CHAPTER 5. IMAGE ANALYSIS
5.3. n-point Correlation Functions
0 10 20 30 40 50
0 10 20 30 40 50 60 70 80 90 100
0.01 0.1 1 10 100
CV(%)
%Passing
Size (mm)
7 8 9 10 CV - 7 CV - 8 CV - 9 CV - 10
Figure 5.18. Grain size distribution for mixtures 7, 8, 9 and 10 and CV’s
From a visual comparison, the gradation curves for the four asphalt mixture groups presented in Figures 5.15, 5.16, 5.17 and 5.18 are very similar. This may suggest that the gradation curves for all mixtures, whether they are made only using virgin material or they contain RAP, TOSS or MWSS recycled material, are very similar.
2-point correlation function for two materials with the same volume fraction but with variability on spatial distribution of particles is presented in Figure 5.19.
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40
0 100 200 300 400 500 600 700 r, pixels
S2
Figure 5.19. 2-pint correlation function for two materials with the same volume fraction and different particles spatial distribution – (Velasquez, 2009)
Very similar algorithms to those proposed by Velasquez (2009) and Velasquez et al., (2010) are adopted in this dissertation to estimate the 2- and 3-point correlation functions of asphalt mixtures. These procedures, described in the next sections of this Chapter, make use of Monte Carlo simulations to approximate the correlation functions of the material. The binary images of the BBR asphalt mixtures specimens obtained in section 5.1 are used for the estimation of the spatial correlation functions. However since computation of the 3-point correlation function was prohibitive for a laptop computational power it was decide to simply the algorithm in order to have at least a one dimensional response of this function.
5.3.1. 2-point Correlation Function
The 2- point correlation function can be computed using a discretized expression of equation [2.17] and binary images of the microstructure of the investigated material. The following discretized version of equation [2.17] was proposed by Berryman (1985) to estimate S2:
M x
i y N
j
y j x i I j i y I
N x y M
x S
1 1
2 (, ) ( , )
) )(
( ) 1 ,
ˆ ( [5.1]
where:
M height of the digital image, and N width of the digital image.
The isotropic 2-point correlation function S2(k) is calculated as the average of the values of S2(x, y) at a fixed radius k, where S2(x, y) is the two-dimensional estimate provided by equation [5.1].
However, since the values of S2(x, y) are generally not known at the specific points of interest, Berryman (1985) introduced the following function:
) sin , cos ˆ ( ) ,
( 2
2 k S k k
S [5.2]
When k·cos and k·sin are not integers, bilinear interpolation can be applied to the right-hand side of equation [5.2]. Knowing the values of [5.2], the average 2-point correlation function can be evaluated as:
k
l k
k l k S
k
S 2
0 2
2 ,4
1 2 ) 1
(
[5.3]
where:
k less than or equal to half of the minimum dimension of the image.
However for high resolution images containing a large number of pixels, a brute force method is computationally very expensive and therefore prohibited to use in an extensive experimental program especially if the computational power available is that of a standard laptop. For this reason an alternative approach is represented by Monte Carlo simulations; this can be used to estimate the 2-point correlation function of heterogeneous materials at a lower computational cost in comparison to brute force methods.
For the 2-point correlation, the algorithm drops vectors of specific length inclined at a random angle in the digital image N number of times and calculates the number of times the end points of the vector are in the phase of interest (Figure 5.20). The procedure is repeated for vectors of lengths varying from zero to half the size of the image. The function rand(), available in MATLAB (2008), was used to randomly generate the location and inclination of each vector.
y
x
randomly selected pixel
random inclination r
are end points in phase of interest?
y
x
randomly selected pixel
random inclination r
are end points in phase of interest?
Figure 5.20. Schematic for 2-point correlation function algorithm – (Velasquez et al., 2010)
Velasquez (2009) and Velasquez et al., (2010) using a two-phase randomly generated material with 1 = 0.5 determined its 2-point correlation function (Figure 5.21) showing, as expected, that the correlation function has the material volumetric fraction 1 as initial value and instantaneously decays to 12, which is the probability of randomly finding two points in phase 1.
