetic intermediate images is chosen. A reliability guided scheme can serve to define the optimal number of the images in the sequence [4]. In this work, to match the original images in Fig. 1(c) and (d), a total number 𝑁 = 8 has been used (two original images plus six synthetic morphed images). Note that, for clarity of representation, only three intermediate images have been reported in the scheme of Fig. 2, whereas the entire image sequence used for the data processing is available in the supplementary material. (6) Creating the ith intermediate warped image. Starting from the two images at the extremes of the series 𝐼1 = 𝐼 𝑟 and 𝐼𝑁 = 𝐼 𝑡 , 𝑁 − 2 symmetrically warped images are created by mapping the gray level distributions as follows:

2.2.2. Image-morphing method The incremental approach reported above is the most commonly adopted method to track large deformations with DIC and it has been used here with the only purpose to provide a quantitative benchmark for the results obtained with our developed morphing-based methods described as follows. The procedure was built upon the basic requirement for any accurate DIC matching i.e. the existence of a sufficient similarity between the two speckle patterns to be correlated. The method hence aims to split the large difference between two images in N increments by creating a sequence of synthetic images that simulate a gradual deformation L∕R L∕R between IUND and IDEF [4,5] or a gradual rotation between ILDEF and R IDEF [13]. Without loss of generality, the method will be illustrated here (see scheme in Fig. 2) by processing the image pair ILDEF and IR DEF (Fig. 1(c) and (d)). The procedure consists of the following main steps: (1) Images masking. The input to the procedure is represented by two largely deformed images: the reference image Ir (in this case, the ILDEF ) and the target image It (IR ) with a f(xr , yr ) and g(xt , yt ) DEF gray intensity distribution, respectively. A user-defined mask is applied to both images to delimit the matching operation to the sole region of interest. (2) SIFT keypoints matching. The two input images are matched with SIFT [11], a robust and fast feature-based method allowing to obtain n pairs (with n of the order of 103 ) of corresponding keypoints Kr (xrK , yrK ) and Kt (xtK , ytK ) in the reference and target images, respectively. In particular, in each image, the SIFT algorithm automatically extracts the subpixel position of a

⎧ ( ⎪𝑤 𝑖 𝑥 𝑟 + ⎨ ( ⎪𝑤 𝑖 𝑥 𝑡 − ⎩

( 𝑖−1 𝑤𝑟 𝑥 , 𝑁−2 𝑢 𝑟 𝑁−𝑖 𝑡 ( 𝑤 𝑥, 𝑁−2 𝑢 𝑡

) 𝑦𝑟 , 𝑦𝑟 + ) 𝑦𝑡 , 𝑦𝑇 −

( 𝑖−1 𝑤𝑟 𝑥 , 𝑁−2 𝑣 𝑟 𝑁−𝑖 𝑡 ( 𝑤 𝑥, 𝑁−2 𝑣 𝑡

(3)

)) ( ) 𝑦𝑟 = 𝑓 𝑥𝑟 , 𝑦𝑟 𝑓 𝑜𝑟 𝑖 = 2, … , 𝑁∕2 ) . ) ( ) 𝑓 𝑜𝑟 𝑖 = 𝑁∕2 + 1, … , 𝑁 − 1 𝑦𝑡 = 𝑔 𝑥𝑡 , 𝑦𝑡

4

(4)

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Fig. 2. Flow chart of the NURBS-based image-morphing procedure for large deformation analysis with DIC.

All the morphed images used in this work have been generated with triangle-based cubic interpolation of the gray intensity values using the griddata Matlab function. If the smoothed disparities maps were the exact formulation of the transformation between the Ir and It images, the two central images of the series IN/2 and 𝐼𝑁∕2+1 (in this case the I4

and I5 images) would be identical. Actually, since the morphing functions have been obtained from a smooth fit of a sparse set of data (see Fig. 2), they only approximate the actual transformation between Ir and It and thus the two central images will differ especially in the most deformed areas of the image where also the SIFT algorithm failed to find correspondence. This 5

