Optical Techniques That Measure Displacements: A Review of the Basic Principles Part 4

Displacement and spatial resolution of DIC

There are two important aspects of DIC as applied to what is basically a speckle photography method. These two aspects are not specific to DIC as a method to retrieve displacements but are relevant to the currently prevailing methodology applied to the so called white speckles, the resolution in the measurement of displacements, and the spatial resolution. The resolutions in the measurement of displacements of the other techniques that have been considered in this paper depend on the pitch or equivalent pitch of the basic carrier that records the displacements. The application of the Nyquist condition tells us that the maximum spatial frequency that can be retrieved is half the frequency of sampling carrier.

In moire the sampling depends on the pitch p of the carrier, in speckle interferometry the sampling frequency is given by the sensitivity equation (10) and in photographic speckle by equation (15) that defines the equivalent pitch. It is also required that these frequencies are recorded by the sensor of the camera that must have a spatial frequency twice the frequency of the carrier. This subject is not addressed in many papers in the DIC literature applied to white light speckles but it is an important parameter in the displacement resolution of DIC as in all other methods utilized in pattern analysis. Figure 32 (a), [40], illustrates the definition of the equivalent of the speckle radius as the distance from the center of the correlation to the point of one half of the intensity called r. These definitions are statistical and give a statistical estimate of the minimal distance between spots that can be considered as measurable in the selected sub-domain.

Figure 30 (b) illustrates the definition [40] of the equivalent of the speckle radius as the distance from the center of the correlation peak to the point of one half of the intensity called r. The values of r are utilized to define a fine pattern with r slightly larger than one pixel, medium with the radius r=2 pixels and coarse 4 pixels. If one assumes that the minimum distance that can be measured corresponds to the distance of two points that can be separated for a fine pattern the spatial resolution will be 2 pixels, for a medium pattern 4 pixels and for a coarse pattern 8 pixels. These quantities then provide the maximum displacement resolution that statistically can be achieved for the corresponding patterns. This is a point that should be clearly understood in the application of DIC to white light speckles. Signal processing laws are laws that apply to all the type of signals that are utilized independently of the algorithms that one can introduce. Concerning the spatial resolution studies that are described in [40] show that in all the different program utilized in the DIC studies the displacement field inside the sub-set is not defined.

Figure 30. (a) Fine, medium speckle sizes defined as the radius of the autocorrelation factor at 50% of the intensity [40]. 12. Discussion and conclusions

All the OTD methods have potentially the same capability to perform the different operations required to measure displacement either in 2-D or in 3-D and to retrieve shape information. Basically the OTD methods can be separated in two basic categories, techniques that utilize deterministic signals and methods that utilize random signals. Within the techniques that utilize random signals there are two basic types: a) Techniques that use random signals produced by the pattern of interference generated by random surface roughness and b) Artificially generated random patterns or random patterns existing already on the surface from sources other than random interference patterns. The basic difference between utilizing deterministic and random signals is the final signal-to-noise ratios and the decorrelation phenomenon that is caused by statistical structure the wavefronts of random signals. At the same time the signals produced by all the techniques depend on whether the light is coherent or incoherent. The relationship between signals and displacements or metrology is independent of the light coherence, only the range of application is affected by the degree of coherence of the light. Both speckle interferometry and moire interferometry can reach very high accuracy because changes of phase of 2n correspond to the wavelength of light 1. Hence one can arrive to the 10 nm range in the displacement measurement [12].

Table 1

1	2	3	4	5	6	7	8
		p ^m	p ^m	p ^m	p ^m	p ^m	p ^m
x	Theory	0.365	0.413	0.492	0.635	0.925	1.22
	10-6	10-6	10-6	10-6	10-6	10-6	10-6
0.250	166.497	165.000	163.162	162.681	164.021	166.335	166.396
		1.49	3.35	3.81	2.47	0.162	0.10

Despite the possible sources of errors that speckle interferometry may have, it has been verified that in a disk under diametrical compression the computed strains obtained from speckle interferometry, Table 1, are within an error less of 1 % compared to the theoretically computed values. In a disk under diametrical compression it is known that the strains theoretically computed with an ideal concentrated load and the actual case, a disk with a narrow region of contact stresses, experimental and theoretical strains are approximately equal at points located around % of the diameter, Similar agreement in strain values were observed for holographic moire, [20], with errors on the order of 1 %.

