Characterization of Necking Phenomena in High-Speed Experiments by Using a Single Camera

2.1. Formation of Images via Mirrors

Mirrors are very useful tools in the field of vision and their shape can be designed to meet various specifications [11–13]. Several angles of view with the same camera are possible. In this part, theoretical expressions of the transformation matrix, that is, relating the 3D coordinates of a point of the scene and its projection in the image plane are recalled. This is performed within the framework of an orthographical model with mirror, which is an appropriate model for the measurements performed herein since the object size is very small (height: 100 mm, diameter: 100 mm, see Figure 1(a)) in comparison with the distance between the camera and the object (ca. 16 m).

Because of the large optical path between the camera and the object (about 16 m), image formation is split into two stages, illustrated in two dimensions in Figure 1(a). This process is identical for the two mirrors and only the generic case is considered in the sequel. Let be a point in the image scene. It is imaged at point by mirror . The orthogonal projection of onto the image plane is denoted by .

To establish the relationship in a 3D setting, the various transformations depicted in Figure 2 are considered. The reference frame of the scene is related to that of the image. Let be the origin in the image plane. For any point , denotes the vector position in the image frame. The mirror is defined by its normal and its center . The mirror plane ( ) is defined by the following equation for any point belonging to the mirror:

(1)

with .

Point is the orthogonal projection of on , so that

(2)

where is the distance

(3)

Point is the symmetric of with respect to plane and its position is given by

(4)

Using homogeneous coordinates, the above relationship can be written in a matrix form

(5)

The transformation from the mirror image to the camera is a classical problem [11]. Usually, it is decomposed into three elementary operations [14], namely, a projection matrix associated with the orthographic projection model

(6)

where is the magnification coefficient, matrix is expressing the transformation between the retinal coordinates in the retinal plane of the camera (in metric units) and the pixel coordinates in the image, and matrix is associated with the transformation between the camera coordinate system and the reference coordinate system, attached to the object in the present case

(7)

where and are scale factors (horizontally and vertically and and are rotation and translation parameters). The combination of these matrices yields

(8)

with being the homogeneous coordinates of in the reference frame. Last, the sought relationship reads

(9)

where the parameters denote the coefficients of the transformation matrix , whose expression is simplified compared with those obtained in the case of pinhole model [6]. In the present case, for and [14].

2.2. Calibration and Stereoscopic Reconstruction

In the following, the calibration technique that provides the coefficients of matrix is introduced. A calibration target is designed with a collection of known reference points where whose position is determined by using a coordinate measuring machine and an optical microscope leading to a m uncertainty. Their image coordinates are identified. The relationship is exploited to determine using a least squares optimization strategy. The objective function to minimize is defined as

(10)

enforcing the condition [14].

Introducing matrix , the elements of matrix read

(11)

The expected conditions for can be checked as a self-consistent validation of the calibration. It is worth noting that with the model used herein, the coefficients are vanishingly small.

In practice, the calibration is carried out by putting a 3D target near the observed object (Figure 3). The calibration using a planar object successively positioned at various positions [15] is not possible in the present experiment. Only one image is necessary to calibrate the system. This approach (i.e., in situ calibration target) is implemented since lighting is obtained by pyrotechnic flashes that are only used during the experiment itself. The latter further requires that the explosive be introduced only at the very end of the experiment preparation, for safety reasons. Prior to that, the position of various objects (the mirrors in particular) may change slightly because of operator manipulations. Therefore, the calibration target has to remain in the field of view (and hence will be destroyed during the explosion). Therefore, the proposed camera calibration procedure is not "optimal" in the sense that all the field of view is not calibrated. However, it will be shown in Section 4.2 that the distortions remain small, thereby only having a small impact on the quality of the reconstruction.

Once the calibration has been carried out, the coordinates of the considered point in the 3D object frame are overdetermined by the corresponding left and right image coordinates . A least squares minimization is used to relate to , which is written as

(12)

with

(13)

where (resp., ) are the coefficients of the left (resp., right) transformation matrix.