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difference is however small enough to still allow a reliable DIC matching. (7) Correlating the two central images of the series. A regular (8 pixels point spacing) point grid PREF (xREF , yREF ) is taken as reference on the IN/2 image and matched to the 𝐼𝑁∕2+1 image. Given the high similarity between the two images, a low-quality but fast coarseDIC scheme is sufficient to find the correspondence. It is important to remark at this point, that none of the synthetic images will be used for the final matching between the Ir and It images but they will serve solely to calculate two reliable tentative point grids from which initializing the fine-DIC correspondence search. (8) Propagating the correlation analysis with the incremental approach. The DIC correlation analysis is propagated from the two central images to the end images following the two paths IN/2 → I1 and 𝐼𝑁∕2+1 → 𝐼𝑁 (note that, for clarity of representation, Fig. 2 reports the path Ir → It to emphasize that the morphing procedure closely follows the rationale behind the reference updating scheme). If the value of N has been properly chosen, the difference between each contiguous image is enough small to guarantee a reliable (NCCC > 0.98) deformation tracking with an incremental approach. As a result, two corresponding set of points Pr (xrC , yrC ) and Pt (xtC , ytC ) are roughly matched on the Ir and It images. (9) Final DIC-matching between the two largely deformed images. The Pr (xrC , yrC ) grid (Fig. 1(c)) is taken as a reference for the final DIC analysis while Pt (xtC , ytC ) represents the tentative grid from which extracting the initial guess values for the non-linear optimization included in the DIC-fine search scheme described in Section 2.2.1. In particular, a tentative vector 𝒑0 = (𝑢0 , 𝑣0 , 𝑢𝑥0 , 𝑢𝑦0 , 𝑣𝑥0 , 𝑣𝑦0 ) is obtained through Eq. (2) for each point of the reference grid to find the vector 𝒑 = (𝑢, 𝑣, 𝑢𝑥 , 𝑢𝑦 , 𝑣𝑥 , 𝑣𝑦 ) of the parameters that optimally describe the affine transformation of the related subset. Optimization was here performed using the Sequential Quadratic Programming (SQP) algorithm implemented in the fmincon Matlab function. Examples of the parameters distribution before (p0 ) and after optimization (p) is reported in the supplementary material [23]. Finally, Fig. 1(c) and (d) show the two (coarsened) final correlated mesh grids for the deformed indented membrane.

2.2.3. Shape-morphing method A further morphing-based method capable to retrieve the deformation between two largely different configurations is described in this section. This method was developed to cope with those cases in which the undeformed and deformed geometries are reconstructed with two different stereo-imaging systems from two (or more) pairs of stereo images captured under varying illumination conditions and with different magnifications and spatial resolutions. Some cases exist where the adoption of two different stereo-systems is unwanted but unavoidable (e.g. when a slow evolving phenomenon is tracked over a long period of time, [31]), while sometimes it is intentional, as for the pneumatic bulge test case presented in this paper. In this latter case, in fact, the deformed dome-like geometry has been retrieved by resorting to a 360° multicamera system since the regions with sharp slopes would have not been reconstructed with sufficient accuracy and spatial resolution with the standard stereo-system used for measuring the flat undeformed shape. For all such situations we developed a morphing-based method that performs correlation between the two textured reconstructed geometries instead of matching the relative stereo images. To allow a final comparative analysis, the procedure is here described by using the two pairs of images of the indented silicone membrane (i.e. data obtained from the same stereo-system, Fig. 1). The validity of the method in its most general form will be then demonstrated in the last experimental session. The proposed shape-morphing procedure consists of the following main steps: (1) Shapes reconstruction and grey intensity association. The procedure starts from the two 3D textured shapes of the undeformed (Fig. 4(a)) and deformed (Fig. 4(b)) configurations as stereoreconstructed with two imaging-systems in their respective reference frames. For both configurations, at each 3D reconstructed point P(X, Y, Z) is associated the gray intensity value 𝑔𝑠 = 𝑚𝑒𝑎𝑛(𝑓 (𝑥𝑅 , 𝑦𝑅 ), 𝑔(𝑥𝐿 , 𝑦𝐿 )) as interpolated from the original stereo-images. The geometries are preliminarily properly aligned one to each other through a sequence of rigid-body motions with the aid of predefined landmarks. For the considered test case, the two geometries are already centered and self-aligned in the same XYZ reference system with the quasi-flat undeformed membrane fairly lying in the XY plane (see Fig. 4). (2) Shapes flattening. A ‘calibrated’ image for each configuration is obtained by flattening the 3D surfaces onto a plane [32] according to a proper user-defined transformation. In other words, a function is defined that maps the 3D position of a world point P(X, Y, Z, gs) onto the 2D position Pm (x, y, gs) of a virtual sensor (with a given user-defined spatial resolution and magnification factor MF). Fig. 6(a) and (c) show the calibrated images IDflat and IUflat of the deformed and the undeformed axial symmetric configurations flattened through the simple transformation:

