Optical Techniques That Measure Displacements: A Review of the Basic Principles Part 3

3-D Incoherent illumination

The utilization of double viewing system as shown in Figure 17, can be applied to get 3-D information in a similar way to what was described in the preceding sections. The recording of the double illumination in the unloaded and loaded conditions yields the contour of the deformed surface. Likewise in incoherent illumination the superposition of the recorded pattern in the unloaded and loaded conditions will provide the projected displacements. However the sensitivity vector for the general case will be changing from point to point and it will be necessary to utilize the point wise solution of three different projections to get the displacement vector.

Figure 26. Graphical representation of the process of modulation of a carrier

Viewing with telecentric lenses produces an effect similar to a collimated illumination reducing the changes of the sensitivity vector to the z-direction when one wants to get the in-plane displacement.

Process to recover the information from recorded data

Figure 26 shows a cross-section of the signal along the x-axis to simplify the visualization of the relationships between the different variables that must be considered [26]-[30]. In the theory of communications, a carrier wave, or carrier is a waveform (usually sinusoidal) that is modulated (modified) with an input signal for the purpose of conveying information. The carrier wave is usually of much higher frequency than the input signal. The purpose of the carrier is to encode information to transmit it. Phase modulation and amplitude modulation (AM) are used methods to modulate the carrier. In what follows phase modulation will be described. The carriers in the optical techniques applied to Experimental Mechanics can be gratings printed on the surface under analysis. In the case of surface contouring the carriers are projected lines. In methods like speckle or holography the carrier are extracted from surface features existing on the surfaces of the analyzed bodies. These features can also be artificially created. In what follows the analysis of a sinusoidal carrier will be introduced as a mathematical model for all types of carriers. One must remember that by utilizing Fourier transform methodology all the integrable functions can be represented by their expansion in Fourier integrals.

The carrier can be thought of as a sinusoidal function generated by a rotating vector E and the phase of the carrier at a point of coordinate x is defined as the total angle rotated by the vector up to that point. ¥ (x) is the modulation function, a function that encodes the optical paths difference as an angular variable. The total phase of the modulated carrier is the addition of the phase generated by the constant rotation plus the modulation function contribution. The general equation for a modulated carrier is,

Where fc is the frequency of the carrier, a known quantity because it was introduced with certain spatial frequency that can be defined asThe presence of a carrier is required in certain optical methods. In other methods the introduction of a carrier may be useful because it can greatly simplify the process of data processing. Figure 26 shows that the phase change is a linear function of the coordinate x and the modulation function is added to this function, resulting in a total phase that corresponds to the modulated carrier. Hence from (26)

Knowing

one can get the modulation function.

From (26) to (28) and looking at Figure 26 it is possible to see that starting from a given sign convention the presence of a carrier defines the corresponding sign of the modulation function 8(x). This means that the carrier provides a reference frequency that removes the need of knowing where the zero reference order is. This is a problem that arises in the interpretation of fringe systems, as in Photoelasticity. There are several ways that one can determine the phase but all of them are based on the utilization of a trigonometric function that limits the phase retrieval from 0 to 2n. This leads to the process of unwrapping that although for smooth functions it works well, in actual applications it can present difficult practical problems of implementation.

Digital imagine correlation

In DIC the optical process to obtain correlation between signals is replaced by digital procedures [31]-[34]. In DIC displacements are directly obtained from point trajectories and the process of fringe unwrapping is bypassed. The understanding and the interpretation of the basic aspects that relate phase and displacements [26]-[30], is a straightforward process. The theory behind DIC to relate displacements and light irradiances is more complex. DIC is a general technique to extract displacement information from recorded irradiance of deformed bodies. DIC is particularly useful when random carriers are utilized. Hence this particular application will be emphasized in this paper.

In the application of two random carriers with DIC two random signal images are recorded and saved in the memory of a computer. From these two images a small subset is extracted, Figure 27. The subset contains a distribution of gray levels. The cross-correlation between the two subsets is computed. A correlation peak is produced; the position of the peak in the sub-set gives the local displacement of the subset. The height of the peak gives the degree of correlation or similarity of the gray levels of the initial and final configuration of the subset. If the cross-correlation has been normalized to the value of 1, values of the peak near one will indicate a good correlation. In the measure that this peak gets lower values the correlation degrades. Unlike moire or speckle interferometry based on resolved patterns of irradiance levels, DIC is based on a subset of pixels. As a result, information of displacements inside the subset cannot be obtained. This aspect of DIC poses a problem of spatial resolution that must be considered in actual applications. Therefore the ratio of the pixels subset size to the overall size of the region under observation is a very important quantity that determines the spatial resolution of the obtained results. Summarizing, the measured displacements are the displacements of a subset.

