incorrect concave hull for certain alpha but correct result for values > and < alpha

following the recommendation of Sergio Agostinho, I am filing an issue about this topic from the mailing list: http://www.pcl-users.org/incorrect-concave-hull-for-certain-alpha-but-correct-result-for-values-gt-and-lt-alpha-td4040281.html

On Ubuntu 14.04:

I want to use pcl::concaveHull to compute the outline of a pointcloud (in
2d). Attached are the point cloud (ceiling_1.pcd) and the example code I
prepared to reproduce the issue.

I know that concaveHull reconstruct returns a collection of hulls. In my
code, I split this collection into separate point clouds and visualize the
result.

Since I am looking for an outline, I am mainly interested in the outer most
polygon.
Regardless of the inner polygons, I expect one polygon enclosing the whole
point cloud.

My expectations are met for alpha = 0.19 and alpha = 0.21 (one big polygon outlines the whole
cloud). However for alpha = 0.20, i.e. a value BETWEEN two other working
values, the result is completely different from the two previous results:

There are 3 polygons along the outline of the point cloud and there remain
many points that are not included in any of the polygons.

To illustrate my issue, I brought 3 images, please consider only the RED
point cloud (the biggest one in the bottom):

1. correct result with alpha = 0.19 (alpha_19.png)
   
2. wrong result with alpha = 0.20 (alpha_20.png)
   
3. correct result with alpha = 0.21 (alpha_21.png)
   

additional files posted inline since github wouldn't allow me to upload them:

concave_hull_issue.cpp is a full code example with the essentials to reproduce the issue:

#include <iostream>
#include <pcl/io/pcd_io.h>
#include <pcl/point_cloud.h>
#include <pcl/surface/concave_hull.h>
#include <pcl/visualization/pcl_visualizer.h>

int
main (int argc, char** argv)
{
//  float alpha = 0.19;
//  float alpha = 0.20;
  float alpha = 0.21;

  // Load file
  pcl::PointCloud<pcl::PointXYZRGB>::Ptr source_cloud (new pcl::PointCloud<pcl::PointXYZRGB> ());
  pcl::io::loadPCDFile ("ceiling_1.pcd", *source_cloud);

  // project to xy-plane
  for (int i = 0; i < source_cloud->size(); ++i)
  {
    source_cloud->points[i].z = 0.0;
  }

  // concave hull computation
  std::vector<pcl::Vertices> polygons;
  pcl::PointCloud<pcl::PointXYZRGB>::Ptr hull_tmp (new pcl::PointCloud<pcl::PointXYZRGB> ());;
  pcl::ConcaveHull<pcl::PointXYZRGB> concave_hull;
  concave_hull.setInputCloud (source_cloud);
  concave_hull.setAlpha (alpha);
  concave_hull.setDimension(2);
  concave_hull.setKeepInformation(true);
  concave_hull.reconstruct (*hull_tmp, polygons);

  // get point clouds from indexes
  std::vector<pcl::PointCloud<pcl::PointXYZRGB>::Ptr> concave_hull_collection;
  for (int j = 0; j < polygons.size(); ++j)
  {
    pcl::PointCloud<pcl::PointXYZRGB> tmp;
    for (int i = 0; i < polygons[j].vertices.size(); ++i)
    {
     tmp.points.push_back(hull_tmp->points[polygons[j].vertices[i]]);
    }
    tmp.is_dense=false;
    tmp.height=1;
    tmp.width=tmp.points.size();
    concave_hull_collection.push_back(pcl::PointCloud<pcl::PointXYZRGB>::Ptr(new pcl::PointCloud<pcl::PointXYZRGB>(tmp)));
  }



  // Visualization
  pcl::visualization::PCLVisualizer viewer ("Illustration of concave hull");
  // Define R,G,B colors for the point cloud
  pcl::visualization::PointCloudColorHandlerCustom<pcl::PointXYZRGB> source_cloud_color_handler (source_cloud, 255, 255, 255);
  // We add the point cloud to the viewer and pass the color handler
  viewer.addPointCloud (source_cloud, source_cloud_color_handler, "original_cloud");

  viewer.addCoordinateSystem (1.0, "cloud", 0);
  viewer.setBackgroundColor(0.55, 0.55, 0.05, 0);
  viewer.setPointCloudRenderingProperties (pcl::visualization::PCL_VISUALIZER_POINT_SIZE, 3, "original_cloud");

