Manhattan metric

The Manhattan metric provides an alternate method to measure distances between crack intersection points.

In Euclidean geometry, the shortest path between two points is a straight line. The norm of the vector that connects two points is the Euclidean distance. In R², this distance is d =p

(y₂−y₁) + (x₂−x₁). There are some geometries where Euclidean distances cannot be calculated. An example of this is the grid presented in figure 3.7 (a). Staying on the grid from the start node to end node is possible only by travelling either in the x direction or along the y direction. Such a geometry is known as the taxi-cab geometry. A combination of transverse displacements along both x and y direction, which is the essence of the Euclidean distance, is not possible in a taxicab geometry. Since the shortest distance between two points cannot be defined in terms of Euclidean distance, a new metric is defined. This metric is known as the Manhattan metric [56]

m =|x₁−x₂|+|y₁−y₂| (3.12) Hence, the distance between two points in a Cartesian coordinate system can be measured using either the Euclidean distance, or by the Manhattan distance if restricted to moving on a grid. While in the case of the Euclidean distance there is only one path which minimizes the distance between two points, in the case of the Manhattan metric, there are multiple paths that minimize distances between two nodes on a grid.

Moving on a crack pattern is similar to moving along a grid. The crack intersection points

become the nodes of the grid, and the cracks become the connecting lines between nodes.

However, the analogy ends there. Depending upon the crack pattern, each neighbouring pair of nodes or crack intersection points are connected by curves. Hence the distances between these crack intersection points are apporoximately Euclidean. In case of ladder-like crack pattern, these curves are horizontal or vertical cracks similar to a grid, in case of an isotropic crack pattern, these curves can be meandering cracks with curvature.

The general definition for the Manhattan distance, on a crack pattern, between two non -neighbouring crack intersection points is

where di represents the Euclidean distance between two neighbouring crack intersection points. In figure 3.7 (b), d_{M an} is the sum of the d₁ to d₅. The Euclidean distance d_E between the two crack intersection points is presented for comparison.

In figure 3.7 (b),d_{M an} is dependent on the geometry of the crack pattern, whereas d_E is dependent on the location of the crack intersection points in real space. For all pairs of non neighbouring crack intersection points, the condition

d_E≤d_{M an} (3.14)

holds true. On a ladder-like crack pattern, travelling from a crack intersection point on the top of the image to the bottom of the image, the ratio of dE and dM an would be close to 1.

Whereas, in case of an isotropic crack pattern, this is not the case. The ratio of d_Eandd_{M an} is represented

as-s_{M an}= d_E

d_{M an} (3.15)

The parameters_{M an}is calculated for a single pair of crack intersection points. For sinusoidal plates, sM an is calculated for all crack intersection points that form pairs between the top part of the image to bottom

-S_{M an} =

whereN is the number of pairs of crack intersection points.

Taking figure 3.8 (a) as an example,SM an is calculated in the following way

-• Starting with a labeled skeleton image of the crack pattern, the crack intersection points are detected . For each crack intersection point, its neighbours are identified.

• The crack intersection points are sorted according to their location. All crack intersec-tion points that lie in top 1/3 and the bottom 1/3 of the image are stored in a separate arrays. Figure 3.8 (a) shows sorted crack intersection points on a crack pattern.

• An adjacency matrix is calculated for all crack intersection points. This is done by creating an N ×N matrix, where N is the number of crack intersection points. The neighbours for each crack intersection point are identified from the neighbours list and the Euclidean distance between is stored in the adjacency matrix as

-A[i, j] = q

(y_j−y_i) + (x_j−x_i) (3.17) whereA is the adjacency matrix and whereiand jare neighbouring crack intersection points. For the i^th column of A, only those rows are non zeros whose index represent the neighbours of the i^th crack intersection point are non zero, the rest of the rows are zero. Figure 3.8 (b) shows a simple cartoon of whose adjacency matrix

is-A=

• Pairs of crack intersection points are constructed where one crack intersection point belongs to the top part of the image and the second point belongs to the bottom of the image.

• For each pair of crack intersection points, the angle between the substrate and vector connecting the pair of crack points is calculated. If the angle lies in between −5^◦ and 5^◦( where 0^◦ is parallel to the ridges on the substrate) then, the pairs of crack inter-section points combined with the adjacency matrix are put into the MATLAB function

’graphshortestpath’, otherwise the pair is ignored. For each pair of crack intersection points, this function uses the Djikstra algorithm [73] to find the shortest path between two points on a network. ’graphshortestpath’ outputs the sequence of crack intersection points needed to travel in order to get from the starting crack intersection point to the ending crack intersection point. Using Euclidean distances in the adjacency matrix, it also outputs the total distance travelled along the crack pattern. This distance is stored in a separate array. The distance along the crack between pairs of neighbours is calculated for all pairs.

• The Euclidean distance between the pair of crack intersection points is calculated for all pairs. This is divided by the distance acquired using ’graphshortestpath’ and the final result is averaged which gives the value of S_{M an}.

For radial crack patterns, the pairs are sorted by considering crack intersection points which are inside a circle that has a diameter of 1/4 of the image and crack intersection points that are located outside a circle with a diameter 1/3 the size of the image. Both the circles are centred about the image. The pairs are formed by selecting one crack intersection point from the inside region and another crack intersection point from the outside region. The rest of the analysis is carried out in a similar fashion.

In case of ladder-like cracksS_{M an} tends to be close to 1, for isotropic crack patterns the lower limit of the value is 1/√

2.

Figure 3.8: Definition of the Manhattan metric (a) shows the crack intersection points sorted according to their location. A crack intersection point from the top 1/3 of the image and a crack intersection point from the bottom 1/3 are paired and the ratio of the Euclidean distance and Manhattan distance s_{M an} is calculated. This is done for all pairs of crack intersection points in figure (a). Some of the points are missing from the top and the bottom since 10% of points from each edge are ignored to prevent boundary effects. (b) is an example of a connected set of points. The connections of figure (b) are represented in the adjacency matrix equation 3.18.