# 3D displacement measurements using a single camera

Julien Réthoré, Fabrice Morestin, Lionel Lafarge, Paul Valverde

► To cite this version:

Julien Réthoré, Fabrice Morestin, Lionel Lafarge, Paul Valverde. 3D displacement measurements using a single camera. Optics and Lasers in Engineering, 2014, 57, pp.20-27. 10.1016/j.optlaseng.2014.01.009. hal-04710335

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# 3D displacement measurements using a single camera

Julien Réthoré<sup>a</sup>, Fabrice Morestin<sup>a</sup>, Lionel Lafarge<sup>a</sup>, Paul Valverde<sup>a</sup>

<sup>a</sup>Laboratoire de Mécanique des Contacts et des Structures (LaMCoS)

Universités de Lyon / INSA Lyon / CNRS UMR 5259

20 Avenue des Sciences, F-69621 Villeurbanne Cedex, France

## Abstract

Measuring 3D displacements usually requires at least two cameras or 3D images. However, in some espacially constrained experimental circumstances, it is only possible to use a single camera. The present paper develops a methodology, based on digital image correlation that is able to measure the homogeneous 3D displacement of a plane surface using a single camera. Based on Descartes’ theory, a model is elaborated that accounts not only for the magnification change due to the out-of-plane displacement but also for the unfocusing of the object. Several examples are provided that show the ability of the technique to track 3D displacements with various optical setups with different depths of field.

**Keywords:** 3D displacement, digital image correlation, Descartes’ optics, single camera

<footnote>Email addresses: julien.rethore@lamcos.insa-lyon.fr (Julien Réthoré), fabrice.morestin@lamcos.insa-lyon.fr (Fabrice Morestin)</footnote>
# 1. Introduction

Since the earliest developments of Digital Image Correlation (DIC), see for example Sutton et al. (1983), the measurement of 3D displacements from 2D images is one of its most advanced applications. The most popular technique consists in using a multiple (at least 2) camera system that allows for recovering not only the 3D displacement field at the surface of an object but also its shape, see Helm et al. (1996) as an early development of the technique. Called stereo-DIC, this technique has encountered a fantastic gain of popularity during the last decade. Its flexibility, its robustness, and its accuracy are nowadays well established. However, in some particular experimental circumstances, using several cameras reveals practically impossible. Typically, small objects (smaller than 1cm² in area) are difficult to analyse. Further, the surface of interest must be accessible with two different angles of view in order to recover a 3D vision. As an alternative to multiple camera techniques, methods for measuring a 3D displacement using a single camera have been proposed more recently. Alternative methods that consist in adding an optical device between the object and the single camera in order to mimic stereo-vision either using mirrors (Pankow et al., 2010; Besnard et al., 2010) or a biprism (Genovese et al., 2012) have been developed. Other methods use diffraction gratting in order to create virtual images of the object on the same camera (Xia et al., 2013; Pan and Wang, 2013). Some other methods combine DIC and Fringe Projection as in Tay et al. (2004), Nguyen et al. (2011) or Siegmann et al. (2011). However, in Tay et al. (2005) or Quan et al. (2008), only a single camera is used without using any other optical system. The principle of these techniques is to model the in-plane transformation of the image due to a 3D displacement of the object. Then using standard a 2D DIC technique, this transformation is measured and the 3D displacement is further identified from the 2D field. To the authors' knowledge, existing techniques consider optical systems with high depth of field, meaning that the only transformation of the image due to the out-of-plane displacement is an affine transformation in the plane. However, the out-of-plane motion also induced a blurring of the image due to unfocusing of the object. While this effect can be neglected when optical system with high depth of field are used, it becomes critical for low depth of field systems, e.g. using macro-lenses. The purpose of the present paper is to extend the methodology for measuring a 3D displacement using a single camera to the case when the blurring effect cannot be neglected. The paper is organized as follows: first, in Section 2, the general framework of Descartes is briefly described as well the effects of out-of-plane motion, then the elaboration of the proposed methodology is detailed in Section 3 and finally, in Section 4, some example applications show the ability of the technique to extract out-of-plane displacement from images coming from a single camera mounted with different types of lens.
## 2. General framework