0.00 0.10 0.20 0.30 0.40 0.50 0.60
0 100 200 300 400 500
r, pixels S2 (r)
Figure 5.21. 2-point correlation function for phase 1 of a randomly generated material – (Velasquez et al., 2010)
The simplified method was also validated by the same authors comparing the results to the analytical exact solution for the Penetrable Sphere Model (Weissberg 1963; Torquato and Stell 1983; Berryman 1985) and it was found that the results from using the implemented algorithm are in good agreement with the theoretical solution. The number of drops N for the estimation of the correlation function was determined from the Penetrable Sphere model as well using a material with matrix = 0.26 (white phase) as a balance between computational time and results
fluctuation. N > 10,000 provided the best compromise. A comparison of the brute force method and the simplified algorithm for a material containing overlapping particles with particles = 0.26 also indicated that the method based on Monte Carlo simulations provides a good approximation of the spatial correlation function with less computational time (Velasquez, 2009; and Velasquez et al., 2010).
5.3.1.1. 2-point Correlation Function for Asphalt Mixture
The binary images of the different specimens were used to compute the 2-point correlation functions of the aggregate phase. The 2-point correlation functions for each specimen of the seventeen mixtures analyzed in this study are presented in Appendix C. As in the case of volume fraction and gradation the single values of the 2-point correlation function for each specimen is calculated as average of the two larger sides of the specimen (~115 × 11.8 mm). An example of the 2- point correlation function for the six specimen obtained from slice 2 of mixture 7 is presented in Figure 5.22.
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0 1 2 3 4 5
Two-point Correlation Function
r (mm)
S 7-2-3 S 7-2-4 S 7-2-5
S 7-2-6 S 7-2-7 S 7-2-8
Figure 5.22. 2-point correlation function for aggregate phase, mixture 7 slice 2
Table 5.2 presents a comparison between the 1-point correlation function (the volume fraction of aggregate, 1), the calculated 2-point correlation function and the theoretical 2-point correlation function (12) at r=5mm for all the asphalt mixtures. The average and coefficient of
variation of the 2-point correlation function of the aggregate phase for all the different asphalt mixtures are presented in Figures 5.23-5.31.
Table 5.2. 2-point correlation function values comparison
Mixture Volume fraction S2 at r=5mm S2Theoretical at r=5mm Difference at r=5mm
ID - - - %
1 0.758 0.555 0.574 3.44
2 0.749 0.540 0.561 3.78
3 0.764 0.569 0.583 2.44
4 0.754 0.549 0.568 3.43
5 0.755 0.557 0.570 2.25
6 0.768 0.568 0.589 3.67
7 0.758 0.561 0.574 2.27
8 0.752 0.560 0.565 0.94
9 0.767 0.581 0.589 1.27
10 0.753 0.562 0.567 0.78
11 0.758 0.560 0.574 2.43
12 0.770 0.583 0.593 1.60
13 0.774 0.597 0.600 0.41
14 0.776 0.595 0.602 1.23
15 0.775 0.593 0.600 1.16
16 0.769 0.591 0.591 0.04
17 0.757 0.559 0.573 2.38
0 1 2 3 4 5 6 7 8 9 10
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0 1 2 3 4 5
CV (%)
Two-point Correlation Function
r (mm)
Average 1 CV
0 1 2 3 4 5 6 7 8 9 10
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0 1 2 3 4 5
CV (%)
Two-point Correlation Function
r (mm)
Average 2 CV
Figure 5.23. 2-point correlation function for mixtures 1 and 2 and CV
0 1 2 3 4 5 6 7 8 9 10
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0 1 2 3 4 5
CV (%)
Two-point Correlation Function
r (mm)
Average 3 CV
0 1 2 3 4 5 6 7 8 9 10
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0 1 2 3 4 5
CV (%)
Two-point Correlation Function
r (mm)
Average 4 CV
Figure 5.24. 2-point correlation function for mixtures 3 and 4 and CV
0 1 2 3 4 5 6 7 8 9 10
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0 1 2 3 4 5
CV (%)
Two-point Correlation Function
r (mm)
Average 5 CV
0 1 2 3 4 5 6 7 8 9 10
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0 1 2 3 4 5
CV (%)
Two-point Correlation Function
r (mm)
Average 6 CV
Figure 5.25. 2-point correlation function for mixtures 5 and 6 and CV
0 1 2 3 4 5 6 7 8 9 10
