Application of LADAR in the Analysis of Aggregate Characteristics (2012)

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13 3.1 Shape In the paving industry, shape is usually described by sphe- ricity, flatness ratio, and elongation ratio. The FTI system implemented the characterization of these terms following Eq. 3-1 to Eq. 3-4. Sphericity is defined by Eq. 3-1. It describes the overall 3-D shape of a particle. Sphericity has a relative scale of 0 to 1. A sphericity value of 1 indicates that a particle has equal dimensions (cubical or spherical). S D D D s m l = 2 3 Eq. 3-1 Where Ds is the shortest dimension of a particle, Dm is the intermediate dimension of a particle, and Dl is the longest dimension of a particle. The flatness ratio is defined by Eq. 3-2, and the elongation ratio is defined by Eq. 3-3. The FTI direct output of shape characteristics includes all three parameters. The flatness and elongation (FE) ratio can be conveniently calculated from the flatness ratio and the elongation ratio by Eq. 3-4. Aggregates are considered flat or elongated if their flatness ratio or elon- gation ratio is greater than 3:1. Flatness Ratio Eq. 3-2= D Ds m Elongation Ratio Eq. 3-3= D Dm l D Dl s FE Ratio 1 Flatness Ratio Elongation Ratio Eq. 3-4 ( )= × = 3.2 Angularity and Texture The discrete Fourier transform (DFT) is applied for the 3-D surface as a discrete function z(x, y) that is nonzero over the finite region 0 ≤ x ≤ N - 1 and 0 ≤ y ≤ N - 1 in the space domain. The two-dimensional N-by-N DFT and inverse N-by-N DFT relationships are given by Eq. 3-5 and Eq. 3-6 (Wang, 2007; MathWorks, 2010). Z p q z x y e j N xp N yq y N x , ,( ) = ( ) − +  = − = ∑ 2 2 0 1 0 pi piN− ∑ 1 Eq. 3-5 z x y Z p q e j N xp N yq q N p N , ,( ) = ( ) +  = − = ∑ 2 2 0 1 0 pi pi − ∑ 1 Eq. 3-6 f N xx = 2pi Eq. 3-7 f N yy = 2pi Eq. 3-8 Where Z(p, q) is the DFT coefficient matrix in the frequency domain, z(x, y) is the z coordinates of an aggregate surface, j is the imaginary root, and fx and fy are frequencies in x and y directions, respectively. This approach is similar to the 1-D Fourier analysis method developed by Wang et al. (2005). High frequency and small magnitude represent texture; low frequency and large magni- tude represent the gentle slopes and flat planes (of a surface) related to shape; intermediate frequency is related to angularity (Smith, 1999; Wang, 2007). Therefore, a critical step is to deter- mine threshold frequencies that separate shape, angularity, and texture from each other. A matrix of a predetermined size is selected as a roughness matrix to represent a sample of the surface within the bound- ary of the particle. Fast Fourier transform is performed on the matrix in order to determine the roughness factors. Obvi- ously, when the matrix is large, the data may include infor- mation for texture, angularity, and shape; when the matrix is adequately small, the data may reflect the texture only. Fig- ure 3-1 shows the schematic configuration of the roughness matrix. Roughness matrices are selected as square matrices sampled from the aggregate surface with various matrix sizes. C h a p t e r 3 Aggregate Shape, Angularity, and Texture Analysis Methods

14 Angularity factor (AF) and texture factor (TF) are defined by Eq. 3-9 in MATLAB. Angularity Factor AF( ) = ( )( ) == ∑ a p q a q NA p , 0 2 11 0 2 NA b p q a a p q ∑ + ( )( )  ( ) = ( , ,Texture Factor TF )( ) + ( )( )  − == ∑∑ a b p q a q N p N 0 2 11 0 2 Eq. 3-9 AF, where a0 is the average height of z(x, y), a and b are the real and imaginary parts of the coefficients for the 2-D fast Fourier transform (FFT2), N is the size of the z(x, y) matrix, and NA is defined as a threshold frequency, with which a 3-D surface reconstructed using the inverse of FFT2 coefficients that have their frequencies smaller than 2pNA/N in either the x direction or y direction should represent the angularity. The FFT2 coefficients are considered to only contribute to angularity if the mean value of distance between the origi- nal surface and the reconstructed surface using the inverse of FFT2 is greater than 0.2 mm, which is the spatial spacing discernible by the unaided human eye (Brandon et al., 1999). The other Fourier coefficients are considered to contribute to texture only. As given in Eq. 3-9, the size of z(x, y) matrix N has a great influence on both AF and TF values. That is, values of both AF and TF would increase with the increase of z(x, y) matrix size N since the total number of pixels (sampled points) on the x-y plane increases by the order of N 2. Consequently, it is necessary to determine the maximum values of N to quan- tify angularity and texture for aggregates of different sizes. In this project, the area of the roughness matrix ranges from 1.0 mm2 to a maximum of 25% of an aggregate surface, based on the sieve size of the aggregate being analyzed. For instance, to evaluate the angularity and texture of aggregates passing a 1-in. sieve and retaining on a ¾-in. sieve, the maximum roughness matrix area is 90.7 mm2, which is 25% of the area of a hole on a ¾-in. sieve. The center of the roughness matrix is chosen as the center of the aggregate surface. That is, the roughness matrices of varying sizes keep the same center point. This is to evaluate how the roughness changes as the roughness matrix area increases. Figure 3-2 shows how to separate texture from angularity on an aggregate surface by applying the FFT2 method to the roughness matrix. In the first column, an original 3-D sur- face of a roughness matrix (301 × 301 pixel by pixel) is pre- sented (with all the sampled points). Reconstructed surfaces are achieved by conducting an inverse Fourier transform to the Fourier coefficients with different frequencies, shown in the second column. In the third column, the differences Dz between the original surface and the reconstructed surface using different critical frequencies are plotted. The recon- structed surface increasingly approaches the actual angular and textured original surface, and the difference dramatically decreases since there are more high-frequency components included in the reconstructed surface. When using frequency f3 to separate angularity from texture for this original surface shown in the first column, the average value of the absolute Figure 3-1. Schematic configuration of roughness matrix.

