Enseñanza de la inteligencia emocional

Before a statistical model can be built, the meshes must be brought into a one-to-one corre- spondence. In order to do this we define a template meshS = (V,E)and adapt this mesh to the input meshes, so that all the meshes share the same topological structure,E.

We implemented two methods for building a mesh correspondence. The first is based on the method by Praun et al. [62] using parameterisation and the second is based on ICP which is described below.

3.2.1

Iterative Closest Point alignment using multi-level free-form de-

formation

We adapt a reference face model to each subject’s face as outlined in figure 3.3. The two surfaces, the subject and the reference are brought into alignment by translation to a common mean and removing rotational variance between the two meshes. The translation to the mean is found by making the centre of mass of the point sets equal. The meshes are scaled by normalising the deviation from the mean and the required rotation found using SVD on the cross-covariances between pairs of meshes.

The reference mesh topology we use is just an individual selected on the basis of the quality of the scan (i.e. all major features visible). We adapt this to each subject using a standard two stage warping process. The first stage is a feature based warping method, in which manually placed landmarks are used to drive a multi-level free-form deformation (MFFD), which is a hierarchy of B-spline interpolating functions with progressively finer resolution

S T v₁ v₂ v₃ v₄ v5 v₆ v0₁ v0₂ v0₃ v0₄ v0₅ v0₅

Figure 3.2: Surface S is matched to surface T using Iterative Closest Point alignment. Each vertexviis matched to a vertexv0iby searching along the surface normal.

[45, 46]. We have implemented the MFFD warping using a space and search efficient oc- tree data structure. Once the face meshes are in approximate alignment correspondences are found using a standard ray-tracing algorithm. Rays are traced out of the reference mesh from each vertex in both directions along the surface normal and the first intersection (within a maximum radius) with the target face is found using an octree ray-tracing method (see figure 3.2). Not all vertices will find a target, and so these displacements are interpo- lated (again using MFFDs) across the reference mesh. This brings the reference mesh into good alignment with the subject.

3.2.2

Surface alignment using Parameterisation

As with our ICP implementation we delineated a set of points on each of a set of scanned faces. The parameterisation method does not require a predefined template mesh, this is defined procedurally. For each mesh we computed a mapping from the three-dimensional surface of the mesh to a two dimensional plane, w : _R3 → _R2 _{using Floaters’ parame-} terisation [28]. For each internal vertex of the mesh, vi ∈ R3, a corresponding position

Subject

Template

Warp

Refine

Figure 3.3: An outline of the Iterative Closest Point Alignment algorithm. First both meshes are delineated by hand. A template mesh is then warped to the rough shape of the subject mesh using the delineated points. The warp is refined by tracing along the surface normals of the subject model and matching at the point of intersection with the template mesh. Finally the texture is reconstructed by applying an inverse warp to the texture-map.

ui ∈ R2 is found on the plane, such that the mesh is ‘flattened’ on to the surface of the plane in a manner that best preserves the angles between edges of the mesh and the lengths of edges (see section 3.1). Each position within a triangle on the surface of the mesh can be mapped onto the plane by finding its associated triangle in the flattened mesh and de- termining its position in this mesh using the method of barycentres. This produces a set of flattened meshes. These meshes can be brought into correspondence using the delineated points, for each delineated point on the surface of the mesh the corresponding point on the plane as found using the mapping,w. This produces a set of templates in two-dimensions for each input face. An average of these templates was found using Generalised Procrustes alignment [25] and warping between each template and the average generated using Thin Plate Splines (TPS). The application of this warp to flattened meshes brings them into cor- respondence on the plane. A new mesh can be generated by regularly sampling along a predefined grid on the plane in the area of the warped mesh. Using the TPS warp on the vertices of the mesh is possible but is not easily defined within the triangles of the mesh, instead we use an inverse warp on the sample points of the regular grid this allows us to sample from the triangles as if they were warped. For each sample point in the regular grid the triangle of the flattened mesh containing it is located and its corresponding point on the surface of the three-dimensional mesh found by inverting the mapping,w. If no triangle is found containing the sampling point the sample point is marked as missing. A new mesh is constructed by triangulating the located sample points in a ‘chess-board’ pattern with two triangles being formed in each quad.

3.2.3

Results

Figures 3.4a and 3.4b provide a ‘before and after’ view of the remeshing process. The two methods produced similar results in terms of fitting. The ICP method relied on the quality of the original mesh, and produced some spurious fitting around areas of high curvature due to the closest point search finding poor correspondences. The parameterisation method was vulnerable to stretching in mapping the surface to two-dimensions, this results in under- sampling. As stretching often occurs in areas of high curvature, this means that the under- sampling will occur most frequency in areas of most interest. The parameterisation method also lacked a hole-filling method. This deficiency was the main reason the ICP method was the chosen method for producing the correspondences between meshes.

(a) Irregular Mesh (b) Regular Mesh

Figure 3.4: ‘Before and after’ view of a reconstructed mesh. The left hand image shows a ‘wire-frame’ view of a mesh as produced by the scanner. The irregular structure of the mesh is clearly visible as are the holes. The right hand image shows a reconstructed mesh, the regular grid pattern is visible as are the un-patched holes in the mesh.

The average of a set of 106 face models produced by the three-dimensional scanner was produced by aligning these meshes to a ‘reference mesh’ using the ICP fitting method. These aligned models were used to build a face model as described in the next section. The mean of the face models can be found in figure 3.5.