Measuring the difference between two meshes

Computing the geometric difference between two 3D models is a quite common task in mesh processing. In our lab, many years ago (11 !), we developed and freely distributed the standard tool for such task, Metro, whose paper has been cited more than 500 times . While Metro is still a small open source standalone command line program available at our web site, its functionality have been integrated into MeshLab in the filter Sampling->Hausdorff Distance, and they can be used in a variety of ways.
So here is a short basic tutorial.

Start with a mesh (in the following the well known Stanford Happy Buddha (1087716 triangles). Aggressively simplify it in a significant way to just 50k tris (e.g. 1/20 of the original size). Reload the original mesh as a new layer. At this point you should have two approximation of the same shape well aligned in the same space. Toggling the visibility on and off of each mesh you should easily see the difference between the two meshes (tip: ctrl+click over the eye icon in the layers window turn off all the other layers).

Now you are ready to start the Hausdorff distance filter.  First of all remember that the Hausdorff Distance between two meshes is a the maximum between the two so-called one-sided Hausdorff distances (technically not they are not a distance):

These two measures are not symmetric (e.g. the results depends on what mesh you set as X or Y).
In the Hausdorff filters MeshLab computes only the one-sided version

leaving the task of getting the maximum of the two to the user.

Now on the practical side. MeshLab uses a sampling approach to compute the above formula taking a bunch of points over a mesh X and searching for each x the closest point y on a mesh Y. That means that the result is strongly affected on how many points do you take over X and there are a lot of option on for that. A common very simple approach is just to use the vertexes of the highest density mesh as sampling points (e.g. the original Buddha vertexes): to do this simply leave checked only the "vertex sampling" option in the filter dialog and be sure that the number of samples is greater or equal than the vertex number. After a few secs the filter ends writing down in the layers log windows the collected info. Something like:

: Hausdorff Distance computed

: Sample 543652

: min : 0.000000 max 0.001862

: mean : 0.000029 RMS : 0.000083

: Values w.r.t. BBox Diag (0.229031)

: min : 0.000000 max 0.008128

: mean : 0.000126 RMS : 0.000361

For sake of human readability the filter reports the values in the mesh units (whatever they are) and with respect to the diagonal of the bounding box of the mesh that is something you are always able to understand without knowing anything about the model units. For example in this case you can see that the maximum error between the two mesh is approximately 1% of the bbox diag, but on the average the two meshes are almost in the 1/10000 range.

The filter save in the all-purpose quality field of the vertexes of the sampled mesh the computed distance values. To better visualize the error you can simply convert these values (for the high resolution mesh) into colors using the Color->Colorize by quality filter that maps them in to a rather red-green-blue colormap. Usually given the non uniform distribution of the values you have to play a bit with the filter parameters clamping the mapping range to something meaningful (only a few points have the maximum so with a linear mapping of the values over the whole range will result into a almost uniform red mesh. Note that it is a red-green-blu map, so red is min and blue is max, so in our case red means zero error (good) and blue high error (bad).
The next image sequence report just a small detail of the one of the points with higher error. During the simplification we removed some topological noise, (the thin tubes connecting the two side of the hole)  from a Hausdorff point of view it is a rather large error: the points in the middle of the thin tubes has nothing in the simplified mesh that is close to them; so they bring up the maximum error significantly. Luckily enough they represent only a small portion of the whole mesh so the average error remains low.

Note that if you measure the other one-sided Hausdorff distance, that specific mesh portion will not denote any particular error, because in that case you sample the simplified mesh and for each point of the simplified mesh there are points of the original mesh that are quite close to them. In other words, in this case the simplified mesh is close to the original one, but the original one is not close to the simplified one.

Next post will discuss some remaining issues including the sampling of the surface, looking at all the taken samples and the found closest points and how to colorize the low resolution mesh...
Second part of the tutorial here.

Meshing Point Clouds

One of the most requested tasks when managing 3D scanning data is the conversion of point clouds into more practical triangular meshes. Here is a step-by-step guide for transforming a raw point cloud into a colored mesh.

Let's start from a colored point cloud (typical output of many 3D scanning devices), each point has just color and no normal information. The example dataset that we will use is a medium sized dataset of 9 millions of points. Typical issues of such a dataset dataset: it is non uniform (comes from an integration of different datasets), has some strongly biased error (alignment error, some problem during data integration), it comes without normals (hard to be shaded).

1. Subsampling
   
   
   
   As a first step we reduce a bit the dataset in order to have amore manageable dataset. Many different options here. Having a nicely spaced subsampling is a good way to make some computation in a faster way. The Sampling->Poisson Disk Sampling filter is a good option. While it was designed to create Poisson disk samples over a mesh, it is able to also compute Poisson disk subsampling of a given point cloud (remember to check the 'subsampling' boolean flag). For the curious ones, it uses an algorithm very similar to the dart throwing paper presented at EGSR2009 (except that we have released code for such an algorith long before the publication of this article :) ). In the invisible side figure a Poisson disk subsampling of just 66k vertices.
2. Normal Reconstruction
   
   
   
   Currently inside MeshLab the construction of normals for a point cloud is not particularly optimized (I would not apply it over 9M point cloud) so starting from smaller mesh can give better, faster results. You can use this small point cloud to issue a fast surface reconstruction (using Remeshing->Poisson surface reconstruction) and then transfer the normals of this small rough surface to the original point cloud. Obviously in this way the full point cloud will have a normal field that is by far smoother than necessary, but this is not an issue for most surface reconstruction algorithms (but it is an issue if you want to use these normals for shading!).
3. Surface reconstruction
   
   
   
