How to prune triangles in output mesh of Surface Reconstruction of Point Cloud - cgal

Question

I have an open surface represented by a point cloud which I've put through the steps described in the Point Set Processing and Surface Reconstruction tutorial. However, applying any of the 3 discussed Reconstruction algorithms results in a closed mesh with extreme extraneous polygons, especially along the "enclosing dome" of the atmosphere around the point cloud model. See attached pictures at the bottom of this question for visual context.
Is there an accurate, or canonically correct within CGAL, method to prune both the large and small triangles which do not intersect with the original point cloud? I have tried iterating over the faces in the output mesh and removing the faces which satisfy either of two conditions: (1) Any length of a given face is longer than the average spacing computed for the origin point cloud, and (2) The given area of the face is larger than some threshold. Either filtering condition results in a model which is not viewable as a mesh in PLY format given the various viewers I've tried. Here is some code which I've considered, but apparently does not do the job:
for (face_descriptor faced : output_mesh.faces()) {
//std::cout << faced << std::endl;
std::vector<double> lengths;
// Get edges from face descriptor
// - NOTE :> Assume three total half-edges because triangles. Not guaranteed, tho.
for (halfedge_descriptor hed : CGAL::halfedges_around_face( output_mesh.halfedge(faced), output_mesh )) {
vertex_descriptor target_vertex = output_mesh.target(hed);
Point_3& target_point = output_mesh.point(target_vertex);
halfedge_descriptor hed_next = output_mesh.next(hed);
vertex_descriptor target_next = output_mesh.target(hed_next);
Point_3& target_point_next = output_mesh.point(target_next);
double length = CGAL::sqrt(CGAL::squared_distance(target_point, target_point_next));
lengths.push_back(length);
simplex.push_back(target_point);
}
// if edge length greater than limit threshold based on average_spacing,
// mark for removal?
for (double length : lengths) {
if (length > average_spacing) {
output_mesh.remove_face(faced);
//CGAL::Euler::remove_face( output_mesh.halfedge(faced), output_mesh );
std::cout << "Removed face " << faced << " with lengths: ";
// List out the lengths found:
for (double length : lengths) {
std::cout << length << ", ";
}
std::cout << std::endl;
continue;
}
}
}
// clear faces marked as removed
output_mesh.collect_garbage();
//output_mesh.is_valid();
Thanks for your insights in advance! This has had me going in circles for weeks.

Answer 1

Following #sloriot 's comment, using Advancing Front Surface Reconstruction is the correct way of handling this, in addition to the hole-filling and fairing steps.
Due to the contiguous nature of advancing front, however, there are still some extraneous facets generated. A solution we've worked out involves projecting various points onto each facet; calculating the distance to the plane of the facet (simplex) and the barycentric coordinates of each point, as filtering criteria. This allows facets which do not match any point clouds to be removed. It is recommended to do this prior to hole filling/fairing or potentially applying other algorithms from the Polygon Mesh Processing set to perform some additional "fixing" for our application.
Thanks so much for the help!