15 difference between the original surface and the recon- structed surface is less than 0.2 mm; therefore, frequency f3 is the critical frequency for this original surface shown in the first column. The relationship between the AF and the corresponding area of the roughness matrix is plotted in Figure 3-3 for some aggregate particles retaining on ½-in. sieve sizes for seven types of aggregates. By plotting the relationship between AF (or TF) and roughness matrix area (hereinafter referred to as AF plot or TF plot), one will notice that the AF and TF values follow a linear relationship in both AF and TF plots. Further- more, aggregates with more angular surfaces tend to have steeper slopes in the AF plot, and aggregates with rougher surfaces tend to have steeper slopes in the TF or AF plots. Therefore, it is reasonable to define the linear relationship (slope) in the AF and TF plots as the angularity and texture of an aggregate, respectively. 3.3 Surface Area, Volume, and Dimensions As a result of method development, a separate program has been developed to calculate the surface areas, volumes, and dimensions. Due to the complicated nature of stitching the sections of the surfaces, these methods have not been adopted for the final version. This program has the following functional- ities: data input and output, aggregate surface area and volume calculation, aggregate surface visualization, aggregate dimen- sion analysis, and calculation of the shape factor, angularity, and texture. Figure 3-4 presents a flowchart of the FTI program. 3.3.1 Surface Area and Volume Calculation A serial program was developed to analyze and visualize aggregate particles on a MATLAB platform. First, original Original 3-D Surface z(x, y) Reconstructed 3-D Surface Difference Δz = z(x, y) – zi(i = 1, 2, 3, 4) z(x, y) z1 = ifft(zrfft(frequency ≤f1)) Average(abs(Δz)) ≤ 1mm z2 = ifft(zrfft(frequency ≤f2)) Average(abs(Δz)) ≤ 0.5mm z3 = ifft(zrfft(frequency ≤f3)) Average(abs(Δz)) ≤ 0.2mm z4 = ifft(zrfft(frequency ≤f4)) Average(abs(Δz)) ≤ 1 × 10-14 Note: 301 × 301 pixel by pixel in x-y plane, mm in z-axis. Figure 3-2. Illustration of the separation between angularity and texture in roughness matrix.

16 data of aggregate surface point coordinates are read to the computer’s memory. The data format is as follows: 99 101 9999 -0.1571 0.2500 5.0000 -0.3140 0.5000 5.0000 -0.4705 0.7500 5.0000 -0.6267 1.0000 5.0000 -0.7822 1.2500 5.0000 . . . . . . . . . . . . The first row is the number of pixels in the x direction and y direction and the total number of pixels in the x-y plane. The second row until the end records the x coordinate, y coordinate, and z coordinate of each point on the aggregate surface. We can calculate the aggregate surface area and volume from the original data (Figure 3-5). The data must be divided into two groups. One is for the upside surface. The other is for the downside surface. We can pair points (x, y, z coordi- nates) into triangle meshes. Every triangle surface area can be calculated as in Eq. 3-10, and so the total surface area is as calculated as in Eq. 3-11. S x y x y x y i i i i i i i = 1 2 1 1 1 1 1 2 2 3 3 Eq. 3-10 S Si i n = = ∑ 1 Eq. 3-11 0.0 5.0x10-4 1.0x10-3 1.5x10-3 2.0x10-3 0 105 15 20 25 30 0.0 2.0x10-5 4.0x10-5 6.0x10-5 8.0x10-5 Blast Furnace Slag Copper Ore Dolomite Glacial Gravel Crushed Glacial Gravel Rounded Iron Ore Limestone A ng ul ar ity F ac to r Slope is defined as Angularity Slope is defined as Texture Te xt ur e Fa ct or Roughness matrix size (mm2) Figure 3-3. Illustration of angularity and texture in the FTI system. Start Read Aggregate visualization Dimension calculation Shape factors Sphericity Flatness ratio Elongation ratio FE ratio Area & volume calculation Output End Figure 3-4. Flowchart of the FTI program. Horizon projection area (Sp) Mean-point Height (ha) Height (hb) Volume (Va) Volume (Vb) Surface Figure 3-5. Illustration of the surface area and volume calculation.