Incoherent techniques apply to larger deformations, both moire and speckle photography can be applied to a very large spectrum of deformations and specimen sizes. In all cases the most important factor is the quality of the signals that encode the displacement information. Utilizing the speckle pattern method, [45] the displacements and strains along the diameter of a disk under diametrical compression were determined. Data were obtained for the same load with six different carrier pitches, from 1.22 to 0.365 microns (Table 1). All these carrier frequencies satisfy the Nyquist condition both for the required sampling frequency of the carrier and the required sampling of the utilized sensor. The final result indicated that the accuracy achieved in displacements and strains is the same when the Nyquist condition is satisfied, regardless of the carrier frequency utilized. These studies resulted in the formulation of the following principle similar to the Heisenberg indetermination principle of signal analysis [46],

In the above equation AIs is the minimum detectable gray level calling Is the maximum amplitude of the available gray levels. The gray levels in a CCD camera or similar devices are quantized and the maximum theoretical dynamical range (amplitude of the vector) is one half of the total number of gray levels 2n (for n=8, I = 128). The actual dynamic range is smaller than this quantity and Af is the maximum detectable sampling frequency.

Figure 31.Plot of the experimental data that provides a numerical expression for the Heisenberg equation (37). The quantityis defined as:

The practical question to be answered is: what is the minimum displacement information that can be recovered within fringe spacing 8 ? Where 5 is a fringe wavelength; it is evident that there is a finite limit to the subdivision of the fringe spacing. The constant C reflects the whole process to obtain displacement information. The constant C is a function of the optical system, the device used to detect the fringes (CCD camera) and the algorithms used to get the displacement information.

There are many important practical consequences of the principle formulated in (37). This equation is a valuable tool for planning experiments involving fringe analysis. Once the Whittaker-Shannon theorem is applied and the required minimum frequency of the carrier is computed, the next step is to select the carrier that is going to be used. In order to obtain frequency and displacement information it is necessary to maximize the amount of energy levels to encode this information. This implies that the largest portion of the dynamic range of the encoding system should be used to store useful information. By doing this the amount of noise in the signal is minimized. An immediate consequence is the need to increase the visibility of the fringes within the range of options available. Consequently when selecting a carrier the Optical Transfer Function (OTF) and the (MTF), the modulus of the (OTF), of the whole system used to encode the information needs to be taken into consideration. Figure 31 clearly shows the effect of encoding information in gray levels. If it is possible to detect close to 5 of the available 128 gray levels, it is possible to get displacement information that is 1/200 of the spatial frequency of the signal or fringe pitch p. On the other extreme if the minimum detectable gray levels is 45 out of the 128, only 1/20 of the pitch p can be recovered. This is a basic law in the process of encoding displacement information and it is independent of the particular method utilized for fringe analysis.

Consequently whether the illumination is coherent or incoherent the recovery of information is governed by (37). Both incoherent light moire, speckle photography and the white light speckle are particularly suited for the measurement of large displacements. What is called large is relative to the actual field of view. There is a relationship between the actual physical size of the analyzed area and of the sampling frequency required to observe the displacement field in the area. The smaller the area the higher the required sampling frequency and vice versa.

To reduce gray levels intensity there are presently two methods, one method has its foundations on the classical analysis of signals as was developed the Theory of Communications. To avoid problems arising from intensity based analysis and following a common trend in optics the notion of phase is utilized. The other option is the digital image correlation and it utilizes the irradiance as expressed in gray levels in a different form. It relates the displacement vector to the changes in irradiance. To put it in perspective the objective of both methods is to obtain from gray levels a vector field that depends on a tensor field (either the strain tensor or the stress tensor). The classical fringe analysis processing technique operates through projection of the vector displacement in two Cartesian components; let us say u and v which are separately determined although they are components of one entity, the displacement vector. Since the basic selected variable is the concept of phase and hence utilizes trigonometric variables. This particular selection of variables leads to a problem, fringe unwrapping. Fringe unwrapping is based on a simple concept, but the difficulty is in the implementation of this concept, due to presence of noise in the signal that are processed. This is particularly true if one is dealing with random signal as is the case with speckle patterns. Utilizing random signals as a carrier of information another important problem must be faced, the decorrelation problem; DIC bypasses these two obstacles. As shown in Figure 29 DIC searches directly for the displacement vectors in the field and operates directly with intensities as shown in (31) by relating the displacement vector to the changes of the intensity field. In the actual implementation of this approach there are a large number of functions that need to be introduced and optimized and choices must be made in the selection of these functions and in the optimization processes. As said before the choices of functions and optimization processes are made by the software developer; once a package of software is assembled the operation does not require intense involvement of the user. In a few words and utilizing general conclusion presented in [41] DIC is particularly suitable for the observation of displacement fields where the selected pixel size of the subsets Ns is a small quantity compared to the total number of pixels of the observed region, strains are large and low order shape functions can be utilized. It is possible to say that DIC has made a variation of speckle photography, white light speckles, a practical tool in many technical problems of mechanics of materials.