### 2.1. Descartes' relations

Descartes' relations are based on a geometrical analysis of the rays' traces. Figure 1 gives an illustration of a typical configuration. A convergent lens is considered. Two basic rules apply:

1. From the object $AB$, the rays that are parallel to the optical axis go through the focal point $F'$.

2. Rays passing through the lens center $O$ are not modified.

Using these two rules, it is possible to form the image $A'B'$ of the object. If $p$ denotes the distance between the object plane ($OA$), $f'$ the focal distance ($OF'$), then the distance $p'$ between the lens and the image plane ($OA'$) fulfils the following relation:

$$ \frac{1}{p} + \frac{1}{p'} = \frac{1}{f'} \qquad (1) $$

### 2.2. Out-of-plane object motion

Suppose now that the object is moved by $w$ along the optical axis. The first effect of the out-of-plane motion is a variation of the magnification. In the image plane, the out-of-plane motion is thus converted into a scaling of the image. The scale factor will be shown to be proportional to the ratio between the lens to object distance and the displacement i.e. $w/p$. This effect has been exploited in several publications in order to measure: the out-of-plane displacement of the object (by identifying the scaling factor of the induced in-plane motion) Tay et al. (2005) or the topography of an object Huang et al. (2005). Considering an homogenenous displacement or at least a piece wise homogeneous displacement, 3D displacement can be retrieved from the apparent in-plane motion Quan et al. (2008). A second effect of the out-of-plane motion of the object is that the image of the object forms in an other plane than $A'B'$. However, if the image is still captured in this plane, a blurring effect appears. Each point of the object becomes a circular zone on the image plane. Conversely, at each point of the image plane, rays emanating from points located within a circular zone of the object add their contribution. The typical size of this zone is again proportional to the ratio $w/p$ but also to the ratio $p'/p$. In usual optical setups in experimental mechanics, the latter ratio is very small. Meaning that a large depth of field is guaranteed and the blurring effect is negligible. However, in the case when macro-lenses are used $p' \approx p$ then the blurring effect cannot be neglected.

The purpose of the developments proposed herein is to add the blurring effect to the in-plane induced motion in order to extend the application of the out-of-plane displacement measurement to the use of optical setups with reduced depth of field.
### 3. Elaboration

In this section, we are describing how the two effects of out-of-plane motions can be modeled. Then a methodology is proposed to estimate the out-of-plane displacement from two images and the appropriate calibration procedure is detailed.

#### 3.1. Translated image construction

Figure 1 describes the considered system as well as the notations used in the following. Let $f$ be the function describing the grey level of the image of the object captured when the image plane and the object plane are conjugate (state $S_0$) through out the Descartes' relations, i.e. when Equation (1) is fulfilled. In this case, the relation:

$$ \frac{r'}{r} = \frac{p'}{p} \qquad (2) $$

holds. In this equation, $r'$ is distance between point $B'$ and the optical axis and $r$ the distance between point $B$ and the optical axis. The object plane is now moved by $w$ along the optical axis, state $S_1$. $P_w(f)$ will denote the transformation of the image from state $S_0$ to state $S_1$. In the latter state, the object is not conjugate with the image plane which is supposed to be fixed. Then, each point of the image plane is reached by the rays coming from a disk (when a circular aperture is considered) on the object plane. The center of this disk is $O_{S_1}$ the intersection of the translated object plane with the ray passing through the lens center $O$. Its distance $R$ from the optical axis is obtained from

$$ \frac{r'}{R} = \frac{p'}{p+w}. \qquad (3) $$

The radius of the disk is obtained by considering marginal rays. The aperture of the camera is considered as a circle of diameter $\phi$. If $D$ denotes the diameter of the disk in the object plane then we have:

$$ \frac{D}{\phi} = \frac{w}{p}. \qquad (4) $$

Any point in the disk had its grey level directly stored in $f$ when the object plane was conjugate to the image plane. The center of the disk $O'$ was at a distance $R'$ from the optical axis given by

