Assignment 3

This assignment explores mesh smoothing using two variants of discrete Laplacian: Uniform and Cotangent.

1)Experiment 1: Cilinder Random Perturbation

	Shade	Mesh Detail
Original
Perturbed
Uniform Laplacian Smoothing (3rd Iteration)
Cotangent Laplacian Smoothing (3rd Iteration)

Cotangent Laplacian outperforms Uniform Laplacian in both the error to the subjacent surface and to the original mesh.

It is interesting to observe how the Cotangent Laplacian preserves the original anisotropic mesh of the cilinder (I mean, the triangulation is still aligned to the radial and axial directions), while the Uniform Laplacian produces a more regular and isotropic mesh.

The Uniform Laplacian has a stronger smoothing effect (i.e., remove higher frequencies) than Cotangent Laplacian. This phenomena is clearly observed from the respective shade figures, and from the MSE figure.

The first three iterations of the Cotangent Laplacian, led to an effective reduction of the noise added to the mesh. Further smoothing increases the error to the original mesh and to the subjacent surface.

	Shade	Mesh Detail
Original
Perturbed
Uniform Laplacian Smoothing (8th Iteration)
Cotangent Laplacian Smoothing (8th Iteration)

This experiment reaffirms two facts observed in the previous experiment: the Cotangent Laplacian preserves the mesh orientation, and the Uniform Laplacian has a stronger smoothing effect.

It is interesting to observe the shrinking effect produced by smoothing. In these experiments, I only fixed the boundary vertices of the cilinder, while the others are recalculated using the respective discrete Laplacian formula. The shrinking effect is even more notorious in lower resolution meshes (i.e. with lower density of vertices), since the effect of weighting over 1-ring is a stronger push towards the cilinder axis. From the error to the Subjacent Surfaces in the figure above, we deduce that the shrinking effect is stronger for Uniform Laplacian than Cotangent.

	Shade	Mesh Detail
Original
Perturbed
Uniform Laplacian Smoothing (5th Iteration)
Cotangent Laplacian Smoothing (5th Iteration)

Form the MSE figures it is notorious that the error to both Subjacent Surface and Original Mesh increases in higher order for Uniform Laplacian than for Cotangent.

	Shade	Mesh Detail
Original
Perturbed
Uniform Laplacian Smoothing (2nd Iteration)
Cotangent Laplacian Smoothing (2nd Iteration)

The original mesh of the cilinder corresponds to the 5th subdivision from an octahedron. Therefore, the mesh is some anisotropic near the original vertices of the octahedron. This anistropy is still preserved applying Cotangent Laplacian. Instead, Uniform Laplacian led to a less anisotropic mesh, where the vertices of the concentric squares (from the original mesh) are rounded.

It is interesting to observe that the Unifrom Laplacian outperforms Cotangent in both subj. surface and original mesh errors for the first few iterations. Certainly, this is because the sphere geometry and the mesh are more regular and isotropic than the cilinder and its mesh. These conditions initally favors the Uniform Laplacian.

From the shade figures, it is quite clear that the smoothing due to Uniform Laplacian is much higher than for Cotangent. In the case of the sphere, Uniform Laplacian lead to the desired result (except by the shrinking effect), but in meshes with geomtric features (corner and edges) we would loose geometric details.

	Shade	Mesh Detail
Original
Perturbed
Uniform Laplacian Smoothing (4th Iteration)
Cotangent Laplacian Smoothing (4th Iteration)

This experiment reaffirms that the geometry and this particular mesh of the sphere favors Uniform Laplacian. Cotangent Laplacian did not perform as well, but was still competitive.

In order to get a good smoothing effect and dont less geometry neither size, we should reproject the recalculated vertices position onto previous mesh.