Figure 27. Illustration of the cross-correlation of images.

Figure 28 shows a speckle displacement field after all the different subsets of the field have been correlated and merged into trajectories. In one single sentence DIC provides the lines that are tangent to the trajectories of the points of a surface. From the trajectories one can extract the displacement field information. If one performs the above described process of correlation without additional corrections the displacement vectors will have random variations both direction and magnitude from subset-to-subset. In the DIC literature there is a large variety of approaches to the solution of this problem. DIC heavily depends on knowledge based information to introduce corrections to the recovered displacements. These different optimization procedures can be subdivided in two basic groups, methods that operate in the actual space and methods that utilize the FT space, [26]-[30],[35]-[38].

Figure28. Displacement field obtained from a speckle pattern.

A certain region of a deformed surface is analyzed; this region has experienced rigid body translations and rotations due to the deformation of the rest of the body that the observed patch belongs to, plus a local deformation; the object of DIC to obtain the local deformation. One has a given surface that for the sake of simplicity is assumed to be a plane and is viewed in the direction normal to the surface. Furthermore it is assumed that a telecentric system is used to get the image of the surface. In this way it is possible to separate the problems that were analyzed in some detail in preceding sections concerning the image formation, from the problem of image correlation. In this surface one has a certain distribution of intensities that it will be assumed corresponds to the random signal incorporated to the surface and is represented by a function Ii(x,y). A displacement field is applied to the surface and a final distribution of intensities If(x,y) is obtained. It is assumed that the light intensity changes are only a function of the displacement field and as it is the case in all experimental methods noise is present. Noise is indicated as all the changes of intensity that are not caused by the displacement field. The displacement field is defined by the function [39], [41],

From the preceding assumption,

In (22) AI is the change of intensity caused by the rigid body motion plus the local deformation of the analyzed surface. In (30) the assumption that the light intensity is modified only by the displacements is implicit. The term In refers to all other causes of change of intensity. The validity of (30) boils down to the signal to noise ratio. To develop the model one has to postulate that the signal content of In is small and hence can be neglected. The problem to be solved is to find u (x,y) and v (x,y) knowing Ii(x,y) and If(x+u,y+v). The solution of the above problem requires the regularity of the functions u(x,y) and v(x,y) implicit in the Theory of the Continuum. One can formulate the problem as an optimization problem, that is find the best values of these two functions that minimize or maximize a real function, the objective function of the optimization process. There are many criteria that can be utilized for this purpose. One criterion is the minimum squares; the difference of the intensities of the two images must be minimized as a function of the experienced displacements. Calling ® (u,v) the optimization function.

For small u(x,y) and v(x,y) the above expression can be expanded in a Taylor series and limiting the expansion to the first order and using vectorial notation as,

In the above equation r is the spatial coordinate; D(r) is the displacement vector and V is the gradient operator. Equation (32) tells us that the gradient of If provides the following information; the displacement information is associated with the gradient of the intensity distribution. If is a scalar function (light intensity), the gradient is a vector and going back to Figure 28 the vectors displacements are plotted following the vectors joining the centers of correlation peaks of the sub-images. Hence the displacement information can be retrieved following the gradient function of the light intensity.The minimization of the objective functions is then a central problem of the image digital correlation technique. In the technical literature there is a large variety of approaches to this problem. One can utilize criteria other than the minimum squares [42]-[44].

Figure 29. Field for the correlation process. (a) Dotted rectangle, NsxNs sub-element, 5 mesh of the region of interest. (b) Displacement experienced by the sub-image with components u and v.

Let us now look at the overall procedures that are necessary to obtain the displacement field. The region of interest is symbolically represented in Figure 29 by a square region. Figure 29 (a) shows a scheme of computation. There is a region of interest, the big square; in one corner there is a sub-element that has a chosen size of NsxNs pixels and the raster of dots indicate the position of the centroids that form a regular mesh of 5×5 pixels. Figure 29(b) shows how the sub-image is displaced and distorted after the deformation of the sample has taken place. By utilizing the model adopted in (32) and operating in the coordinate-space it is possible to define the vectors displacement in the region of interest, as shown in Figure 28 and 29.Two images are being compared. The reference image represented in Figure 29(a), by square image of NsxNs pixels, and the second one, called the deformed image, represented by the distorted square. The operator chooses the size of the zones of interest, the sub-sample by setting the size Ns so that NsxNs pixels are considered. To map the whole region of interest, the second parameter to choose is the separation 5 between two consecutive sub-samples. The parameter 5 defines the mesh formed by the centers of each sub-sample used to analyze the displacement field (Figure 29). Different strategies can be applied to retrieve the full field. Let us concentrate in the fundamental operation, the extraction of the information from a sub-sample. This aspect of the problem will be covered by utilizing an approach that is followed by a large number of contributors to this method.