Using the same reference grid Pr (xrC , yrC ), the sequence of operations listed above were used to correlate the image pair ILDEF / ILUND thus calculating the undeformed left point grid (Fig. 1(a)). Finally, since ILUND and IR were not very dissimilar, a standard DIC corUND relation analysis was performed to find the undeformed right point grid (Fig. 1(b)). The high level of deformation existing between the image pairs in Fig. 1 can be easily inferred by looking at the optimized distributions of the deformation gradient components of the shape function reported in Fig. 3 (the more obvious distributions of the u and v parameters are reported in [23]). It is evident, in fact, how the images undergo to large local normal and shear strains for either the pair IR / DEF ILDEF (as regards the gradients of the displacement component along the parallax direction) and IR / IR . Parameters with one order of DEF UND magnitude difference are instead observed for the nearly flat surface of the undeformed configuration (first row of Fig. 3). The analysis was completed by computing the 3D undeformed and deformed geometries (Fig. 4), and the full surface deformation and strain maps (Fig. 5). Plots in Fig. 5 reproduce the expected distributions of deformation and strain (up to 120%) while revealing at the same time a certain degree of asymmetry in the load application. In fact, as it can be also inferred from the asymmetry of the deformed images in Fig. 1 and from the frame sequence in [23], the indenter was not positioned exactly at the membrane center and it moved not perfectly perpendicularly to it. Of course this occurrence had no influence on the validity and on the accuracy of the proposed data processing scheme.

𝑟𝑚 (𝑥, 𝑦) =

√

𝑅2 − 𝑍 2 ⋅ 𝑀𝐹 ; 𝜃𝑚 (𝑥, 𝑦) = 𝜃(𝑋, 𝑌 , 𝑍 )

(5)

where (rm , 𝜃 m ) are the polar coordinates of the image point Pm in the virtual sensor coordinate system (centered at the image center), and (𝜃, 𝜑, R) are the spherical coordinates of the corresponding 3D measured point P in the world coordinate system. In this study, calibrated images of 800 × 800 pixels with 𝑀𝐹 = 10 pixel∕mm have been synthetized. The term ‘calibrated’ is used here to indicate that the information of the original set of 3D coordinates (X, Y, Z) is interpolated to the pixel positions (x, y) of the synthetic image (using the TriScatteredInterp Matlab function) and stored throughout the entire data processing procedure for later retrieval. (3) Image SIFT matching. The two images so obtained (Fig. 6(a) and (c)) are matched with SIFT yielding to find two sets of corresponding keypoints. The related disparities 𝑊𝑢𝐷 (𝑥𝐷𝐾 , 𝑦𝐷𝐾 , 𝑑𝑢) and 𝑊𝑣𝐷 (𝑥𝐷𝐾 , 𝑦𝐷𝐾 , 𝑑𝑣), after NURBS fitting, represent an ap6

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Fig. 3. Plots of the distribution of the displacement gradients after optimization for the three matched pairs of images in Fig.1. First row: parameters for the 𝐼𝑈𝑅𝑁𝐷 ∕𝐼𝑈𝐿𝑁𝐷 image pair. 𝑅 𝐿 𝑅 ∕𝐼𝐷𝐸𝐹 image pair. Third row: parameters for the 𝐼𝐷𝐸𝐹 ∕𝐼𝑈𝑅𝑁𝐷 image pair. Note the large difference between the values of the gradients for the Second row: parameters for the 𝐼𝐷𝐸𝐹 deformation component along the parallax direction as obtained with the same stereo angle on a quasi-flat surface (first row) and a complex shape surface (second row).

