Harris 3D: a robust extension of the Harris operator for interest point detection on 3D meshes 965
For 3D meshes, several approaches have been proposed,
most of which have tried to extend the detectors proposed
for images. After the SIFT method proposed by Lowe [17],
a number of extensions have been presented which use
Difference-of-Gaussians (DoG) as interest point detector.
Castellani et al. [3] applied the DoG detector over vertices
in scale space obtained with successive decimations of the
original shape. Vertices with high response in their DoG op-
erator are selected as interest points. In the same way, Zou
et al. [27] proposed to build a geodesic scale space, and sub-
sequently to apply DoG detector on that space for detect-
ing interest points on a surface. Also, Zaharescu et al. [26]
assumed that the vertices of an 3D object have associated
information such as curvature or photometric properties.
Defining a discrete Difference-of-Gaussians operator, the
authors applied this operator on the function defined by
the associated information over a manifold. This approach
showed good results in matching of 3D models sequences.
On the other hand, the geometric diffusion theory can be
used for detecting interest points on surfaces. The diffusion
process reveals interesting characteristics from the intrinsic
geometry of a surface which can be exploited to detect out-
standing structures. As a 3D surface property related to the
diffusion process on a manifold, the Laplace–Beltrami op-
erator has been also used to detect interest points. Hu and
Hua [9] defined the geometric energy of a vertex as function
of the eigenvalues and eigenvectors of the Laplace–Beltrami
spectrum of a given object. Vertices where the energy is a
maximum are considered as interest points. In addition, the
energy provides the scale where the selected vertices are in-
teresting. The selected interest points were used in a match-
ing task with promising results. On the other hand, Sun et al.
[24] defined the Heat Kernel Signature as a temporal domain
restriction of the Heat Kernel on a manifold, which is re-
lated to the Laplace–Beltrami spectrum. In 3D meshes, each
vertex has an associated signature. A vertex is selected as
interest point, when for large time values, its signature has a
maximum with respect to the neighbor vertices.
Similarly, Zou et al. [28] proposed to build a scale space
of the surface geometry via the surface Ricci flow which
satisfies a set of desired properties for a multi-scale repre-
sentation. The authors applied the Ricci flow over a metric
of the surface based on edge lengths. The scale space is rep-
resented as a matrix with curvature values calculated from
the set of diffused metrics. Then, the Laplacian of a vertex
is computed using the cotangent schema using the curvature
as associated values. A vertex is considered as an interest
point if it is an extreme of the Laplacian in the 1-ring neigh-
borhood and neighboring scales.
Also, curvature-based methods have been proposed.
Gelfand et al. [5] described an interest point detector based
on a new descriptor called the integral volume descriptor.
For each vertex, the amount of volume in the intersection
of a ball centered in the vertex and the 3D object describes
an interesting local measure. The authors showed that this
quantity is closely related to the curvature in the vertex. Ver-
tices with uncommon integral values are selected as interest
points. Also using curvature in vertices, Ho and Gibbins [8]
suggested a measure called the curvedness measure in or-
der to describe the geometric information in a vertex. The
curvedness is calculated from the principal curvatures of a
vertex. This measure can be calculated in different scales by
selecting different neighborhood sizes which are used to fit
quadratic patches over which curvatures are computed. Ver-
tices with extremal values in its curvedness, with respect to
neighboring points and scales, are selected as interest points.
Differently, Liu et al. [15] proposed a Monte Carlo strat-
egy to select a random set of points on a surface with each
point having the same probability to be chosen. These points
were used in partial shape retrieval. The assumption be-
hind this proposal is that the vertices of a shape are sam-
ples of the original surface and the tasks that use them can
be affected by shape tessellations. Similarly, Shilane and
Funkhouser [22] considered random points on a 3D surface,
selecting only those points that contribute to improve the re-
trieval performance. With a training phase, it was possible to
assign a predicted distinction value to each selected point in
the 3D collection and thus, using that values to assign new
ones to points of a new shape.
As another approach, the mesh saliency defined by Lee
et al. [14] has proven to be a robust feature to many 3D ap-
plications. The process to compute the mesh saliency of a
3D object begins calculating a Gaussian-weighted average
of the mean curvature on a surface. Each vertex in an ob-
ject is thus associated with the difference of such average
in different scales, which is the saliency of that vertex. Ver-
tex with the highest saliency can be considered as interest
points.
Conformal parameterization has also been used to pro-
pose interest points detectors. Methods based on conformal
parameterization [10, 20] transform a 3D surface into a 2D
parameterization that can be seen as an image. A 3D to 2D
mapping is said to be conformal if angles are preserved.
Once an image is computed, interest points can be detected
on it, and subsequently these are mapped back to the 3D
domain.
Finally, Mian et al. [18] related the repeatability of key-
points (extracted from partial views of an object) with a
quality measure based upon principal curvatures.
3 Interest points detection
Harris and Stephens [
7] proposed an interest points detector
for images. Their method is a popular technique due to its
strong invariance to rotation, scale, illumination variation,