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0 1 2 3 4 5
CV (%)
Two-point Correlation Function
r (mm)
Average 7 CV
0 1 2 3 4 5 6 7 8 9 10
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0 1 2 3 4 5
CV (%)
Two-point Correlation Function
r (mm)
Average 8 CV
Figure 5.26. 2-point correlation function for mixtures 7 and 8 and CV
0 1 2 3 4 5 6 7 8 9 10
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0 1 2 3 4 5
CV (%)
Two-point Correlation Function
r (mm)
Average 9 CV
0 1 2 3 4 5 6 7 8 9 10
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0 1 2 3 4 5
CV (%)
Two-point Correlation Function
r (mm)
Average 10 CV
Figure 5.27. 2-point correlation function for mixtures 9 and 10 and CV
0 1 2 3 4 5 6 7 8 9 10
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0 1 2 3 4 5
CV (%)
Two-point Correlation Function
r (mm)
Average 11 CV
0 1 2 3 4 5 6 7 8 9 10
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0 1 2 3 4 5
CV (%)
Two-point Correlation Function
r (mm)
Average 12 CV
Figure 5.28. 2-point correlation function for mixtures 11 and 12 and CV
0 1 2 3 4 5 6 7 8 9 10
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0 1 2 3 4 5
CV (%)
Two-point Correlation Function
r (mm)
Average 13 CV
0 1 2 3 4 5 6 7 8 9 10
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0 1 2 3 4 5
CV (%)
Two-point Correlation Function
r (mm)
Average 14 CV
Figure 5.29. 2-point correlation function for mixtures 13 and 14 and CV
0 1 2 3 4 5 6 7 8 9 10
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0 1 2 3 4 5
CV (%)
Two-point Correlation Function
r (mm)
Average 15 CV
0 1 2 3 4 5 6 7 8 9 10
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0 1 2 3 4 5
CV (%)
Two-point Correlation Function
r (mm)
Average 16 CV
Figure 5.30. 2-point correlation function for mixtures 15 and 16 and CV
0 1 2 3 4 5 6 7 8 9 10
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0 1 2 3 4 5
CV (%)
Two-point Correlation Function
r (mm)
Average 17 CV
Figure 5.31. 2-point correlation function for mixture 17 and CV
The 2-point correlation functions calculated for asphalt mixtures materials (Figures 5.23 - 5.31) behave similarly for all the seventeen asphalt mixtures investigated. The value of the correlation function does not fluctuate as the distance (r) increases. The 2-point correlation function starts at approximately aggregate and smoothly drops to aggregate2 as also shown by the difference between the computed and theoretical value (Table 5.2). For each mixture, no large coefficients of variation are observed between the correlation function measured from the twelve BBR beam sspecimens. The maximum coefficient of variation calculated is 5.13%.
5.3.2. 3-point Correlation Function
Several approaches are available to compute the 3-point correlation function. The use of brute force applied to the discretized version of equation [2.21] is however prohibitive in terms of computational time. Berryman (1985) proposed a simplified algorithm based on Monte Carlo
simulations and on a set of lattice commensurate triangles to approximate the 3-point correlation function. He showed that, by using the symmetries of the 3-point correlation function, computational time can be reduced. For an isotropic and statistically homogeneous material, the 3-point correlation function does not depend on the location or orientation of the triangle but only on the size and shape of it. A procedure was proposed by Berryman (1985) to define the set of triangles used for the estimation of the spatial correlation function. Three integer l, m, and n are used to characterize each triangle (Figure 5.32). In order to avoid calculating the probability of the same triangle more than once (redundancy) the following conditions are imposed:
The length of the longest side of the triangle l is not larger than half of the size of the image,
A local coordinate system (x′, y′) with origin at the vertex formed by the intersection of the longest and shortest side of the triangle is a defined,
The longest side l of the triangle is place along the x′ axes. Then, with respect to this local coordinate system, the second and third vertex of the triangle are located at (l, 0) and (m, n), respectively.