$$ R' = R \frac{p'}{p} = r'(1 + \frac{w}{p}). \qquad (5) $$

Here is the first effect mentioned in the previous Section: compared to the image obtained in state $S_0$, a scale factor $w/p$ is applied to the image. Note that if this scale factor can be estimated (e.g. by in-plane digital image correlation) then the out-of-plane displacement can be derived from the scale factor as proposed in
Again considering marginal rays, the diameter of the disk is

$$D' = D \frac{p'}{p} = \phi \frac{w p'}{p^2}. \qquad (6)$$

Here is the second effect, for one point of the image acquired in state $S_1$, the grey level is an averaged of the grey level at all the points of the image acquired in state $S_0$, i.e. $f$ over a disk of diameter $D'$ and center $O'$. This is written, in a polar co-ordinates system $(\rho, \alpha)$ centered at $O'$ (see Figure 2):

$$P_w(f)(x, y) = \frac{4}{D'^2\pi} \int_{\rho=0}^{\rho=D'/2} \int_{\alpha=-\pi}^{\alpha=\pi} f(\bar{x} + \rho \cos(\alpha), \bar{y} + \rho \sin(\alpha)) \rho d\rho d\alpha. \qquad (7)$$

where $(x, y)$ are the cartesian co-ordinates of any point and $(\bar{x}, \bar{y})$ the co-ordinates of the disk center $O'$ in the reference image $f$:

$$\bar{x} = (x - x_c)(1 + \frac{w}{p}) + x_c \qquad (8)$$

and

$$\bar{y} = (y - y_c)(1 + \frac{w}{p}) + y_c, \qquad (9)$$

where $(x_c, y_c)$ are the co-ordinates of the intersection between the optical axis and the image plane. In the numerical implementation, this integral is performed using $n_g \times n_g$ equally spaced and equally weighted integration points. These integration points are distributed over a square that contains the disk over which the integration must be performed. Points that fall out of the disk are affected a zero weight. The number of integration points $n_g$ is chosen as $2E(D'/2)$ where $E$ states for the integer part of a real number.

### 3.2. Image correlation

It is now supposed that once the image $f$ conjugate to the object has been captured, the sample is moved and an other image $g$ is captured. The value of the displacement applied to the sample when $g$ was acquired is now searched for. For this purpose, the optical flow equation is used. This equation written between the actual image $g$ and the transformed image $P_w(f)$ reads:

$$P_w(f)(x, y) = g(x, y). \qquad (10)$$

A non-linear least-squares technique is used to obtain an estimation of the out-of-plane displacement $w$. $w$ is thus obtained through:

$$w = \operatorname{Arg Min} \int_{(x,y) \in ROI} [P_w(f)(x, y) - g(x, y)]^2 dx dy \qquad (11)$$
In this equation, ROI defines the Region Of Interest of the image over which the computation is performed. If a uniform displacement w is thought then, from an initial guess, the solution increment dw is obtained by the following formula:

$$dw = \frac{\int_{(x,y) \in ROI} \frac{\partial P_w}{\partial w}(x,y) [P_w(f)(x,y) - g(x,y)] dxdy}{\int_{(x,y) \in ROI} \frac{\partial P_w}{\partial w}(x,y) \frac{\partial P_w}{\partial w}(x,y) dxdy} \quad (12)$$

The iterative process is run until convergence is reached, i.e. for example when $|\frac{dw}{w}| < 1.e-3$. The efficiency of the procedure strongly depends on the evaluation of $\frac{\partial P_w}{\partial w}$. This derivative formally reads:

$$\frac{\partial P_w}{\partial w}(x,y) = -\frac{8}{D'^3\pi} \frac{\phi p'}{p^2} \int_{\rho=0}^{\rho=D'^2/2} \int_{\alpha=-\pi}^{\alpha=\pi} f(\bar{x} + \rho \cos(\alpha), \bar{y} + \rho \sin(\alpha)) \rho d\rho d\alpha + \frac{2}{D'\pi} \int_{\alpha=-\pi}^{\alpha=\pi} f(\bar{x} + \frac{D'}{2} \cos(\alpha), \bar{y} + \frac{D'}{2} \sin(\alpha)) \left[ \frac{\phi p'}{p^2} + \frac{r'}{p} (\cos(\theta) \cos(\alpha) + \sin(\theta) \sin(\alpha)) \right] d\alpha. \quad (13)$$

