This article describes shape analysis to analyze and process geometric
shape
A shape is a graphics, graphical representation of an object's form or its external boundary, outline, or external Surface (mathematics), surface. It is distinct from other object properties, such as color, Surface texture, texture, or material ...
s.
Description
''Shape analysis'' is the (mostly) 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 a
boundary representation
In solid modeling and computer-aided design, boundary representation (often abbreviated B-rep or BREP) is a method for representing a 3D shape by defining the limits of its volume. A solid is represented as a collection of connected surface ...
is used to describe the object with its boundary (usually the outer shell, see also
3D model
In 3D computer graphics, 3D modeling is the process of developing a mathematical coordinate-based representation of a surface of an object (inanimate or living) in three dimensions via specialized software by manipulating edges, vertices, and ...
). However, other volume based representations (e.g.
constructive solid geometry) or point based representations (
point cloud
A point cloud is a discrete set of data Point (geometry), points in space. The points may represent a 3D shape or object. Each point Position (geometry), position has its set of Cartesian coordinates (X, Y, Z). Points may contain data other than ...
s) can be used to represent shape.
Once the objects are given, either by modeling (
computer-aided design
Computer-aided design (CAD) is the use of computers (or ) to aid in the creation, modification, analysis, or optimization of a design. This software is used to increase the productivity of the designer, improve the quality of design, improve c ...
), by scanning (
3D scanner
3D scanning is the process of analyzing a real-world object or environment to collect three dimensional data of its shape and possibly its appearance (e.g. color). The collected data can then be used to construct digital 3D models.
A 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 a ''shape 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.
A ''complete shape descriptor'' is a representation that can be used to completely reconstruct the original object (for example the
medial axis
The medial axis of an object is the set of all points having more than one closest point on the object's boundary. Originally referred to as the topological skeleton, it was introduced in 1967 by Harry Blum as a tool for biological shape reco ...
transform).
Application fields
Shape analysis is used in many application fields:
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archeology
Archaeology or archeology is the study of human activity through the recovery and analysis of material culture. The archaeological record consists of artifacts, architecture, biofacts or ecofacts, sites, and cultural landscapes. Archaeolo ...
for example, to find similar objects or missing parts
*
architecture
Architecture is the art and technique of designing and building, as distinguished from the skills associated with construction. It is both the process and the product of sketching, conceiving, planning, designing, and construction, constructi ...
for example, to identify objects that spatially fit into a specific space
*
medical imaging
Medical imaging is the technique and process of imaging the interior of a body for clinical analysis and medical intervention, as well as visual representation of the function of some organs or tissues (physiology). Medical imaging seeks to revea ...
to understand shape changes related to illness or aid
surgical planning
Surgery is a medical specialty that uses manual and instrumental techniques to diagnose or treat pathological conditions (e.g., trauma, disease, injury, malignancy), to alter bodily functions (e.g., malabsorption created by bariatric surgery ...
*
virtual environments or on the
3D model market to identify objects for copyright purposes
*security applications such as
face recognition
A facial recognition system is a technology potentially capable of matching a human face from a digital image or a Film frame, video frame against a database of faces. Such a system is typically employed to authenticate users through ID verif ...
*entertainment industry (movies, games) to construct and process geometric models or animations
*
computer-aided design
Computer-aided design (CAD) is the use of computers (or ) to aid in the creation, modification, analysis, or optimization of a design. This software is used to increase the productivity of the designer, improve the quality of design, improve c ...
and
computer-aided manufacturing
Computer-aided manufacturing (CAM) also known as computer-aided modeling or computer-aided machining is the use of software to control machine tools in the manufacturing of work pieces. This is not the only definition for CAM, but it is the most ...
to 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 to ''congruency'', meaning that congruent shapes (shapes that could be translated, rotated and mirrored) will have the same descriptor (for example
moment or
spherical harmonic
In mathematics and Outline of physical science, physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields. The tabl ...
based descriptors or
Procrustes analysis
In statistics, Procrustes analysis is a form of statistical shape analysis used to analyse the distribution of a set of shapes. The name '' Procrustes'' () refers to a bandit from Greek mythology who made his victims fit his bed either by stretch ...
operating on point clouds).
Another class of shape descriptors (called ''intrinsic'' shape descriptors) is invariant with respect to
isometry
In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' me ...
. 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 the
Laplace–Beltrami spectrum
A spectrum (: spectra or spectrums) is a set of related ideas, objects, or properties whose features overlap such that they blend to form a continuum. The word ''spectrum'' was first used scientifically in optics to describe the rainbow of co ...
(see also
spectral shape analysis).
There are other shape descriptors, such as ''graph-based'' descriptors like the
medial axis
The medial axis of an object is the set of all points having more than one closest point on the object's boundary. Originally referred to as the topological skeleton, it was introduced in 1967 by Harry Blum as a tool for biological shape reco ...
or the
Reeb graph that 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.
See also
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List of geometric shapes
Lists of shapes cover different types of geometry, geometric shape and related topics. They include mathematics topics and other lists of shapes, such as shapes used by drawing or teaching tools.
Mathematics
* List of mathematical shapes
* List ...
*
Spectral shape analysis
*
Discrete Morse theory Discrete Morse theory is a combinatorial adaptation of Morse theory developed by Robin Forman. The theory has various practical applications in diverse fields of applied mathematics and computer science, such as configuration spaces, homology com ...
*
Discrete differential geometry
Discrete differential geometry is the study of discrete counterparts of notions in differential geometry. Instead of smooth curves and surfaces, there are polygons, meshes, and simplicial complexes. It is used in the study of computer graphics ...
*
Topological data analysis
In applied mathematics, topological data analysis (TDA) is an approach to the analysis of datasets using techniques from topology. Extraction of information from datasets that are high-dimensional, incomplete and noisy is generally challenging. TDA ...
*
Equidimensional
References
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External links
The Princeton Shape Benchmark*
Shape Analysis using the Laplace-Beltrami spectrum*{{cite journal , first=S. , last=Loncaric , title=A survey of shape analysis techniques , journal=Pattern Recognition , volume=31 , issue=8 , pages=983–1001 , date=1998 , doi=10.1016/S0031-2023(97)00122-2 , bibcode=1998PatRe..31..983L , url=
Image processing
Digital geometry
Topology
Differential geometry