  // display results of concave cull as line polygons
  std::vector<float> color(3);
  for (int i = 0; i < concave_hull_collection.size(); ++i)
  {
    std::stringstream ss;
    ss << i;
    std::string id = ss.str();
    color[0] = i == 0 ? 1.0f : 0.0f;
    color[1] = i == 1 ? 1.0f : 0.0f;
    color[2] = i == 2 ? 1.0f : 0.0f;

    // Add polygon to the visualizer
    viewer.addPolygon<pcl::PointXYZRGB>(concave_hull_collection[i], id);
    viewer.setShapeRenderingProperties(pcl::visualization::PCL_VISUALIZER_REPRESENTATION, pcl::visualization::PCL_VISUALIZER_REPRESENTATION_WIREFRAME, id);
    viewer.setShapeRenderingProperties(pcl::visualization::PCL_VISUALIZER_LINE_WIDTH, 3, id);
    viewer.setShapeRenderingProperties(pcl::visualization::PCL_VISUALIZER_COLOR, color[0], color[1], color[2], id);
  }

  while (!viewer.wasStopped ()) { // Display the visualiser until 'q' key is pressed
    viewer.spinOnce ();
  }

  return 0;
}

ceiling_1.pcd is the point cloud I am referring to:

    # .PCD v0.7 - Point Cloud Data file format
    VERSION 0.7
    FIELDS x y z rgb normal_x normal_y normal_z curvature
    SIZE 4 4 4 4 4 4 4 4
    TYPE F F F F F F F F
    COUNT 1 1 1 1 1 1 1 1
    WIDTH 2673
    HEIGHT 1
    VIEWPOINT 0 0 0 1 0 0 0
    POINTS 2673
    DATA ascii
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    -4.6310601 1.36316 2.3722799 0 0.00215212 0.021439785 -0.99976778 0
    -4.2324643 1.238575 2.3739452 0 -0.0011073062 0.013056955 -0.99991423 0.00060540735
    -4.3406048 1.3288299 2.37466 0 0.0020572289 0.015665758 -0.99987525 0
    -4.4316602 1.3805201 2.3748503 0 0.0043869331 0.020588847 -0.99977833 0.00073827279
    -4.2255702 1.34266 2.3749299 0 -0.0011074777 0.018059788 -0.99983627 0.00044712552
    -5.2421298 1.35638 2.3722699 0 -0.0017510155 0.016445693 -0.99986321 0
    -5.3381896 1.3589801 2.372752 0 -0.0047652717 0.016814217 -0.99984729 0.00045631768
    -5.1434898 1.3255601 2.37222 0 -0.00046880785 0.018670347 -0.9998256 0.0010806549
    -4.9729404 1.3836201 2.37291 0 0.0020497295 0.021991722 -0.99975604 0
    -5.0370007 1.4102401 2.3732901 0 0.0037134429 0.021551643 -0.99976087 0
    -5.1756506 1.4281155 2.3736126 0 0.0021880006 0.01971451 -0.99980325 0
    -4.8191152 1.3553581 2.3736107 0 -0.00064392545 0.022734515 -0.99974132 0.00095489871
    -4.7368503 1.3523501 2.3733399 0 -0.0028021291 0.019191314 -0.99981189 0
    -4.8303905 1.41679 2.3745401 0 0.0048480658 0.024115426 -0.99969745 0
    -5.5239201 1.3825101 2.3748 0 -0.0067006624 0.019517768 -0.99978703 0.00040107319
    -

Start with the supplied concave_hull_issue.cpp reproducer and ceiling_1.pcd, using pcl::ConcaveHull with dimension 2 and alpha values 0.19, 0.20, and 0.21. Trace the implementation behind pcl/surface/concave_hull.h and compare the returned polygons and omitted points. Done means the intermediate alpha produces the expected outer polygon consistently with the neighboring values.