In this equation $(r', \theta)$ are the polar co-ordinates of any $(x, y)$ point with respect to the optical axis. Two terms appear in the expression of $\frac{\partial P_w}{\partial w}$, the first one corresponds to the variation of the integral relative to a change in the disk's area and the second one is related to the flow of light intensity through the disk's contour when its shape is modified according to a change in $w$. An solution increment $dw$ is then computed and the solution is updated. It is supposed above that there is no constrast neither brightness variation. In pratice, such variations may occur and they must be accomodated in some way. Constrast and brightness variations could be introduced as two additional parameters of the minimization to fall in the scope of the zero mean normalized sum of square differences criterion. It is preferred herein to normalize a priori (without considereing the normalization parameters as unknown of the minimization problem) the transformed image $P_w(f)$ after it has been computed. The normalization consists in shifting (same mean value) and scaling (same standard deviation) the grey level histogram of the transformed image so that it matches the histogramm of $g$.

### 3.3. Influence of in-plane motion

In the previous paragraph, it has been supposed that no in-plane displacement occurs. It is actually not the case and these effects must be incorporated for a robust estimation of the out-of-plane displacement. If a homogeneous in-plane translation $(u, v)$ is considered then the transformation of the initial image reads

$$P_U(f)(x, y) = \frac{4}{D'^2\pi} \int_{\rho=0}^{\rho=D'^2/2} \int_{\alpha=-\pi}^{\alpha=\pi} f(\bar{x} - u + \rho \cos(\alpha), \bar{y} - v + \rho \sin(\alpha)) \rho d\rho d\alpha. \quad (14)$$

The 3D displacement vector is now obtained as the result of the least-squares minimization of the scatter between the actual image $g$ and the transformed image $P_U(f)$:

$$(u, v, w) = \operatorname{Arg Min} \int_{(x,y) \in ROI} [P_U(f)(x,y) - g(x,y)]^2 dxdy \quad (15)$$
Then from an intial value of the displacements **U** = (u, v, w), the solution increment d**U** is obtained by solving a linear system

$$
\mathbf{M}\mathbf{d}\mathbf{U} = \mathbf{b}. \tag{16}
$$

If the terms corresponding to u, respectively v and w, refers to indice 1, respectively 2 and 3, then the
components of matrix **M** are

$$
M_{ij} = \int_{(x,y) \in \text{ROI}} \frac{\partial P_U}{\partial U_i}(x,y) \frac{\partial P_U}{\partial U_j}(x,y) dx dy, \quad (17)
$$

and the components in vector **b** have the following expression:

$$
b_i = \int_{(x,y) \in \text{ROI}} \frac{\partial P_U}{\partial U_i}(x,y) [P_U(f)(x,y) - g(x,y)] \, dxdy. \quad (18)
$$

The derivative of the transformed image P<sub>U</sub>(f) with respect to the three components of the displacement are invoked in the linear system to solve. Concerning the derivative with respect to the out-of-plane displacement w, the expression of Equation (13) is modified in order to account for u, v and for the offset of the optical center:

$$
\frac{\partial P_U}{\partial U_3}(x, y) = -\frac{8}{D'^3\pi} \frac{\phi p'}{p^2} \int_{\rho=0}^{\rho=D'/2} \int_{\alpha=-\pi}^{\alpha=\pi} f(\bar{x} - u + \rho \cos(\alpha), \bar{y} - v + \rho \sin(\alpha)) \rho d\rho d\alpha \\
+ \frac{2}{D'\pi} \int_{\alpha=-\pi}^{\alpha=\pi} f\left(\bar{x} - u + \frac{D'}{2} \cos(\alpha), \bar{y} - v + \frac{D'}{2} \sin(\alpha)\right) \left[\frac{\phi p'}{p^2} + \frac{r'}{p}(\cos(\theta)\cos(\alpha) + \sin(\theta)\sin(\alpha))\right] d\alpha. \tag{19}
$$

