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on purely geometric measurements without consider-
ing human visual perception; thus they are not cor-
related with the human perception. On the other
hand, perceptually-based metrics incorporate mech-
anisms of HVS. Comprehensive surveys on general
mesh quality assessment methods can be found in
(Bulbul et al., 2011) and (Lavou
´
e and Mantiuk, 2015);
whereas surveys on perceptual quality metrics are
presented in (Lin and Jay Kuo, 2011) and (Corsini
et al., 2013).
Image-based perceptual metrics operate in 2D
image space by using rendered images of the 3D
mesh while evaluating the visual quality. These met-
rics generally employ HVS models such as Contrast
Sensitivity Function (CSF), which maps spatial fre-
quency to visual sensitivity. Most common image
quality metric is Visible Difference Prediction (VDP)
method which produces a 2D local visible distortions
map, given reference and test images (Daly, 1992).
Similarly, Visual Equivalence Detector method out-
puts a visual equivalence map which demonstrates
the equally perceived regions of two images (Rama-
narayanan et al., 2007).
Curvature and roughness of a surface are widely
employed for describing surface quality. GL1 (Karni
and Gotsman, 2000) and GL2 (Sorkine et al., 2003)
are roughness-based metrics that use Geometric
Laplacian of the mesh vertices. Lavoue et al. (Lavou
´
e
et al., 2006) measure structural similarity between
two mesh surfaces by using curvature for extracting
structural information. This metric is improved with
a multi-scale approach in (Lavou
´
e, 2011). Two def-
initions of surface roughness are utilized for deriv-
ing two error metrics called 3DW PM1 and 3DWPM2
(Corsini et al., 2007). Another metric called FMPD is
also based on local roughness derived from Gaussian
curvature (Wang et al., 2012). Curvature tensor differ-
ence of two meshes is used for measuring the visible
errors between two meshes (Torkhani et al., 2014). A
novel roughness-based perceptual error metric, which
incorporates structural similarity, visual masking, and
saturation effect, is proposed by Dong et al. (Dong
et al., 2015). There are also recent studies that lever-
age machine learning methods for mesh quality as-
sessment (Yildiz et al., 2018). A metric specific to the
validation of human body models is also proposed in
(Singh and Kumar, 2017).
The literature survey shows that most of the ex-
isting visual quality metrics do not take the temporal
effects into account. Moreover, they are mostly con-
cerned with the global quality of the meshes rather
than the local visibility of distortions. These issues
were already addressed by (Yildiz and Capin, 2017).
The main objective of this study is to remove the ne-
cessity for a spatiotemporal volume in that method;
thus making the pipeline fully object-space.
3 APPROACH
In this mesh-based approach, almost the same steps
in (Yildiz and Capin, 2017) exist with several adap-
tations for 3D. The method is applied on the mesh
vertices, not on the spatiotemporal volume represen-
tation. However, this introduces a restriction for the
reference and test meshes to have the same number of
vertices, since the computations are done per vertex.
The steps of the method are displayed in Figure 1.
Frames for reference and test animations go through
the same processing pipeline and the difference be-
tween these results gives us the per vertex visible dif-
ferences prediction map. Details of each step are ex-
plained below.
3.1 Preprocessing
In this step, illumination calculation and vertex veloc-
ity estimation are performed as in the spatiotemporal
volume approach. Instead of the spatiotemporal vol-
ume calculation, Manifold Harmonics Basis (MHB)
are computed and stored to feed the Channel Decom-
position step of the proposed approach.
Illumination Calculation. Vertex shades are com-
puted using Phong reflection model with only diffuse
and ambient components. Most of the user experi-
ments for measuring the visual quality of 3D meshes
in the literature, use such a simple shading scheme.
Calculation of MHBs. Calculation of MHBs is a
costly operation since it requires eigen-decomposition
of the mesh Laplacian. Fortunately, once they are
computed; there is no need to recalculate them.
For a triangle mesh of n vertices, a function basis
H
k
, called MHB is calculated. The k
th
element of the
MHB is a piecewise linear function with values H
k
i
defined at i
th
vertex of the surface, where k = 1...m
and i = 1...n (Vallet and L
´
evy, 2008). MHB is com-
puted as the eigenvectors of discrete Laplacian of
¯
∆
whose coefficients are given in Eq. 1.
¯
∆
i j
= −
cotβ
i j
+ cotβ
0
i j
q
|v
∗
i
||v
∗
j
|
(1)
where β
i j
and β
0
i j
are two angles opposite to edge de-
fined by vertices i and j, v
∗
refers to the circumcentric
dual of simplex v, and |.| denotes the simplex volume.
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