2.3. Registration by Digital Image Correlation

The reconstruction is possible only if the points of the right image correspond to those of the left image. It becomes necessary to register spatially and temporally all the points. This is carried out by resorting to Digital Image Correlation (DIC), which consists in following the position of a random pattern in a sequence of images. This technique has the advantage of offering a much denser field of reconstruction than that provided by point tracking. Position uncertainty of the imagepoints is less than 0.1 pixel for the present applications. An example of speckle and grid in the case of cylinder expansion is shown in Figure 4. DIC principle is to register the gray levels of two images, one being the reference and the other the deformed one, with . The brightness conservation is given by

(14)

The technique used herein is global and consists in expanding the displacement onto a basis of (known) functions

(15)

where are the sought parameters associated with basis vectors . The displacement field is then found by carrying out the global minimization of the following functional:

(16)

by resorting to multiscale linearizations/corrections [16] using the following Taylor expansion up to the first order of

(17)

so that linear systems have to be solved

(18)

where and are the partial derivatives with respect to and . System (18) is written in a more compact way as

(19)

where is the vector containing the coefficients to be determined. In the following, the so-called Q4-DIC [16] is used in which a mesh made of 4-noded quadrilateral (Q4) elements are used for which a bilinear interpolation is used to describe the displacement field in each element. The main interest is that continuity of the displacement field is introduced, which offers larger robustness and a greater number of measurement points for the same uncertainty level [16].

4.1. Resolution

A resolution calibration target, similar to a Foucault pattern, is put in place of the object. The former consists of 6 small plates joined together to form a  mm plate. The pattern consists of a succession of horizontal and vertical lines with varying thicknesses ranging from 0.6 mm to 1.4 mm with a 0.1 mm step. The center of the plate is then put in place of the observed object (located at a distance of 16 m from the camera) and at an angle of with respect to the optical axis to estimate the depth of field.

The 25 images acquired by the camera have been compared visually. In view of the absence of any variation, only one image of the sequence was analyzed. The step of digitization is m in the film plane so that the physical size of one pixel is equal to m in the object plane. The image is then filtered out to remove luminous heterogeneities caused by an imperfect lighting. The result is shown in Figure 6.

The resolution of the optical chain is sought to assess the minimum size that can be observed, and the size of the random pattern to be deposited for an optimal observation. To quantify this size, contrast of each line is analyzed to deduce the cut-off frequency corresponding to 50% of the dynamic range. The latter is obtained by analyzing a zone close to the edges of the plate and by rescaling the amplitudes between 0 and 1. Local contrast is obtained in an identical way for horizontal and vertical lines.

To increase the signal-to-noise ratio, the local contrast of each set of lines of the resolution target is obtained by averaging the realigned pattern (after corrections of residual rotations ensuring perfect horizontality or verticality of the transitions). This average is performed over 50 pixels for the vertical direction and 1,000 pixels for the horizontal one. In Figure 7(a), the change of the gray level is shown for two sizes, namely, 1.4 mm and 1 mm, with a loss of contrast appearing for the smallest marking size.

In Figure 7(b), variations of contrast that would be obtained if the calibration target was seen through a linear optical system of Gaussian transfer function of Full Width at Middle Height (FWMH) ranging from 0.5 to 3.9 mm are plotted in solid lines. These values are given for magnifications of about 20. The experimental curve lies between the lines corresponding to an FWMH of 1.1 and 1.3 mm, meaning that it will be difficult to distinguish correctly elements smaller than these two sizes. Thus, the speckle deposited onto the cylinder must have at least a diameter of 1.2 mm. This characterization is useful for the realistic synthesis of images discussed in Section 5.1.

4.2. Lens Distortions

Because of the use of a rotating mirror framing camera and many mirrors between the object and the camera, the estimation of optical distortions is an important step in the experiments. Techniques utilized to determine the distortions of the whole optical system require the acquisition of one image per secondary lens. It is different from procedures followed to analyze quasistatic experiments [17] or even for the case of a dynamic test [18]. Moreover, because of the complexity of the optical chain, it is not possible to project the distortions found onto a polynomial basis as generally performed [6, 17]. In the present case, the frame-to-frame distortions were found negligible with respect to that of the whole optical chain.

The proposed approach to correct for optical distortions consists in resorting to a random numerical texture that is printed onto a plate by laser engraving. The plate and its support are put on a stool. Then, pictures of the plate are shot by the camera. A correlation computation is run between the digital reference and the first image acquired by the camera. This computation, carried out by the technique presented in Section 2.3, provides a displacement field containing all the information concerning magnification, in- and out-of-plane rotation (a first-order approximation can be used in the present case since the distance of the object to the camera is very large), and distortions. To account for these different components, the displacement field is projected onto the following basis

(20)

The distortion field corresponds to the displacement residuals. The type of image obtained in the present case is shown in Figure 8(a) for a theoretical image shown in Figure 8(b). The junction between the two mirrors gives rise to spurious results. Consequently, the distortions are evaluated independently for the left and right mirrors (Figures 9(a) and 9(b)). The amplitudes of the distortions remain small, namely, mean value in the centipixel range, standard deviation in the pixel range, and no particular pattern is caused by the piece of adhesive tape seen in Figure 8(a). Consequently, the influence of optical distortions on the stereoscopic reconstruction of the object is neglected. The fields of distortions are similar for the 25 images of the sequence [19], thereby proving that the frame-to-frame distortions are of secondary influence. Last, no artifacts related to digitization (e.g., discontinuities between successive lines perpendicular to the scan direction) were observed in the analyses performed in the present section. If any, they remain very small in comparison with those induced by the optical chain.