Concerning the in-plane displacement, they do not influence the size of the integration disk. However, their
variation produces a motion of the integration disk. This results in a variation of the transformed image
accounting for the flow of the light intensity through the disk’s contour when it is moving along x or y axis.
This leads to the following expressions

$$
\frac{\partial P_U}{\partial U_1}(x, y) = -\frac{2}{D'\pi} \int_{\alpha=-\pi}^{\alpha=\pi} f\left(\bar{x} - u + \frac{D'}{2}\cos(\alpha), \bar{y} - v + \frac{D'}{2}\sin(\alpha)\right) \cos(\alpha) d\alpha \quad (20)
$$

and

$$
\frac{\partial P_U}{\partial U_2}(x, y) = -\frac{2}{D'\pi} \int_{\alpha=-\pi}^{\alpha=\pi} f\left(\bar{x} - u + \frac{D'}{2}\cos(\alpha), \bar{y} - v + \frac{D'}{2}\sin(\alpha)\right) \sin(\alpha) d\alpha. \quad (21)
$$

In the formulation elaborated herein, (u, v, w) are unknown displacements to be measured. Several additional
parameters are invoked in the formulas: (x<sub>c</sub>, y<sub>c</sub>) the co-ordinates of the optical center of the optical system
but also p, p' and φ that characterize the geometry of the optical setup. It is noted that p' and φ appear in
the expressions above only as φp'/p². Two independent parameters are chosen in order to describe the system
: a₁ = 1/p and a₂ = φp'/p². In addition to these parameters, (x<sub>c</sub>, y<sub>c</sub>) also describe the optical system.
### 3.4. Calibration

It is not practically feasible to obtain direct robust estimates of $(x_c, y_c)$, $p$, $p'$ and $\phi$. A calibration procedure is thus designed in order to estimate the meaningful parameters $(x_c, y_c, a_1, a_2)$. For this purpose, starting from an initial image conjugate to the object ($f$), a series of $N_c$ images {$h$}$_{j=1..N_c}$ (at least one) are captured for given values {$w_j^c$}$_{j=1..N_c}$ of the out-of-plane displacement and no in-plane displacement. For all these images, there is no in-plane displacement, the out-of-plane displacement is known and the tranformed image $P_{w_j^c}(f)$ is supposed to register with the corresponding image $h_j$:

$$ (x_c, y_c, a_1, a_2) = \operatorname{Arg Min} \sum_{j=1..N_c} \int_{(x,y) \in ROI} \left[ P_{w_j^c}(f)(x,y) - h_j(x,y) \right]^2 dx dy. \quad (22) $$

Following the same route as for estimating the displacements in the previous section, an iterative solution algorithm is initiated until convergence on the calibration parameters is reached. A linear system is solved at each iteration in order to update the parameter values. The linear system to solve is similar to what was obtained in the previous paragraph and the derivation of the mathematical expression is not detailed herein. While only rigid-body translations have been considered herein, the proposed methodology is not restricted to such simple kinematics. The present study is proposed with the examples below as a proof of concept which we will further investigate to obtain arbitrary 3D displacement fields.

## 4. Examples

### 4.1. Motion of a light-box

An home-made light box was especially designed to validate the procedure proposed in the paper. A transparent surface on which a speckle pattern was applied, using spraid black paint, is placed between a light source and the camera. Then the object (the transparent plate with the black paint) is moved along the optical axis and in a plane parallel to the camera plane. The light source is attached to the plate. Hence there is no light variation induced by the plate motion.