17 where Si is the area of the ith triangle, S is total surface area, n is the number of triangles, and (xi1, yi1), (xi2, yi2), and (xi3, yi3) are coordinates of the three points on the ith triangle. The total volume (Vi) is the sum of volume (Va) between the top surface and the horizontal projection surface and volume (Vb) between the bottom surface and the horizontal projection surface for each element, as illustrated in Figure 3-5. The calculation methods of V, Va, and Vb are shown by Eq. 3-12, Eq. 3-13, and Eq. 3-14. V V Vi a b= + Eq. 3-12 V S h V S ha p a b p b= × = × Eq. 3-13 V Va b i i n , = = ∑ 1 Eq. 3-14 3.3.2 Aggregate Visualization Once the aggregate surface points are available, we have three options for aggregate visualization: mesh, color, and light sur- face, as illustrated in Figure 3-6, Figure 3-7, and Figure 3-8. The visualization helps us make qualitative assessments. 3.3.3 Aggregate Dimension Analysis One of the important aggregate characteristics is three dimensions, shown in Figure 3-9. The three dimensions are Figure 3-6. Mesh surface. Figure 3-7. Color surface.

18 orthogonal to each other. The first maximum dimension is the longest one of the aggregate. We can measure all distances of any pair of points on the surface, and the longest distance is the first maximum dimension. Pi (xi, yi, zi) and Pj (xj, yj, zj) are any of the two points on the aggregate surface. The distance between the two points is Dij: D x x y y z zij j i j i j i= −( ) + −( ) + −( )2 2 2 Eq. 3-15 D Dl i j n ij= ≤ ≤max ,1 Eq. 3-16 where (xi, yi, zi), (xj, yj, zj) are coordinates of points Pi and Pj, Dij is the distance between points Pi and Pj, Dl is the longest length, and n is the total number of the points on the surface. In the next step we determine the intermediate dimension, which is perpendicular to both the longest dimension and the shortest dimension. Figure 3-10 illustrates the way we calculate the intermediate dimension. All the vertical planes Figure 3-8. Light surface. x (mm) y (mm) z (mm) Figure 3-9. Three dimensions.

19 shown in Figure 3-10 are slices perpendicular to the longest dimension. We measure all the distances between the points intersecting the plane and the surface of the particle and get the longest one. The maximum of all the longest distances on all the slices is the second maximum dimension. The point Pr(xr, yr, zr) is a joint point of the first maximum axis and the normal plane containing the second dimension. The direction of the first maximum axis is (a, b, c) or (xm2 - xm1, ym2 - ym1, zm2 - zm1) (end points of the longest dimen- sion). So the normal plane is: a x x b y y c z zr r r−( ) + −( ) + −( ) = 0 Eq. 3-17 a x x b y y c z zm m m m m m= − = − = −2 1 2 1 2 1, , Eq. 3-18 where (xr, yr, zr) is the cross point between the longest dimen- sion and a plane perpendicular to the longest dimension, (a, b, c) is the direction along the longest dimension, and (xm1, ym1, zm1) and (xm2, ym2, zm2) are the coordinates of two ends on the longest dimension. Given (a, b, c) and (xr, yr, zr) calculated using the afore- mentioned method, we can get every point coordinate on the plane D shown in Figure 3-10. There is a curved line of intersection between surface M and slice plane D. The maxi- mum distance between two points on the curved intersection line is the longest length on the slice plane. The intermedi- ate dimension is the maximum value among all these longest lengths on slice planes. Finally, the shortest dimension is calculated as follows. The shortest dimension is perpendicular to both the lon- gest dimension and the intermediate dimension. As shown in Figure 3-11, plane E is perpendicular to the intermediate dimension, and it is perpendicular to plane D with an inter- section line. Obviously, any line on plane E is perpendicular to both axis A and axis B. The longest length G between two points on the intersection line F is defined as the shortest dimension. Slice plane D perpendicular to the longest dimension The longest length on slice planes The axis along the longest dimension Aggregate surface M Figure 3-10. The intermediate dimension. Axis A along the longest dimension Axis B along the intermediate dimension Plane D is perpendicular to axis The longest length G on plane E Aggregate surface M Plane E is perpendicular to axis B, with an intersection line of F between plane E and surface M. Figure 3-11. The shortest dimension.