Shape analysis (digital geometry)

This article describesshape analysisto analyze and process geometricshapes.

Description

Shape analysisis the (mostly)^{[clarification needed]}automatic analysis of geometric shapes, for example using a computer to detect similarly shaped objects in a database or parts that fit together. For a computer to automatically analyze and process geometric shapes, the objects have to be represented in a digital form. Most commonly aboundary representationis used to describe the object with its boundary (usually the outer shell, see also3D model). However, other volume based representations (e.g.constructive solid geometry) or point based representations (point clouds) can be used to represent shape.

Once the objects are given, either by modeling (computer-aided design), by scanning (3D scanner) or by extracting shape from 2D or 3D images, they have to be simplified before a comparison can be achieved. The simplified representation is often called ashape descriptor(or fingerprint, signature). These simplified representations try to carry most of the important information, while being easier to handle, to store and to compare than the shapes directly. Acomplete shape descriptoris a representation that can be used to completely reconstruct the original object (for example themedial axistransform).

Application fields

Shape analysis is used in many application fields:

archeologyfor example, to find similar objects or missing parts
architecturefor example, to identify objects that spatially fit into a specific space
medical imagingto understand shape changes related to illness or aidsurgical planning
virtual environmentsor on the3D model marketto identify objects for copyright purposes
security applications such asface recognition
entertainment industry (movies, games) to construct and process geometric models or animations
computer-aided designandcomputer-aided manufacturingto process and to compare designs of mechanical parts or design objects.

Shape descriptors

Shape descriptors can be classified by their invariance with respect to the transformations allowed in the associated shape definition. Many descriptors are invariant with respect tocongruency,meaning that congruent shapes (shapes that could be translated, rotated and mirrored) will have the same descriptor (for examplemomentorspherical harmonicbased descriptors orProcrustes analysisoperating on point clouds).

Another class of shape descriptors (calledintrinsicshape descriptors) is invariant with respect toisometry.These descriptors do not change with different isometric embeddings of the shape. Their advantage is that they can be applied nicely to deformable objects (e.g. a person in different body postures) as these deformations do not involve much stretching but are in fact near-isometric. Such descriptors are commonly based on geodesic distances measures along the surface of an object or on other isometry invariant characteristics such as theLaplace–Beltrami spectrum(see alsospectral shape analysis).

There are other shape descriptors, such asgraph-baseddescriptors like themedial axisor theReeb graphthat capture geometric and/or topological information and simplify the shape representation but can not be as easily compared as descriptors that represent shape as a vector of numbers.

From this discussion it becomes clear, that different shape descriptors target different aspects of shape and can be used for a specific application. Therefore, depending on the application, it is necessary to analyze how well a descriptor captures the features of interest.

References

De Floriani, Leila;Spagnuolo, Michela (2007).Shape Analysis and Structuring.Springer.ISBN 978-3540332640.
Delfour, Michel C.; Zolésio, J.P. (2001).Shapes and Geometries: Analysis, Differential Calculus, and Optimization.SIAM.ISBN 978-0898714890.
Application of Shape Analysis.9-ème Colloque Franco-Roumain de Mathématiques Appliquées: 28 août–2 septembre 2008, Braşov, Roumanie: livre des résumés. University of Transilvania. 2008.ISBN 978-973-598-341-3.

External links

The Princeton Shape Benchmark
Kazhdan, M.; Funkhouser, T.; Rusinkiewicz, S. (2003)."Rotation invariant spherical harmonic representation of 3D shape descriptors"(PDF).SGP '03: Proceedings of the 2003 Eurographics/ACM SIGGRAPH symposium on Geometry processing.pp.156–164.doi:10.2312/SGP.SGP03.156-165/156-165(inactive 1 November 2024).ISBN 978-1-58113-687-6.{{cite book}}:CS1 maint: DOI inactive as of November 2024 (link)
Shape Analysis using the Laplace-Beltrami spectrum
Loncaric, S. (1998). "A survey of shape analysis techniques".Pattern Recognition.31(8):983–1001.Bibcode:1998PatRe..31..983L.doi:10.1016/S0031-2023(97)00122-2.