This article describes shape analysis to analyze and process geometric
shape
A shape or figure is a graphics, graphical representation of an object or its external boundary, outline, or external Surface (mathematics), surface, as opposed to other properties such as color, Surface texture, texture, or material type.
A pl ...
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 el ...
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 any surface of an object (inanimate or living) in three dimensions via specialized software by manipulating edges, vertices, an ...
). However, other volume based representations (e.g.
constructive solid geometry
Constructive solid geometry (CSG; formerly called computational binary solid geometry) is a technique used in solid modeling. Constructive solid geometry allows a modeler to create a complex surface or object by using Boolean operators to combi ...
) or point based representations (
point cloud
Point or points may refer to:
Places
* Point, Lewis, a peninsula in the Outer Hebrides, Scotland
* Point, Texas, a city in Rains County, Texas, United States
* Point, the NE tip and a ferry terminal of Lismore, Inner Hebrides, Scotland
* Point ...
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 data on its shape and possibly its appearance (e.g. color). The collected data can then be used to construct digital 3D modelling, 3D models.
A 3D scanner can ...
) 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 recogn ...
transform).
Application fields
Shape analysis is used in many application fields:
*
archeology
Archaeology or archeology is the scientific 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 landscap ...
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 constructing building ...
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 rev ...
to understand shape changes related to illness or aid
surgical planning
Surgical planning is the preoperative method of pre-visualising a surgical intervention, in order to predefine the surgical steps and furthermore the bone segment navigation in the context of computer-assisted surgery.
The surgical planning is mos ...
*
virtual environments
Virtual reality (VR) is a Simulation, simulated experience that employs pose tracking and 3D near-eye displays to give the user an immersive feel of a virtual world. Applications of virtual reality include entertainment (particularly video g ...
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 capable of matching a human face from a digital image or a video frame against a database of faces. Such a system is typically employed to authenticate users through ID verification services, and wo ...
*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
Moment or Moments may refer to:
* Present time
Music
* The Moments, American R&B vocal group Albums
* ''Moment'' (Dark Tranquillity album), 2020
* ''Moment'' (Speed album), 1998
* ''Moments'' (Darude album)
* ''Moments'' (Christine Guldbrand ...
or
spherical harmonic
In mathematics and 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.
Since the spherical harmonics form a ...
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'' ( el, Προκρούστης) refers to a bandit from Greek mythology who made his victims fi ...
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'' mea ...
. 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 (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of colors i ...
(see also
spectral shape analysis Spectral shape analysis relies on the spectrum (eigenvalues and/or eigenfunctions) of the Laplace–Beltrami operator to compare and analyze geometric shapes. Since the spectrum of the Laplace–Beltrami operator is invariant under isometries, it ...
).
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 recogn ...
or the
Reeb graph
A Reeb graphY. Shinagawa, T.L. Kunii, and Y.L. Kergosien, 1991. Surface coding based on Morse theory. IEEE Computer Graphics and Applications, 11(5), pp.66-78 (named after Georges Reeb by René Thom) is a mathematical object reflecting the evoluti ...
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
*
List of geometric shapes
Lists of shapes cover different types of 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 of two- ...
*
Spectral shape analysis Spectral shape analysis relies on the spectrum (eigenvalues and/or eigenfunctions) of the Laplace–Beltrami operator to compare and analyze geometric shapes. Since the spectrum of the Laplace–Beltrami operator is invariant under isometries, it ...
*
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 compu ...
*
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, ge ...
*
Topological data analysis
In applied mathematics, topological based 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 challengin ...
*
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