4.6. EVALUATION OF THE SOFTWARE
Therefore two groups of comparisons are presented:
• a comparison with artificial created strain field,
• a comparison with a commercial software.
4.6.1 Artificial strain fields
In order to test the capabilities of the proposed software, the creation of artificial images with an imposed strain field would help. This can be done mainly in two ways: deforming the real images or creating an artificial images. The first approach can be difficult as it could be difficult do really deform according to user-defined strain field. Moreover the quality of the image would play an important role in the capability of the software to understand the field correctly.
The second approach seams to be more reliable and powerful as can easily create different kind of deformations or strain fields and any kind of images. In order to create images with an imposed deformation and also with random distribution of gray-scale values (to simulate white and black dots on the specimen surface) a procedural texture6 should be used. In particular a pseudo-random appearance noise distribution is preferable because it allows the application of strain fields without losing the correspondence of the colors. In fact a pseudo-random distribution just looks like random. However at a given coordinate, even with many runs of the pseudo-random generator, when there is no applied strain field the generated color for one pixel will be always the same.
The Perlin noise function fulfills all the aforementioned constrains. It was developed in 1983 by Ken Perlin to give a natural looking to computer effects in the film Tron by Walt Disney Pictures. Since that time it has been used in many occasions. Skipping the mathematical description, it is just important to know that Perlin noise is based on the sum of functions describing the color of the image, at different frequencies (i.e. at different level of detail as big frequency means coarse description and low frequency means many details). The final result appears in Figure 4.33.
Deforming Figure 4.33 according to the following function
f (x, y) = x + a· p · sin
� 2 · π ·x
p
�
(4.39)
will lead to a image with a sinusoidal strain field along x (and constant along y). Figure 4.34 shows the comparison of the artificially created image with Phoenix c� results. As it is obvious from this figure, the correspondence between these two series of data is very high. In the particular proposed example the result has been a series of values with the mean of −1.4 · 10−4 pixels and a standard deviation of 67 · 10−4 pixels, this is a good evidence for the precision of Phoenix c�
6A procedural texture is an artificial image created using an algorithm aimed to obtain a realistic representation of natural elements.
4.6. EVALUATION OF THE SOFTWARE
Figure 4.33: Example of image created by Perlin noise algorithm
software, as one ten-thousandth pixel is much lower than the smaller movement measurable by this technique.
In an other attempt and in order to understand the limit of the software, a set of images created using a translational function (in the form of f(x, y) = x + k) have been analyzed.
The analysis always starts from the same original image and not from any deformed image (i.e.
Figure 4.33). The function moves the field only along the x axis of a certain amount of pixels.
The mean value and the standard deviations are listed in Table 4.5.
Table 4.5: Mean and standard deviation of different test at small translations Translation [pixel] Mean value [pixel] Standard deviation [pixel]
1 1.000 0.00042
0.1 0.120 0.0031
0.01 0.0135 0.0012
0.001 0.0014 0.00046
0.0001 0.00014 0.00015
The results of Table 4.5 are obtained with a completely flat field, i.e. a constant value. In a more realistic case the displacements are very small and also changing from point to point. In order to understand how the software may react to this scenario different sinusoidally deformed images have been created and analyzed (i.e. functions like in Equation (4.39) but where a assumes different values). The results are shown in Figure 4.35.
Comparing Figure 4.35(a) and Figure 4.34, these two figures are very similar to each other and
4.6. EVALUATION OF THE SOFTWARE
!"
! "#
Figure 4.34: DIC analysis on images created by Perlin noise algorithm
just the amplitude is one half in graph of Figure 4.35(a) respect to Figure 4.34. In Figure 4.34 the field is well followed, apart a small difference at the peaks of the sinusoidal curve. Figure 4.35(b) starts to show a higher difference, again at the function peaks. Anyway the shape is still followed and the errors are still relatively low. This behavior is even more visible in Figure 4.35(c) where an irregular amplitude appears as well. The last graph, Figure 4.35(d), shows an extremely irregular distribution of the values that are following the sinusoidal shape. Considering very small applied amplitude, even this result is a good evidence for Phoenix c� abilities.
Introducing an error parameter Ψ that is calculated as follows:
Ψ =� 1 a2 ·
�
uDIC(x) − a · sin� 2 · π ·x
p
��2
(4.40)
It is possible to define the error for each of the four analyzed cases (a is the amplitude and p is the period). The resulting values are presented in Table 4.6.
Table 4.6: Error parameter Ψ for different amplitude values Amplitude a [pixel] Ψ
1 0.16
0.1 2.2
0.01 5.4
0.001 10.4
Similar results can be achieved by a bi-dimensional displacement function.
4.6. EVALUATION OF THE SOFTWARE
!
(a) Amplitude of 1 pixel
(b) Amplitude of 0.1 pixels
(c) Amplitude of 0.01 pixels
(d) Amplitude of 0.001 pixels
Figure 4.35: Tests of the reliability of the software at very small displacement fields
4.6.2 Vic-2D
Artificial images are very good as they allow the user to create easily even very complex displacement and strain fields and check the quality of the algorithm or the proposed software.
However in this case errors in the illumination, specimen mounting, not good painting, etc.
are not considered. This may lead to erroneous results. In order to test the software in a more irregular environment, i.e. where the images are not as perfect as when they are created by a function, the authors decided to compare the results of Phoenix c� with a commercial software. The software here used is Vic-2D from Correlated Solutions Inc., a company from South Carolina (USA) leader in the development of non-contacting measurement solutions (see http://www.correlatedsolutions.com).
The company produces software in the field of laser shearography, videostroboscope systems and also digital image correlation. For this last technology the company offers a solution for bi-dimensional measurements, Vic-2D, and a solution also for three-dimensional measurements, Vic-3D. This is a tool that uses two cameras to measure the object shape, displacements and full-field strains in three dimensions.
The proposed software for two-dimensional solution is Vic-2D. On the website of the company it is stated that this software is able to measure strains from 500 microstrains till 500% and from