Fuzzy Methods in Nonlinear Signal Processing: Part II—Applications - Nonlinear Signal and Image Processing: Theory, Methods and Applications - page 103

(a) (b) (c) (d)

(e) (f) (g) (h)

FIGURE 3.26
Smoothing of fan disk and gear: (a) and (e) original surfaces; (b) and (f) noisy surfaces; (c) and
(g) results of Taubin's linear anisotropic filtering method; (d) and (h) FMED v -based results.

filtering ( 4) and shown in Figure 3.24c and g. Note that the surface

normals processed by the FMED
v

σ = 0 .

are very close to the original surface nor-
mals. Finally, the reconstructed surfaces, obtained using the vertex update
procedure discussed next, are shown in Figure 3.24d and h. These results
demonstrate that the FMED filter is effective at removing noise from surface

normals, while preserving desired features, such as crisp edges.
The vertex positions can be reconstructed based on the smoothed surface
normals using the least-square error (LSE) method proposed in Reference 44
and adopted in Reference 43. Under this methodology, the vertex positions
are updated as 44

v

x i ← n f n t f (

+ λ − )

x i x j x i , (3.18)

j ∈ i ∗ f ∈ F ij

where x i and x i are the current and updated vertex position vectors, respec-
tively, and n f is the surface normal of face f . Also, i ∗ denotes the neighbor-
hood of vertex i , i.e., the set of vertices that are connected to vertex i by an
edge, and F ij denotes the set of faces that contains the edge . For compar-

ison, Figure 3.25 and Figure 3.26 show the surface smoothing results using
Taubin's 44 and the FMED v -based methods operating on a cube and fan disk.
Taubin's method diffuses edges and corners. Furthermore, this method signif-
icantly reduces the size of triangles at edges and corners. The FMED v method,
in contrast, not only removes most of the noise, but also preserves features

{ i, j }