5.4 Feature Descriptors

Given a set of feature points (or, in the case of a dense descriptor, all the points on the shape), a local descriptor is then computed. There exists a plethora of different shape descriptor algorithms, which depend very much on the representation in which the shape is given (e.g. mesh or point cloud), the kind of information available and its quality (e.g. sampling resolution), and the application in mind. A descriptor can be characterized by its (i) “informativity”, i.e., the information content that can be used for shape discrimination; (ii) invariance to some class of shape transformations, such as deformations or noise, (iii) computational complexity, (iv) compactness of descriptor and complexity of comparison of two descriptors (e.g. in shape matching applications). In addition, if the descriptor is used in combination with a feature detector, its sensitivity to feature location (detector repeatability) might be important. There are many tradeoffs between these properties that can be made in feature-based shape analysis applications.

5.4.1 A Taxonomy

Descriptors can be categorized as geometric or photometric (or both, referred to as hybrid), depending whether they rely only on the 3D geometry of the shape, or also make use of the texture. Some photometric descriptors can be adapted to work with geometric information, where some geometric property (e.g. curvature) is used in place of the texture [87]. A wide variety of geometric quantities such as local patches [58], local moments [24] and volume [34], spherical harmonics [73], and contour and edge structures [45, 63] trying to emulate comparable features in images, can be used for geometric descriptors.

Multiscale descriptors (e.g. [20, 68, 81]) look at the shape at multiple levels of resolution, thus capturing different properties manifested at these scales. This is opposed to single scale or scalar descriptors, such as conformal factor [6].

Descriptors which are not altered by global scaling of the shape are called scaleinvariant. Such an invariance can in some cases be achieved by shape normalization; a better approach is to build scale-invariance into the descriptor construction.

Because typically a descriptor operates locally around the feature point, feature descriptors are usually not very sensitive to non-rigid deformations of the shape. Nevertheless, there exist several geometric descriptors which are based on intrinsic properties of the manifold and thus theoretically invariant to isometric deformations by construction. Examples of intrinsic descriptors include histograms of local geodesic distances [10, 66], conformal factors [6], some settings of [87], and heat kernels [20, 81]. Such descriptors are called intrinsic and also isometry-invariant or bending-invariant.

Another type of transformation, of interest in practical applications, is changes in topology, manifested as the presence of holes, missing parts, or changes in connectivity. Descriptors insensitive to such changes (typically, a simpler case of point-wise connectivity change) are referred to as topology-invariant.

aInvolving mesh rasterization

bAssuming geodesic distances. Different invariance properties can be achieved using diffusion or commute-time distances

cPoint-wise connectivity changes have only a local effect and do not propagate to distant descriptors

dIf photometric texture is used; in general, depending on the texture choice

eTriangulation-dependent

fDefined for shapes with fixed topology (e.g. watertight)

gCan be made approximately invariant using small support

hThe use of photometric information can reduce the sensitivity to topological noise compared to HKS

iCan be made scale invariance as in [44]

Finally, some authors [34] make a distinction between high-dimensional (or rich) and low-dimensional descriptors. The former refers to descriptors providing a fairly detailed description of the shape properties around the point such as [5, 39], while the latter computes only a few values per point and typically are curvature-like quantities such as shape index [42] and curvedness [43]. We find this division somewhat misleading, as there is no direct relation between the descriptor “richness” and dimensionality (recent works in computer vision on descriptor hashing and dimensionality reduction [79] demonstrate that rich descriptors such as SIFT can be compactly represented in much lower dimensions without losing much information). The question whether the “richness” of a descriptor is sufficient depends in general on the application and the data.

Table 5.2 summarizes the theoretical properties of known descriptors, some of which are detailed in what follows. The invariance properties of many descriptors were evaluated quantitatively in the SHREC robust feature detection and description

benchmark [12], testing the descriptor variability under simulated transformations of different types (non-rigid bending, different types of noise, holes, etc.).

We devote particular attention in this section to different varieties of the recently introduced heat kernel signatures, which we consider to be one of the most versatile descriptors currently available, possessing provable invariance properties, as well as a promising and interesting field for future research.

5.4.2 Curvature-Based Descriptors (HK and SC)

The simplest and perhaps earliest shape descriptors based on curvature (also referred to as HK descriptors) were introduced by Besl [8, 9]. The combination of the mean curvature H = 12 (κ1 + κ2) and the Gaussian curvature K = κ1 · κ2 allow the classification of the type of a local surface patch as saddle valley (K, H < 0), saddle ridge (K < 0, H > 0), concave or convex cylinder (K = 0, H < 0 and K = 0, H > 0, respectively), concave or convex ellipsoid (K > 0, H < 0 and K > 0, H > 0, respectively), or plane (K = H = 0). The values of H and K depend on the shape scale.

Koenderink and van Doorn [43] defined a different descriptor (referred to as SC) which decouples the type and strength of local shape curvature as follows: The

lar local shape type is at a point. Planar shapes have indeterminate shape index and can be determined from the curvedness C = 0.

Both the HK and SC descriptors make use of the mean curvature, which is not intrinsic and hence not deformation invariant.

5.4.3 Spin Images

The spin image descriptor [2, 3, 39] represents the neighborhood of a point on a shape by fitting an oriented coordinate system at the point. The local system of cylindrical coordinates at point x is defined using the normal and tangent plane: the radial coordinate α defined as the perpendicular distance to the line through the surface normal n(x), and the elevation coordinate β, defined as the signed perpendicular distance to the tangent plane. The cylindrical angular coordinate is omitted because it cannot be defined robustly and unambiguously on many surface patches, such as those where the curvature is the similar in all directions.

A spin image is a histogram of points in the support region represented in α, β coordinates. The bins can be in linear or logarithmic scale. The support region is defined by limiting the range of the values of α and β (thus looking at points y within some distance from x) and requiring that cos−1 n(x), n(y) < ε (limiting