Fig. 4. The textured reconstructed geometries of the undeformed (a) and deformed (b) configurations for the circular silicone membrane undergoing axisymmetric indentation.

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Fig. 5. Experimental results for the silicone membrane subjected to axisymmetric indentation. Plots of the displacement vector components u, v, w and of total in-plane displacement √ 𝑑𝑇 𝑂𝑇 = 𝑢2 + 𝑣2 (a)–(d). Plots of the first and second principal Green’s strains (e)-(f).

Fig. 6. Calibrated images of the flattened geometries used for DIC registration with the shape-morphing method. The deformed image IDflat (a), the deformed image after morphing IMflat (b) and the undeformed image IUflat (c). Correlated (coarse) mesh grids are superimposed to the images.

proximate description of the warping transform of IDflat to IUflat over the entire ROI. (4) Image morphing. A simple (i.e. low-order polynomial) function is chosen to fit the warping information obtained from the previous step. The morphing function may assume different forms depending on the variables selected for its expression. In this case, given the axial symmetry of the problem, we assumed 𝜃𝑚 = 𝜃 and closely fit the 𝑟𝑚 = 𝑟𝑚 (𝜑) curve obtained from the previous step with a 3th order polynomial function (see [23]). This function, applied to the original (X, Y, Z) data of the deformed configuration, allowed to create a morphed calibrated image IMflat (Fig. 6(b)) very close to IUflat (compare the two superimposed correlated grids in Fig. 6(b) and (c)). (5) Image DIC matching. A reference point grid is defined over the ROI of IUflat and accurately matched to IMflat with DIC. (6) Correlated 3D point clouds reconstruction. The information 𝑋 = 𝑋(𝑥, 𝑦), 𝑌 = 𝑌 (𝑥, 𝑦) and 𝑍 = 𝑍(𝑥, 𝑦) associated to the calibrated images is interpolated at the non-integer positions

of the two correlated grids, thus yielding to obtain the two (undeformed and deformed) 3D point clouds needed for the final calculation of deformation and surface strains. 2.2.4. Comparison of experimental results obtained with the three different approaches The results obtained from the same set of images as processed with the classical reference-updating scheme (Section 2.2.1) and with the two morphing based approaches reported above have been quantitatively compared by extracting the displacement distributions along two diametric profiles of the deformed membrane. Although the data obtained with the reference-updating scheme is not error-free (a matching error accumulation through the 34 images of the sequence may have been occurred), it was taken here as a reference for evaluating the relative performances of the developed one-step procedures with respect to a classical multi-step scheme. Fig. 7 reports the plots of the displacement components w and u of a point set lying on the 𝑦 = 0 plane and the plot of the displacement component v of another point set at 𝑥 = 0. 8

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Fig. 7. Plots of the displacement profiles along a diameter as obtained with the standard DIC-incremental approach and with the two different morphing methods presented in this paper. Note that the w and the u distributions are related to points at 𝑦 = 0 and the v distribution to points at 𝑥 = 0 (refers to plots in Fig.5).

