3D Shape Analysis in a Nutshell - Mathematical Tools for Shape Analysis and Description - page 9

first step in the pipeline of other shape analysis solutions, such as shape description and re-
trieval. For example, in chapters 4 and 8 we will see how differential geometry provides tools
for the identification of significant morphological features: curvature can be used to char-
acterize surface features such as tips, pits, mounts, and blends, whereas the critical points of
the height function can be used in digital terrain modeling to identify salient features such
as peaks and valleys.
• Segmentation: How can we decompose a 3D model into its significant parts? Shape seg-
mentation serves to decompose an object into parts which are perceptually relevant, or which
have a particular meaning in a given context, for example, the base, shaft and capital in a col-
umn in 3D cultural heritage applications. Again, geometric modeling can provide tools to
derive meaningful shape segmentations. For example, in chapter 5 we will see how spectral
analysis gives the basis to define a library of deformation-invariant and multi-scale shape
segmentations which are well-aligned with perceptual properties and intuition.
• Semantic labeling (annotation): How can we assign meaningful textual information to
objects and their parts? Semantic annotation is the automatic, or semi-automatic, label-
ing of objects (or parts of objects) with a tag describing their content. It requires deriving
high-level information from low-level properties. In chapters 4 and 12 we will see how the
analysis of the geometry of parts, together with the study of topological relationships be-
tween shape parts (such as adjacency), gives cues to the automatic recognition and labeling
of functional parts of man-made 3D shapes, for example, the tagging of vase parts with
labels such as container, handle, base, tip.
• Registration (alignment, correspondence finding): How can we align features of 3D
models? Finding correspondences between discrete sets of points on different models has
applications in object tracking, surface completion, statistical shape modeling, symmetry
analysis, shape interpolation, attribute transfer. It can also be part of a shape matching
pipeline. In chapter 6 , we will show how the study of transformations between mathemati-
cal spaces (isometries and Möebius transformations) will give rise to an efficient algorithm
for discovering dense sets of point correspondences between deformable objects.
• Description: How can we communicate what a 3D object looks like, or what is its mean-
ing? Shape descriptions aim to find concise yet informative signatures of shape models.
ese signatures are machine-understandable indexes to the informative content of 3D
models. ey enable shape matching, retrieval, and classification. For example, in chap-
ter 5 , it will be shown how a concise shape signature can be derived from the solution of
the heat equation, which represents the amount of heat transferred from a point to another
point in a given amount of time. e simulation of physical processes on surfaces and the
use of spectral analysis techniques produces a signature which is multi-scale, deformation
invariant, and robust to noise.