The process begins with a discrete and normalized version of (32).The deformed coordinates are obtained from the initial coordinates by Taylor’s series expansion,

For example the Taylor’s series is terminated in the first order. Although higher orders can be introduced, it is easier and more convenient to explain the basic ideas of this particular approach to DIC by utilizing the first order. The meaning of the above equation can be better grasped by looking at Figure 29, where u and v contain components of the rigid body displacement of the sub-sample, and the derivatives express the effect of the local deformations in the displacement field.

To make sure that the distribution of intensities in one subset is continuous and has continuous derivatives the light distribution, I(x,y) is interpolated utilizing (i.e. bicubic-spline) an expansion of the light intensity, as chosen by many authors that have contributed to DIC. The relationship between the displacement field and the gradient of the intensity field comes from (32). This equation indicates that the displacement field is associated with the gradient of the intensity field. To get displacement information from the image intensity distribution one replaces the bicubic spline expression in a normalized expression of (33). After this substitution the optimization of (33) requires the solution of a non linear system of equations. This brings additional complications but there are many methods that were developed and can be applied in this case.

There is a large variety of software packages for DIC. These packages depend fundamentally on the specific choices of the correlation coefficient C defined in this paper by (33). The next step is to select a function that defines the displacement field in a subset. This function is called the shape function 9, and on the optimization algorithms and interpolation functions that are needed to compute sub-pixel displacements from images that were obtained with specific pixel resolutions. One very important aspect that is quite often not referred to in the literature is that no matter how complex your algorithm is no gain of information can be achieved if this information does not already exist in your primary data, the gray levels. These levels depend on satisfying the Nyquist condition in connection to both the frequencies recovered and on the sampling of the gray levels by the camera sensor.

Summarizing the first basic concept that is clearly shown in Figure 27, the comparison of the distribution of gray levels coming from two images (initial and final) provides a measure of the mechanical displacements experienced by a surface. The analysis of the intensity distribution is done on sub-set images and following the structure of electronic image sensor these sub-images are squares of Nsx Ns pixels. This is the basic foundation of DIC that separates it from the other methods that measure displacements.

The second basic development is connected with the description of the displacement field in the sub-set. This second basic aspect in DIC heavily depends on knowledge based information; a function 9 is introduced that describes the displacement field of the sub-set domain; following the nomenclature of Finite Element, 9 is called the shape function. There are several shape functions 9 utilized in DIC: 9 constant that corresponds to a rigid body motion of the sub-image; 9 linear or affine transformation, 9 quadratic and it is possible to include higher orders. The next fundamental development is embodied in (33-35), that relate the optical flow to the kinematic variables that depend on the choice of 9 and can be represented by a vector P. Having posed the problem in terms of optical and mechanical variables the next step is to relate both set of variables. This is an inverse problem, which says knowing Ii(x,y) and If(x,y) find 9, that is determine the vector P that best accounts for the observed optical flow. This connection between the two sets of variables is represented by (32) and is embodied in (33). This equation implies two choices, the first choice is utilization of a truncated Taylor’s expansion of the displacement field to the first order term or higher order terms. The second choice is the selection of minimum squares criterion for the optimization procedure implicit in (31). This is the approach followed by the majority of the authors in the field and for most of the commercial packages that are available. However as pointed out before there are other optimization mechanisms that can be utilized.

The theoretical framework described above to connect displacements to light intensity is unique to DIC and separates it from all the other techniques that were previously described. The inverse problem is formulated, the main variables set up, and the next step is to solve the optimization problem. The problem is formulated in terms of minimum squares hence it is a non linear problem. The symbol O (P) represents the solution of the problem. The solution is the sum of the leading term, plus additional terms: P( l) indicates a linear approach to the displacement vector, P(q) indicates a quadratic solution and one can utilize successive higher order terms.

In (28)P0(C), indicates a constant, P(l) indicates a linear term, P(q) indicates a quadratic term. The higher order terms of the power series become smaller as the order of the terms increase. This optimization is achieved utilizing nonlinear iterative optimization algorithms, such as first gradient descent, Newton-Raphson, or Levenberg-Marquard.

To summarize DIC in a few sentences, although the actual approach to the solution of obtaining displacements from light intensity is complex and requires a number of choices, the actual choices are made by the developer of the software. Once a package of software is put together the operation of the software is pretty much automatic. This has made DIC a very popular choice for experimental mechanics. Users should be cautious however that the Nyquist condition must always be satisfied otherwise the results obtained will have no value.