superplastic aluminum sheet blank subjected to a pneumatic bulge test [33]. In this test, a circular blank is interposed between a die and a blankholder that have been pre-heated up to the test temperature. A gas pressure acts on one side of the blank forcing the material to copy the die geometry. A die with a cylindrical cavity is commonly used for the material characterization [34]. In this work, we tested a superplastic aluminum sheet (commercial name ALNOVI-U) of 1.35 mm thickness, at 500 °C. A die cavity with a 45 mm diameter and a constant gas pressure of 0.8 MPa, which corresponds to an average strain rate of approximately 7 × 10−3 s−1 , were used. Given the geometrical constraints imposed by the testing rig [34] and considering the severe experimental conditions, it was not possible to access the sample to optically track the deformation during the inflation. The sole chances to perform a measurement were limited to before (flat undeformed configuration) and after forming (dome-like deformed configuration, see Fig. 8). This case was thus particularly suited to test our one-step large deformation analysis. The optical measurement presented a series of further challenges mainly related to the lighting and to the application of a suitable speckle pattern. The dome-like shape and the high reflectivity of the smooth metallic surface, in fact, made difficult to find a suitable illumination direction that did not create a diffusing bright spot somewhere on the imaged curved surface. This problem was addressed by averaging the gray level from a large number of images of the same surface portion as captured under different illumination directions as it will be later better explained. As regards the speckle pattern, after several failed attempts employing a high-temperature resistant paint (that detached because of the large deformation) and a permanent marker (that dissolved for the high temperature), we used a B1 graphite pencil to manually apply a coarse speckle pattern that remained fairly unaltered during the test. To increase the contrast, prior to patterning, the aluminum blank surface was bleached with a solution of sodium hydroxide that, from previous tests [35], demonstrated to not affect the material superplastic behavior. The most important issue, however, concerned the proper selection of a video stereo-arrangement capable to guarantee the accurate reconstruction of the entire deformed shape. When imaging largely bulged samples through a standard stereo-video system, in fact, the most peripheral areas may be partially occluded from the view or sampled with poor spatial resolution thus disallowing their accurate reconstruction. Most of the DIC studies available in the literature, in fact, report full-

Fig. 8. Picture of an aluminum sheet blank before and after a pneumatic bulge test showing the large degree of deformation suffered by the sample and the final rupture at the dome apex.

Plots reveal a satisfactory overlapping of the experimental data with an absolute error of 0.06 ± 0.07 mm, 0.03 ± 0.04 mm and 0.01 ± 0.02 mm for the w, u and v components calculated with the image-morphing method, and a corresponding absolute error of 0.05 ± 0.03 mm, 0.02 ± 0.01 mm and 0.02 ± 0.02 mm for the shape-morphing method. A few points with a maximum shape deviation (distance along the Z direction) of 0.31 mm were found in the regions with the largest values of the ux and uy deformation gradients (see second raw in Fig. 3 and error plots in [23]) and relatively low values of the NCCC. Elsewhere, shape deviations from the reference geometry (incremental method) are of the order of 10−2 mm. Finally, the two deformed geometries calculated with the morphing-based methods developed in this work overlap with a maximum absolute error of 8.5 ⋅ 10−2 mm [23]. 2.3. Deformation measurement on a bulge specimen obtained with superplastic forming The final experiment planned for this study aimed to apply the developed method to the full-surface deformation measurement of a 9

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Fig. 9. Picture of the stereo-DIC set-up used for all the experimental tests reported in this study. In this case, the bulge sample is being tested. The sample is mounted on a rotation stage fixed on a 45° bracket with a conical calibration target surrounding it for the automatic merge of the measured points clouds over the full 360°. The upper inset shows a typical image of the series.

Fig. 10. Picture of the bulge sample tested in this study (a) and its full 360° textured geometry reconstructed with Stereo-DIC (b).

surface data only for geometries with mild slopes (e.g. [20]) or limit the measurement to the pole region in the case of large out-of-plane deformation (e.g. [36]). In this work, since no time-resolved measurements were entailed, the problem was addressed by adopting a virtual multicamera system (Fig. 9, [37]). In particular, the dome was mounted onto a rotation stage fixed to the optical bench through a 45° bracket. This special arrangement allowed to perform 360° imaging while providing at the same time an inclined view of the bulged sample that yielded to obtain a fairly uniform spatial resolution along the full profile of the dome (see inset of Fig. 9 showing a typical image of the series). Since a large number of frames were captured over the full 360° (at an angular spacing of about 1°), an incremental approach was adopted to perform spatial tracking between the two 20° spaced stereo images for each of the 21 partially overlapping patches used to reconstruct the deformed geometry. In particular, the patches were automatically reconstructed and merged in the same global reference system by using

the camera pose information provided by the conical calibration target surrounding (and rotating with) the sample (Fig. 9). For further details on the virtual multi-camera system calibration and related metrological performances, the interested reader may refer to [37]. Finally, the virtual multidirectional lighting achieved through sample rotation allowed to attenuate the brightest areas of the sample surface by averaging the gray scale information from overlapping reconstructed areas. A picture of the bulge sample tested in this study, together with its reconstructed textured 3D geometry is reported in Fig. 10. The textured undeformed geometry was reconstructed with the standard stereo arrangement. The deformation analysis was performed by following the shape-morphing procedure described in Section 2.2.3. Prior to data analysis, a coordinate transformation was needed to align the two geometries in the same global reference frame. To this scope, three landmarks chosen among the speckles on the external clamping area of the specimen were matched between the two configurations and reconstructed in the