To avoid calculating the correlation function for the same triangle more than once then:
2 / l
m [5.4]
ml n
m2 22 [5.5]
y′
x′ l
m n y′
x′ l
m n
Figure 5.32. Schematic of triangles used for calculation of 3-point correlation function – (Velasquez et al., 2010)
Each triangle defined by the integers (l, m, n) was randomly dropped N number of times in the digital image and the number of times (Nhits) the three vertices of the triangle were in the phase of interest was counted (Figure 5.23). The value of the 3-point correlation function for that specific triangle is:
N N n m l
S3(, , ) hits/ [5.6]
are triangle vertices in phase of interest?
y
x
random inclination n l
m
are triangle vertices in phase of interest?
y
x
random inclination n l
m
Figure 5.33. Schematic of Monte Carlo simulations for 3-point correlation function – (Velasquez et al., 2010)
The 3-point correlation function was applied by Velasquez (2009) and Velasquez et al., (2010) to study the effect of different sizes of beams of asphalt mixture specimen and the microstructure of the material. The 3-point correlation function for a randomly generated material and the penetrable sphere material was calculated using a set of triangles with the same shape but different sizes. The shape of the triangle was defined by L = 3 pixels and M = N =1 pixel and the size of the triangle was determined by a factor p, that varied from 1 to half the size of the image. The set of triangles used to calculate S3 was determined by the following triplets:
) , , ( ) , ,
(l m n p LM N [5.7]
It was found that S3 can capture differences in the microstructure of two completely different materials, similar to what 2-point correlation function (S2) does. For the random material, the spatial correlation function starts at the volumetric fraction of the matrix phase (i.e. white pixels) and then instantaneously drops to matrix3. The authors also showed that, as expected, the 3-point correlation function of the random material reflects no patterns in the internal structure of the material and that the 3-point correlation function for the penetrable spheres model
behaves similarly to previous research (Berryman, 1985) when N = 100,000 triangle drops are used during the simulations.
5.3.2.1. 3-point Correlation Function for Asphalt Mixture
The same binary images of the asphalt mixtures BBR beams used for the calculation of the 2-point correlation function of the different specimens were used also for the computation of the 3-point correlation function. First a simplified algorithm based on the conditions stated above, and very similar to what was proposed by Velasquez (2009) and Velasquez et al., (2010) was coded in MATLAB using the random number generator function for the Monte Carlo simulations. This algorithm should generate a three-dimensional output. However it was found that this is prohibitive for a computational power available in a standard laptop and thus this approach was dropped. It was decided to select only a single triangle with fixed proportion and varying dimension from one point to half of the size the smaller side of the specimen larger face (~115 × 11.8 mm). The set of triangles defined by L = 2, M = 1, N = 1, and p = 0 - 32, was selected. This helped to drastically reduce the computation time form more that 5 days for a single simulation to few minutes. It must be underlined that this single triangle procedure does not provide the entire 3-point correlation function; however, for the purpose of this dissertation it was considered a good compromise between computation power and an appropriate response.
An example of the 3- point correlation function for the six specimen obtained from slice 2 of mixture 7 is presented in Figure 5.34.
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0 4 8 12 16 20 24 28 32
Three-point Correlation Function
p (size factor)
S 7-2-3 S 7-2-4 S 7-2-5
S 7-2-6 S 7-2-7 S 7-2-8
Figure 5.34. 3-point correlation function for aggregate phase, mixture 2 slice 2
Table 5.3 presents a comparison between the 1-point correlation function (the volume fraction aggregate, 1), the calculated 3-point correlation function and the theoretical 3-point correlation function (13) for all the asphalt mixtures at p=32. The average and coefficient of variation of the 3-point correlation function of the aggregate phase for all the different asphalt mixtures are presented in Figures 5.35-5.43. Appendix C contains the plots for the 3-point correlation functions for each specimen of the seventeen mixtures analyzed in this dissertation.
