5.3. Mapper on 3D Shape Database The top row shows the rendering of one model from each of the 7 classes. The bottom row shows the same model colored by the E1 function (setting p = 1 in equation 4–1) computed on the mesh.

In this section we apply Mapper to a collection of 3D shapes from a publicly available database of objects [SP]. These shapes correspond to seven different classes of objects (camels, cats, elephants, faces, heads, horses and lions). For each class there are between 9 and 11 objects, which are different poses of the same shape, see Figure 7. This repository contains many 3D shapes in the form of triangulated meshes. We preprocessed the shapes as follows. Let the set of points be P. From these points, we selected 4000 landmark points using a Euclidean maxmin procedure as described in [dSVG04]. Let this set be Y = { Pi, i ∈ L}, where L is the set of indices of the landmarks.

Choosing Landmark points: MaxMin data points. In this very simplified case my data points lie in a two-dimensional plane. Normally data points are high dimensional. For example, I may be comparing the expression or thousands of genes in tumor cells to healthy cells using microarray data. OR I might be comparing politicians voting records. Or I might be comparing the stats of basketball players. These three applications were all, by the way, published by Lum et al this past February in Nature’s Scientific Reports. I have included a link to their paper on my youtube site. http://www.nature.com/srep/2013/130207/srep01236/full/srep01236.html 1.) choose point l1 randomly

Choosing Landmark points: MaxMin data points. In this very simplified case my data points lie in a two-dimensional plane. Normally data points are high dimensional. For example, I may be comparing the expression or thousands of genes in tumor cells to healthy cells using microarray data. OR I might be comparing politicians voting records. Or I might be comparing the stats of basketball players. These three applications were all, by the way, published by Lum et al this past February in Nature’s Scientific Reports. I have included a link to their paper on my youtube site. http://www.nature.com/srep/2013/130207/srep01236/full/srep01236.html 1.) choose point l1 randomly

Choosing Landmark points: MaxMin data points. In this very simplified case my data points lie in a two-dimensional plane. Normally data points are high dimensional. For example, I may be comparing the expression or thousands of genes in tumor cells to healthy cells using microarray data. OR I might be comparing politicians voting records. Or I might be comparing the stats of basketball players. These three applications were all, by the way, published by Lum et al this past February in Nature’s Scientific Reports. I have included a link to their paper on my youtube site. http://www.nature.com/srep/2013/130207/srep01236/full/srep01236.html 2.) If {l1, …, lk-1} have been chosen, choose lk such that {l1, …, lk-1} is in D - {l1, …, lk-1} and min {d(lk, l1), …, d(lk, lk-1)} ≥ min {d(v, l1), …, d(v, lk-1)}

Choosing Landmark points: MaxMin data points. In this very simplified case my data points lie in a two-dimensional plane. Normally data points are high dimensional. For example, I may be comparing the expression or thousands of genes in tumor cells to healthy cells using microarray data. OR I might be comparing politicians voting records. Or I might be comparing the stats of basketball players. These three applications were all, by the way, published by Lum et al this past February in Nature’s Scientific Reports. I have included a link to their paper on my youtube site. http://www.nature.com/srep/2013/130207/srep01236/full/srep01236.html 2.) If {l1} chosen, choose l2 such that {l1} is in D - {l1, …, lk-1} and min {d(l2, l1)} ≥ min {d(v, l1)}

Choosing Landmark points: MaxMin data points. In this very simplified case my data points lie in a two-dimensional plane. Normally data points are high dimensional. For example, I may be comparing the expression or thousands of genes in tumor cells to healthy cells using microarray data. OR I might be comparing politicians voting records. Or I might be comparing the stats of basketball players. These three applications were all, by the way, published by Lum et al this past February in Nature’s Scientific Reports. I have included a link to their paper on my youtube site. http://www.nature.com/srep/2013/130207/srep01236/full/srep01236.html 2.) If {l1, …, lk-1} have been chosen, choose lk such that {l1, …, lk-1} is in D - {l1, …, lk-1} and min {d(lk, l1), …, d(lk, lk-1)} ≥ min {d(v, l1), …, d(v, lk-1)}

Choosing Landmark points: MaxMin data points. In this very simplified case my data points lie in a two-dimensional plane. Normally data points are high dimensional. For example, I may be comparing the expression or thousands of genes in tumor cells to healthy cells using microarray data. OR I might be comparing politicians voting records. Or I might be comparing the stats of basketball players. These three applications were all, by the way, published by Lum et al this past February in Nature’s Scientific Reports. I have included a link to their paper on my youtube site. http://www.nature.com/srep/2013/130207/srep01236/full/srep01236.html 2.) If {l1, l2} have been chosen, choose l2 such that {l1, …, lk-1} is in D - {l1, …, lk-1} and min {d(l3, l1), d(l3, l2)} ≥ min {d(v, l1), d(v, l2)}

Choosing Landmark points: MaxMin data points. In this very simplified case my data points lie in a two-dimensional plane. Normally data points are high dimensional. For example, I may be comparing the expression or thousands of genes in tumor cells to healthy cells using microarray data. OR I might be comparing politicians voting records. Or I might be comparing the stats of basketball players. These three applications were all, by the way, published by Lum et al this past February in Nature’s Scientific Reports. I have included a link to their paper on my youtube site. http://www.nature.com/srep/2013/130207/srep01236/full/srep01236.html 2.) If {l1, …, lk-1} have been chosen, choose lk such that {l1, …, lk-1} is in D - {l1, …, lk-1} and min {d(lk, l1), …, d(lk, lk-1)} ≥ min {d(v, l1), …, d(v, lk-1)}

In order to use this as the point cloud input to Mapper, we computed distances between the points in Y as follows. First, we computed the adjacency matrix A for the set P by using the mesh information e.g. if Pi and Pj were connected on the given mesh, then A(i, j) = d(Pi,Pj), where d(x, y) is the Euclidean distance between x, y ∈ P. Finally, the matrix of distances D between points of Y was computed using Dijkstra’s algorithm on the graph specified by A.

In order to apply Mapper to this set of shapes we chose to use E1(x) as our filter function (setting p = 1 in equation 4–1), see Figure 7. In order to minimize the effect of bias due to the distribution of local features we used a global threshold for the clustering within all intervals which was determined as follows. We found the threshold for each interval by the histogram heuristic described in Section 3.1, and used the median of these thresholds as the global threshold.

The output of Mapper in this case (single filter function) is a graph. We use GraphViz to produce a visualization of this graph. Mapper results on a few shapes from the database

1. Mapper is able to recover the graph representing the skeleton of the shape with fair accuracy. As an example, consider the horse shapes. In both cases, the three branches at the bottom of the recovered graph represent the font two legs and the neck. The blue colored section of the graph represents the torso and the top three branches represent the hind legs and the tail. In these examples, endowing the nodes of the recovered graph with the mean position of the clusters they represent would recover a skeleton. 2. Different poses of the same shape have qualitatively similar Mapper results however different shapes produce significantly different results. This suggests that certain intrinsic information about the shapes, which