The camera is mounted first with a 35 mm lens and then with a 85 mm lens. Figure 3 shows the reference images captured with the two lenses. The aperture is kept to 2.4 in order to minimize the depth of field and the working distance is kept the same (about 1 m). Two other cameras are used in order to analyze the motion of the plate using stereo-DIC. This additional system will be used for validation purposes. For comparison purposes again, an in-plane digital image correlation technique will be used to estimate the out-of-plane displacement. The latter ignores the blurring effect due to the out-of-plane displacement. The advantages and the limitations of this approach compared to the proposed one will be emphasized.
### 4.1.1. Using a 35 mm lens

First a 35 mm lens is mounted on the camera. Figure 4 shows images for a ±10 mm out-of-plane displacement. In this optical configuration, the influence of the out-of-plane displacement is difficult to detect by a naked eye. The blurring effect is low and thus all the three approaches listed above are supposed to give the same results. Out-of-plane as well as in-plane displacements are measured using the three methods mentioned above:

* 3D-DIC refers to the results obtained using 2 cameras and a stereo-DIC technique.

* 2D-DIC refers to a method inspired by Tay et al. (2005) that converts an in-plane affine transformation into an out-of-plane displacement.

* I-DIC refers to the technique proposed herein that integrates the two effects of the out-of-plane displacement into a DIC algorithm.

A series of images with different positions of the light box are treated. For the 2D-based methods the calibration is performed using image number 30, that was captured at an out-of-plane displacement of -10 mm. Concerning the out-of-plane displacement, the results are shown by Figure 5. It is demonstrated that all the three methods perfectly agree. Considering 3D-DIC as the reference measurement, it is thus shown that the geometrical model used in the formulation of the 2D based methods is correct. On Figure 6, all the three components of the displacement obtained by 3D-DIC and I-DIC are plotted. Again a perfect agreement is obtained what demonstrates the ability of the proposed I-DIC method for measuring 3D homogeneous displacements. To confirm these results, for an out-of-plane displacement of ±10 mm, Figure 4 shows the actual images, the reconstructed images $P_U(f)$ and the scatter between these two images. For representation purposes, the latter have been rescaled otherwise they appear completely black, the mean difference being about 2 grey levels. This enforces the confidence in the reconstruction of the translated image.

### 4.1.2. Using a 85 mm lens

Second, a 85 mm lens is mounted on the camera. Using this lens, the depth of field is slightly lower than in the previous case when a 35 mm lens was used. The reference image is shown by Figure 3(b). On Figure 7, the comparison between the actual images and the reconstructed images for out-of-plane displacements of ±10 mm is proposed. The scatter between the two images is also depicted on this Figure. The mean scatter is again about 2 grey level what demonstrates a satisfying reconstruction of the actual image from the reference one even in this case with a lower depth of field. On Figure 8, the in-plane displacements and
out-of-plane displacement obtained from 3D-DIC and I-DIC are compared. Again, a excellent agreement
between the two methods is obtained.

4.2. Using a macro-lens

In this example a macro-lens is used with a large magnification. This type of lens is known to have a
very low depth of field as the distance between the object and the lens is of the same order of magnitude
as the distance between the lens and the image plane. The large magnification and the low depth of field of
this type of lens make the images captured very sensitive to out-of-plane displacements. In the experiment,
a small mirror (12 mm in diameter) is fixed on the head of a piston that is translated in order to increase the
pressure of a lubricant which rheology is to be characterized. High pressure range (up to 30 GPa) may be
investigated and thus the design of the chamber and the piston is subjected to extremely severe constraints.
Using only one camera mounted with a macro-lens is the only acceptable optical setup for observing the
motion of the piston.

The experiment proposed herein is performed in a validation stage of the design of the testing device.
The piston is fixed on a micro-metric moving platform. Using this platform a prescribed displacement is
imposed to the piston while a series of images is captured. The reference image is shown by Figure 9. Durst
was settled on the lens before image acquisition. A mask is computed from an image of a white object and
the pixels where durst was settled are ignored in further computations. On Figure 10, the results obtained
by the 2D-DIC approach and the proposed I-DIC approach are compared to the prescribed displacements.
Satisfying results are obtained using the proposed method whereas for large out-of-plane displacements, the
2D-DIC approach fails at predicting correctly w. This is due to the low depth of field of the optical setup used
in this application which induced an intensive blurring of the image. Indeed, one can see on Figures 11(a)
and 11(d) the actual images for an out-of-plane displacement of -4 mm and -8 mm. Compared to the
reference image in Figure 9, these images were intensively transformed due to the out-of-plane displacement.