Table 5.3. 3-point correlation function values comparison
Mixture Volume fraction S3 at p=32 S3Theoretical at p=32 Difference at p=32
ID - - - %
1 0.758 0.424 0.435 2.55
2 0.749 0.410 0.421 2.57
3 0.764 0.442 0.446 0.86
4 0.754 0.421 0.428 1.68
5 0.755 0.430 0.431 0.22
6 0.768 0.441 0.452 2.58
7 0.758 0.435 0.435 0.05
8 0.752 0.434 0.425 2.08
9 0.767 0.451 0.452 0.08
10 0.753 0.438 0.427 2.69
11 0.758 0.433 0.435 0.44
12 0.770 0.453 0.457 0.70
13 0.774 0.472 0.464 1.53
14 0.776 0.467 0.467 0.00
15 0.775 0.467 0.465 0.42
16 0.769 0.466 0.455 2.51
17 0.769 0.466 0.455 2.51
0 1 2 3 4 5 6 7 8 9 10
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0 4 8 12 16 20 24 28 32
CV (%)
Three-point Correlation Function
p (size factor)
Average 1 CV
0 1 2 3 4 5 6 7 8 9 10
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0 4 8 12 16 20 24 28 32
CV (%)
Three-point Correlation Function
p (size factor)
Average 2 CV
Figure 5.35. 3-point correlation function for mixtures 1 and 2 and CV
0 1 2 3 4 5 6 7 8 9 10
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0 4 8 12 16 20 24 28 32
CV (%)
Three-point Correlation Function
p (size factor)
Average 3 CV
0 1 2 3 4 5 6 7 8 9 10
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0 4 8 12 16 20 24 28 32
CV (%)
Three-point Correlation Function
p (size factor)
Average 4 CV
Figure 5.36. 3-point correlation function for mixtures 3 and 4 and CV
0 1 2 3 4 5 6 7 8 9 10
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0 4 8 12 16 20 24 28 32
CV (%)
Three-point Correlation Function
p (size factor)
Average 5 CV
0 1 2 3 4 5 6 7 8 9 10
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0 4 8 12 16 20 24 28 32
CV (%)
Three-point Correlation Function
p (size factor)
Average 6 CV
Figure 5.37. 3-point correlation function for mixtures 5 and 6 and CV
0 1 2 3 4 5 6 7 8 9 10
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0 4 8 12 16 20 24 28 32
CV (%)
Three-point Correlation Function
p (size factor)
Average 7 CV
0 1 2 3 4 5 6 7 8 9 10
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0 4 8 12 16 20 24 28 32
CV (%)
Three-point Correlation Function
p (size factor)
Average 8 CV
Figure 5.38. 3-point correlation function for mixtures 7 and 8 and CV
0 1 2 3 4 5 6 7 8 9 10
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0 4 8 12 16 20 24 28 32
CV (%)
Three-point Correlation Function
p (size factor)
Average 9 CV
0 1 2 3 4 5 6 7 8 9 10
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0 4 8 12 16 20 24 28 32
CV (%)
Three-point Correlation Function
p (size factor)
Average 10 CV
Figure 5.39. 3-point correlation function for mixtures 9 and 10 and CV
0 1 2 3 4 5 6 7 8 9 10
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0 4 8 12 16 20 24 28 32
CV (%)
Three-point Correlation Function
p (size factor)
Average 11 CV
0 1 2 3 4 5 6 7 8 9 10
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0 4 8 12 16 20 24 28 32
CV (%)
Three-point Correlation Function
p (size factor)
Average 12 CV
Figure 5.40. 3-point correlation function for mixtures 11 and 12 and CV
0 1 2 3 4 5 6 7 8 9 10
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0 4 8 12 16 20 24 28 32
CV (%)
Three-point Correlation Function
p (size factor)
Average 13 CV
0 1 2 3 4 5 6 7 8 9 10
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0 4 8 12 16 20 24 28 32
CV (%)
Three-point Correlation Function
p (size factor)
Average 14 CV
Figure 5.41. 3-point correlation function for mixtures 13 and 14 and CV
0 1 2 3 4 5 6 7 8 9 10
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0 4 8 12 16 20 24 28 32
CV (%)
Three-point Correlation Function
p (size factor)
Average 15 CV
0 1 2 3 4 5 6 7 8 9 10
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0 4 8 12 16 20 24 28 32
CV (%)
Three-point Correlation Function
p (size factor)
Average 16 CV
Figure 5.42. 3-point correlation function for mixtures 15 and 16 and CV
0 1 2 3 4 5 6 7 8 9 10
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0 4 8 12 16 20 24 28 32
CV (%)
Three-point Correlation Function
p (size factor)
Average 17 CV
Figure 5.43. 3-point correlation function for mixture 17 and CV
The average 3-point correlation functions computed for asphalt mixtures materials (Figures 5.35 - 5.43) has a similar pattern for all the seventeen asphalt mixtures considered in this thesis.
No large fluctuations on S3 are observed as the size of the triangle p increases. S3 begins at
aggregate and smoothly drops to aggregate3. The maximum coefficient of variation measured for S3
is 5.86%.