   Once rough normals are available Poisson surface reconstruction is a good choice. Using the original point cloud with the computed normals we build a surface at the highest resolution (recursion level 11). Roughly clean it removing large faces filter, and eventually simplify it a bit (remove 30% of the faces) using classical Remeshing->Quadric edge collapse simplification filter (many implicit surface filters rely on marching cube like algorithms and leave useless tiny triangles).
4. Recovering original color
   
   
   
   Here we have two options, recovering color as a texture or recovering color as per-vertex color. Here we go for the latter, leaving the former to a next post where we will go in more details on the new automatic parametrization stuff that we are adding in MeshLab. Obviously if you store color onto vertexes you need to have a very dense mesh, more or less of the same magnitudo of the original point cloud, so probably refining large faces a bit could be useful. After refining the mesh you simply transfer the color attribute from the original point cloud to the reconstructed surface using the vertex attribute transfer filter.
   
   
5. Cleaning up and assessing
   
   
   
   The vertex attribute transfer filter uses a simple closest point heuristic to match the points between the two meshes. As a side product it can store (in the all-purpose per-vertex scalar quality) the distance of the matching points. Now just selecting the faces having vertices whose distance is larger than a given threshold we can easily remove the redundant faces created by the Poisson Surface Reconstruction.

This pipeline is only one of the many possible way of ending up into a nice mesh. For example different choices could have been done for step 2/3. There are reconstruction algorithms that do not need surface normals, like for example the "Voronoi Filtering" that is an interpolating reconstruction algorithm (e.g. it build up only triangles on the given input points) but usually these filters works better on very clean datasets, without noise or alignment errors. Otherwise on noisy datasets it is easy that they create a lot of non manifold situations. Final thanks to Denis Pitzalis for providing me this nice dataset of a Cypriot theater.

On the storage of Color in meshes

A very short post to clarify a bit the ways in which people can store color (and other) information on a mesh. AFAIK there are mostly three ways to keep color:

• Per-vertex: each vertex stores a color. A triangular face can have vertexes of different color and inside a triangle the color is linearly interpolated. Color is smooth across the surface.
  
• Per face: each face has a distinct color. You can easily see the discontinuity of colors among the faces (no interpolation is usually done.
• As texture: the most general way that is reasonably decoupled from the mesh itself. You only need a good parametrization.

In MeshLab most of the algorithms (painting, color processing etc.) manage per-vertex color, but a few conversion tools are provided.

• per-vert -> per -face
  
• per-face -> per-vert
• texture -> per-vert

A per-vert -> texture is strikingly missing :). To be added in the very near future. Obviously assuming the previous existence of a parametrization...
A fourth technique could be mentioned, keeping stuff per wedge, i.e. for each corner of the face we can store different colors (or other attributes), but this approach is rather unused (it can be simulated by duplicating vertices).
Below a few images showing the difference between the three modes on a small (40k tri) mesh; respectively: no color (to give you an idea of mesh density), color by texture, per vertex color, per face color.

Creating Voronoi Sphere

February 2016 update. You can make it online without even installing anything, by using the new browser based version of MeshLab: www.meshlabjs.net 

MeshLab is quite useful for a lot of classical mesh processing tasks, but sometimes it can be used for more weird things. A few weeks ago, after stumbling upon the cool Shapeways 3D printing service I uploaded there a few artsy mathematical sculptures that I created with MeshLab. Here is how I did this one, called Voronoi sphere.
It is a double Voronoi diagram, in the sense that there is a coarse Voronoi diagram over the sphere surface but also the surface that creates the edges of this diagram has been carved to create another finer Voronoi diagram. Such a shape is really very light and thin but much more robust that you could imagine.

1. Start from a sphere (file->new->Sphere),
   
   
   
   
   
2. Refine it using Filter>Remeshing>Loop Subdivision surfaces. repeat without shame (lowering the edge threshold parameter) until it becomes reasonably well tessellated. 300k faces are enough.
   
   
   
   
3. Create some well distributed samples over the surface using
   Filter>Sampling>Poisson Disk Sampling. 50 points are a good choice. Apparently the filters does nothing, but if you reveal the layer panel (guess the icon in the toolbar :)), you can see that there are two layers. Make invisible the first layer and switch the rendering mode to points: you will see the well distributed Poisson samples (hint: alt+mouse wheel change the drawn size of the points).
   
   
4. Create the actual Voronoi diagram by simply choosing the filter
   Color>Voronoi Vertex Coloring. As reported in the top of the parameter window, this filter, given a mesh M and a point-set P, project the points of P over M and color each vertex of M according to the geodesic distance from these projected points. Marking the backdistance flag in the parameter window the filter computes the distance from the borders of the Voronoi diagram instead of the projected sites itself. This filter, beside coloring the mesh, writes on each vertex of the mesh the distance value itself, in the all-purpose attribute named 'quality'.
   
   
   
5. Make the mesh layer active, and start the Select by vertex Quality filter. enable the preview option and enable visualization of selected faces. Play with the slider until you get something similar to the image on the right; in practice, exploiting the quality value stored onto the vertices that code the distance from the border of the Voronoi diagram we have just selected the faces very near to these borders.
   
   
   
   
   
6. Apply the Filter>Selection>Invert Selection and then delete the selected faces. Edges are probably quite jaggy, so apply a couple of times the simplest of all the smoothing filter, the old classical laplacian filter (Filter>Smoothing>Laplacian Smoothing).

Now stop and save the mesh. Next post will show you how to continue by transforming the current mesh, that is a surface, into a solid object ready to be printed. In the meantime if you like the sculpture, you can buy a small (10 cm) and cheap (less than 20$) copy of this sculpture here.