5.1. Detection Level

Before applying the stereo-correlation procedure to an experimental case, it is important to evaluate the size and amplitude of defects that can be detected. The hydrodynamic code HESIONE [25] predicts the shape of the specimen at different stages of evolution. For any instant of time, , the predicted surface is projected onto the actual surface by least squares minimization.

Figure 10(a) shows one picture of the specimen in its reference state. The surface texture is artificially created by mimicking laser marking (i.e., with parallel rays) based on a computerized pattern. The picture of the surface deformed by the computed displacement field, and onto which the original surface marking has been projected, is shown in Figure 10(b). In addition to the smooth displacement field, some additional perturbations are superimposed to check the resolution of the analysis. They would correspond to localized "bumps" of various diameters (0.5, 1, 5, 10, and 30 mm) and amplitudes (0.125, 0.25, 0.5, 1, 2.5, and 5 mm) as illustrated in Figure 10(c). A total of 15 different perturbations are introduced.

Based on the knowledge of the transformation matrix, each point of the 3D surface is projected onto the two image planes to create synthetic left and right stereoscopic image pairs as close as possible to experimental images. When compared with the experimental geometry, the mean distance between the projection into the image of a known 3D point and the corresponding image-point extracted in the image is equal to pixels. The blurring effect of the entire optical chain is taken into account through convolution with a Gaussian filter. 16 image pairs are generated, one of them (reference) containing no perturbation. Two examples of left-right pairs are shown in Figures 11(a)–11(d) and 11(b)–11(e). To appreciate the effect of the bumps on the images, the same figure shows the difference between two similar images with and without the perturbations. Figures 11(c) and 11(f) correspond, respectively, to the left and right views. It is to be emphasized that no noise has been added to the images in order to focus on detection issues.

A DIC analysis was performed on those artificial images, based on the same choice of parameters as the one used in the experiment, namely, pixel elements are selected based on the signal-to-noise ratio. A comparison between the measured and prescribed displacements for each bump allows for the evaluation of the resolution. To carry out this analysis, the measured and imposed shapes are unfolded onto a plane as suggested by Luo and Riou [26].

Figure 12(a) shows the prescribed perturbation for the easiest cases (amplitudes of 2.5 mm and 5 mm, left and right, respectively, for a 30 mm diameter bump), while Figure 12(b) is the measured shape. In spite of a large noise affecting the shape of the bump, this perturbation is rather well captured by DIC computations. Figures 12(c) and 12(d) correspond to smaller perturbations (amplitudes of 125 m and 250 m, left and right respectively, for a 5 mm diameter). Although the perturbations are detected, their sizes and amplitudes cannot be estimated reliably. The reason for this lies in the intrinsic resolution of the DIC analysis performed here with elements of size 16 pixels or 2.9 mm. Thus the entire bump can fit in a two-element wide square. A summary of the results is presented in Figures 13(a) and 13(b) where measured amplitudes and diameter, respectively, normalized by the prescribed counterpart, are shown for all tested cases.

It is concluded that for very severe experimental conditions (rotating mirror high-speed camera at 16 m optical distance from the specimen, magnification of 22, small radius of curvature, and poorly contrasted surface texture), the limit of detection of such bumps is of the order of 5 mm, and a minimum size of about 10 mm is needed to allow for a reliable quantification of the perturbation. Moreover, 250  m amplitudes are resolved reliably for those conditions. These conclusions hold for a fixed element size of 16 pixels. Smaller elements lead to too noisy measurements to secure the determination, whereas larger elements are too coarse. This level is to be compared with the 3D reconstruction uncertainty achieved herein. A level of m is estimated by randomly perturbing the position of the calibration points, and reconstructed points with realistic values [19].

Possible improvements involve drastic changes in the experimental set-up. CCD camera could offer images in digital format directly, thus limiting the digitation step in the analysis. However, access to similar pixel sizes still represents a technical challenge. A better resolution could also be obtained through a higher magnification, at the expense of a smaller frame.

5.2. Large Displacement Handling

In the context of detonics, very high strain levels between consecutive images have to be captured. This fact is a major difficulty for DIC. A specific procedure has been designed to allow for a much more robust analysis in this context. As a side benefit, displacement fields appear to be less subjected to noise.