Figure 11 again confirms the ability of the transformation P_U(f) to account for the scaling effect as well
as for the blurring effect induced by the out-of-plane displacement. Indeed, the correlation residuals (see
Figure 11(e) and 11(f)) between the actual images and the reconstructed images are really low, 3.6 and 5.5
grey levels in average, even in the last case when very large distortions of the image are to be recovered.

5. Conclusion

A methodology for measuring the 3D displacements of an object using a single camera is proposed.
Contrary to existing methods (Tay et al., 2005; Quan et al., 2008), it is proposed herein to account for
unfocusing of the object when it is subjected to an out-of-plane motion. Then, the method is not limited,
as shown in the example, to high depth of field lenses.

The general framework is the optics of Descartes and the model that is elaborated for reconstructing a
blurred image from a reference sharp image invokes geometrical considerations only. A procedure is also
derived from the theoretical model to calibrate the only four parameters required for the 3D measurements.

Comparisons with stereo-DIC and existing techniques that ignore unfocusing are carried out. It is
obtained that for high depth of field systems, the proposed methodology allows for similar performances.
The proposed method is also capable to provide for robust 3D displacements for low depth of fields systems
whereas techniques that do not consider unfocusing reveal less robust and less accurate in this case.

Future works will be focused on the potential extension of the method to non-planar object and to the
extraction of field data.
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<figure><img src="image_2.png" /><figcaption>Figure 1: Usual and translated configurations of a convergent lens in the framework of Descartes’ optics.</figcaption></figure>
<figure><img src="image_3.png" /><figcaption>Figure 2: Notations for the case when the object plane is not conjugated with the image plane.</figcaption></figure>
<figure><img src="image_4.png" /><figcaption>Figure 3: Reference images of the ligh box with a 35 mm lens (a) and with a 85 mm lens (b).</figcaption></figure>
<figure><img src="image_5.png" /><figcaption>Figure 4: For the case of the ligh box with a 35 mm lens: (a,d) images for an out-of-plane displacement of ±10 mm, (b,e) the corresponding reconstructed images $P_U(f)$, and the scatter (c,f). Note that in the latter, white color corresponds to 10 grey levels of the 255 levels in the initial image.</figcaption></figure>
<figure><img src="image_6.png" /><figcaption>Figure 5: Out-of-plane displacement for the test using a 35 mm lens: 3D-DIC refers to the displacement obtained using 2 cameras and a stereo-DIC technique, 2D-DIC refers to the conversion of an in-plane in an out-of-plane displacement and I-DIC refers to the proposed integrated technique.</figcaption></figure>
<figure><img src="image_7.png" /><figcaption>Figure 6: For the test using a 35 mm lens, comparison of the components of the displacement obtained by 3D-DIC and I-DIC.</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>Figure 7: For the case of the ligh box with a 85 mm lens: (a,d) images for an out-of-plane displacement of ±10 mm, (b,e) the corresponding reconstructed images $P_U(f)$, and the scatter (c,f). Note that in the latter, white color corresponds to 10 grey levels of the 255 levels in the initial image.</figcaption></figure>
<figure><img src="image_9.png" /><figcaption>Figure 8: For the test using a 85 mm lens, comparison of the components of the displacement obtained by 3D-DIC and I-DIC.</figcaption></figure>
<figure><img src="image_10.png" /><figcaption>Figure 9: Reference image for the exmaple using a macro-lens.</figcaption></figure>
<figure><img src="image_11.png" /><figcaption>Figure 10: For the test using a macro-lens, comparison of the components of the displacement obtained by 2D-DIC and I-DIC.</figcaption></figure>
<figure><img src="image_12.png" /><figcaption>Figure 11: For the example with a macro-lens: (a,d) image for an out-of-plane displacement of about -4 mm and -8 mm, (b,e) the corresponding reconstructed images $P_U(f)$, and the scatter (c,f). Note that in the latter, white color corresponds to 10 grey levels of the 255 levels in the initial image.</figcaption></figure>