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Fig. 11. Calibrated images of the aluminum bulge sample used for DIC registration. The original deformed image IDflat (a), the deformed image after shape-morphing IMflat (b) and the undeformed image IUflat (c). DIC correlated (coarse) mesh grids are superimposed to images. Note that the most external portion of the clamping area was not included in the ROI considered for the DIC analysis of the deformed configuration (panels (a) and (b)).

Fig. 12. Plots of sixteen reconstructed 𝑍 = 𝑍(𝑟) profiles of the bulge sample superimposed to the profile of the FE model used as flattening function (a). Plots of three different shapemorphing functions as obtained from the FE simulation data (FEM), from the landmarks position measurement (EXP-dots) and from the measurement with DIC (EXP-DIC). Note that the FEM data intentionally simulate a slightly different test case as better explained in the text.

Fig. 13. A reconstructed coarse set of 3D corresponding points superimposed to the undeformed (a) and deformed (b) geometries showing the large extension of the region interested by the measurement.

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Fig. 14. Experimental results obtained from the shape-morphing DIC-measurement on the aluminum bulge sample. Plots of the u, v and w displacement components (a)–(c). Plots of the Green’s first and second principal strains (d)-(e) and of the anisotropic ratio 𝐷𝐴 = 𝐸1 ∕𝐸2 (f).

corresponding reference frames. After a manual coarse alignment aimed to find an initial guess, an optimization-based routine was run to find the coordinate transformation parameters for the rigid motion (i.e. the components of the rotation matrix and of the translation vector) that minimized the Euclidean distance between the two set of corresponding landmarks. Unfortunately, it was not possible to achieve a ‘perfect’ overlapping over the clamping area since the specimen suffered a certain degree of deformation exerted by the pliers used for removing the dome from the fixture when the material was still warm and deformable (note the deviation from flatness in the clamping area for some of the profiles reported in Fig. 12(a)). However, it should be remarked that, although the measured deformation data may contain a certain amount of rigid body motion, it still serves to extract a correct Green’s strain information [25]. Fig. 11(a) and (c) report the calibrated images of the deformed and undeformed configurations, respectively, showing the severe bulging at the image center and the speckle pattern degradation at the dome pole due to the poor lighting conditions but mostly to the print left from the probe of the position transducer used to track the out-of-plane displacement during the inflation [34]. As regards the selection of a proper warping function to generate the calibrated image in Fig. 11(b), we explored two different problem-specific options that yielded to obtain comparable results. The most practical choice was to evaluate the actual deformation undergone by the sample from measuring the 3D position, before and after the test, of a group of landmarks drawn to this scope along a diameter of the sheet blank (inclined dots line in Fig. 11). Such a choice obviously yielded to obtain an intermediate image very close to the undeformed one (compare the so evaluated deformation function with that measured afterwards with DIC in Fig. 12(b)). Hence, to demonstrate that the morphing law does not need to reproduce the exact transformation between the two images as long as it reduces the large difference between them, we used the displacement information obtained from the FE simulation of a bulge test carried out under different experimental