The principle of the method is simply to initialize the DIC analysis, which is in the present case an iterative procedure, by a prior determination of the displacement field obtained via a simulation of the experiment. This allows for a convergence of the displacement determination into the deepest minimum, and avoids trappings into secondary minima. Let us note that a multiscale strategy is adopted in the DIC analysis for the same purpose of limiting secondary minima trappings [16]. However, at the largest scales, the contrast of the images is significantly reduced and hence some nodes or zones may be polluted by such artifacts. In contrast, a fair prior estimate of the displacement field, which may still be inaccurate, requires DIC to address only the remaining corrections displacements and hence can be tackled with less coarsened images. As this procedure only affects the initial displacement field, it does not affect the final one at convergence as can be checked by perturbing this prior determination, and checking that the final determination is unaffected. This robustness allows for some tolerance on the quality of this first displacement field, and hence, small effects such as the motion of the cylinder axis are neglected.

In the DIC procedure, the deformed image is corrected at each iteration by the current determination of the displacement field. Hence, any displacement field can be used at initialization, and no subsequent modification of the DIC algorithm is needed. When the initial field is close to the final solution, convergence is fast and accurate. The only operation needed is to transform the 3D displacement field from HESIONE code [25] into both the right and left 2D displacement fields used for the time registration of the DIC procedure. This projection is performed as in the previous subsections using the transformation matrices introduced in Section 2 once the 3D frames used in the code and in the stereo-correlation procedure coincide. This is achieved by a least squares minimization between the initial experimental and numerical shapes. A mispositioning error of the order of 1 mm is achieved. Figure 14 illustrates the effect of the prior correction of a late image onto reference one. It can be seen that most of the displacement has been accounted for, and only small differences remain to be determined (by DIC). When this prior determination is not taken into account, a larger element size (24 pixels) has to be chosen to limit the uncertainty level. With the present initialization, a 16 pixel element can be dealt with.

5.3. Experimental Reconstruction

After the experiment, the film composed of 25 acquisitions is developed and the 25 images are digitized independently from each other. From the fixed elements of the scene (yellow paper in Figure 1(a) or the calibration target shown in Figure 14), repositioning of images is performed by adjusting a translation correction so that the elements remain motionless [19].

Reconstruction of the surface is given in Figure 15 for three different times. In these figures, the result of the stereo-correlation is shown in blue, while the computed one (given by HESIONE) is in red. Apart from some slight positioning differences between experimental and computed geometries, the two surfaces superimpose quite well for the entire duration of the experiment. Displacement corrections of at most pixels need to be measured. With the multiscale algorithm used herein, this level is easily measured. Moreover, in the last picture, a clear defect can be identified, which is interpreted as a local thinning of the specimen, that is, an example of necking. The image pair shown in Figure 16 supports that statement.

In order to investigate on a quantitative basis the onset of necking, it is proposed to base the analysis on the standard deviation of the normal surface displacements. This standard deviation can be seen as a measure of the roughness of the expanding surface, which is expected to remain small for a uniform strain of the surface, and to display a sudden increase when necking (at least necking that can be captured by the DIC analysis, i.e., at a large enough scale). The standard deviation is expected to be sensitive to the sampled surface as the lower part of the specimen is subjected to a larger strain. Thus the standard deviation is estimated over three regions, namely, first globally over the entire field of analysis, second over a central zone (where edge effects are avoided), and third on a zone located at the bottom of the specimen where necking is seen to occur first.

The first two standard deviations are shown as functions of time (through the image number) in Figure 17(a). A steady increase is observed with an acceleration taking place for the two last stages, although the level of fluctuation of this curve makes this last observation questionable. The central zone shows a smaller roughening, which is consistent with the lower strain level reached in this zone. Let us emphasize the fact that the displacement field is computed from the reference image to the current one, and hence the steady increase cannot be attributed to a cumulation of measurement errors. Note also that the initial image has already a significant roughness because it has suffered a significant expansion prior to be captured in the first image. The third standard deviation computed over the bottom part of the specimen is shown in Figure 17(b). The sudden acceleration of the standard deviation for the last images seems to depart clearly from the previous steady evolution and supports the previous discussion on a detectable necking occurring in this zone. This also supports the idea that the global standard deviation is affected by the necking of the lower part, and that the sudden rise is actually meaningful. Moreover, the a priori uncertainty is estimated to amount to 0.5 pixel, or 90  m. The impact of image noise has been estimated to amount to an additional 100  m [19]. The sum of these uncertainties is well below the fluctuation level reported in those graphs.