conditions (0.4 MPa gas pressure at the same temperature, 500 °C). As expected, in this second case, the intermediate image (Fig. 11(b)) is close but slightly deformed with respect to the undeformed image in Fig. 11(c) (compare the correlated grids superimposed to the images). Following the procedure detailed in Section 2.2.3, the two calibrated images in Fig. 11(b) and (c) were matched with DIC. Although the appearance of the speckle patterns was quite different, after some image brightness and contrast adjustment, satisfactory values for the NCCC were obtained almost everywhere with the exception of a few sparse points (especially close to the dome pole, see plot of NCCC distribution over the ROI in [23]). The dense correlated grids then served to reconstruct > 6200 point pairs used for the subsequent deformation analysis. Fig. 13 shows two coarse sets of DIC matched points superimposed to the two textured geometries (note the large extension of the measurement area made possible by the 360° measurement). Finally, Fig. 14 reports the experimental results in terms of displacement and strain maps showing the typical distributions of a bulge test. In particular, from the plot of the degree of anisotropy (ratio between the two principal strains) it is interesting to notice how the large extension of the surface covered by the measurement allowed to capture the gradual vanishing of the equibiaxial stress–strain state (obtained at the pole where a nearly emispherical cup is formed) that occurs when moving towards the clamping area where a large difference exists between the (allowed) meridional strains and the (constrained) circumferential strains. 3. Concluding remarks This study aimed to explore the possibility to perform spatial or temporal DIC matching between two largely deformed images through the generation of a series of synthetic images simulating a gradual change in the viewing direction [13] or a gradual change in the deformation level [4,5]. The one-step temporal matching procedure has also been 12

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References

extended to the case in which the correspondence search is performed between two textured 3D geometries. This latter approach was applied to a bulge test sample obtained with superplastic sheet metal forming allowing to calculate strain levels above the 150% over the entire region of the sample outside the clamping area. The potential relevance of a full-field strain measurement for this special application is undoubted [21,38]. A bulge test rig working at high temperatures is commonly equipped with a position transducer that allows the acquisition of the dome height during the test. After testing, the sample is sectioned and its thickness is measured at the dome apex. Then, on the basis of the membrane theory and related assumptions (e.g. on the dome shape) the strain is calculated a posteriori from the information on the thickness and/or on the dome height [34]. With a full-field optical measurement such the one presented in this work, a complete map of the deformation can be obtained without resorting to any simplifying assumptions, thus providing important information on the potential material anisotropy (which is often neglected at these temperature conditions) and/or heterogeneity (such that due to the selective microstructural alteration caused by a thermo-mechanical process (see e.g. [39,23]). Although the developed shape-morphing procedure has been illustrated with two applications involving axial-symmetric geometries, it can be potentially extended to any geometry after selecting a proper shape flattening strategy among the large variety available in the literature on the subject (see e.g. [32] and reference therein). Therefore, also considering the inherent robustness of the SIFT and of the NURBS frameworks, the method is deemed of potential general applicability under different experimental settings. On the other hand, it should be also acknowledged that this work reports an initial investigation into the potential of a novel methodology and, as such, there is scope for further improvements. A first limitation of our study concerns the method used for assessing the metrological performances of the measurement. It should be specified, in fact, that the data results from the incremental method were here taken as a benchmark with the only aim of comparing the commonly adopted multi-step approach with the proposed one-step method, but that they do not represent the ground truth of the measurement. Interestingly, the two morphing-based procedures brought to results more close to each other than to those obtained with the incremental approach (see error maps in [23]) and this should warn against the presence of a possible error accumulation in the multi-step data analysis. This occurrence could be likely due to the large number of the images of the temporal sequence and to the fact that the deformation increment between two subsequent steps was chosen a priori as a 2 mm Z-step of the indenter and not on the basis of a reliability guided approach [4]. A more rigorous error analysis could be performed by resorting to FE-generated synthetic 3D stereoimages under predefined displacement and noise levels analogously to what has become a common practice for the 2D cases. It should also be mentioned that, while the generation of the synthetic images does not represent a critical step of the procedure since it marginally affects only the definition of the initial guess for the final matching between the two extreme configurations, this latter stage of the analysis could be further improved according to the state of the art on the DIC method. In particular, a second-order shape function may be a more appropriate choice to describe large heterogeneous deformations [40] and the zero-normalized cross correlation (ZNCC) coefficient is expected to cope more effectively with both scaling and offset of lighting [41]. Although the limitations listed above, the results obtained from this preliminary study still reveal an important potential of the morphing-based methods for all those applications (e.g. in biomechanics) where large portions of complex shaped objects undergo large and heterogeneous deformations.

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