Shape analysis
Encyclopedia
This article describes shape analysis to analyze and process geometric shapes
.
The shape analysis described here is related to the statistical analysis of geometric shapes
, to shape matching and shape recognition. It applies purely to the geometry of an object, not to the structural analysis
that deals with predicted behaviour of mechanical parts.
is used to describe the object with its boundary (usually the outer shell, see also 3D 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 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
transform).
or spherical harmonic
based descriptors or Procrustes analysis
operating on point clouds).
Another class of shape descriptors (called intrinsic shape descriptors) is invariant with respect to isometry
. 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
.
There are other shape descriptors, such as graph-based descriptors like the medial axis
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.
Shape
The shape of an object located in some space is a geometrical description of the part of that space occupied by the object, as determined by its external boundary – abstracting from location and orientation in space, size, and other properties such as colour, content, and material...
.
The shape analysis described here is related to the statistical analysis of geometric shapes
Statistical shape analysis
Statistical shape analysis is a geometrical analysis from a set of shapes in which statistics are measured to describe geometrical properties from similar shapes or different groups, for instance, the difference between male and female Gorilla skull shapes, normal and pathological bone shapes, etc...
, to shape matching and shape recognition. It applies purely to the geometry of an object, not to the structural analysis
Structural analysis
Structural analysis is the determination of the effects of loads on physical structures and their components. Structures subject to this type of analysis include all that must withstand loads, such as buildings, bridges, vehicles, machinery, furniture, attire, soil strata, prostheses and...
that deals with predicted behaviour of mechanical parts.
What is shape analysis?
Shape analysis is the mainly 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 representationBoundary representation
In solid modeling and computer-aided design, boundary representation—often abbreviated as B-rep or BREP—is a method for representing shapes using the limits...
is used to describe the object with its boundary (usually the outer shell, see also 3D model). However, other volume based representations (e.g. constructive solid geometry
Constructive solid geometry
Constructive 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 combine objects...
) or point based representations (point clouds
Point cloud
A point cloud is a set of vertices in a three-dimensional coordinate system. These vertices are usually defined by X, Y, and Z coordinates, and typically are intended to be representative of the external surface of an object....
) can be used to represent shape.
Once the objects are given, either by modeling (computer-aided design
Computer-aided design
Computer-aided design , also known as computer-aided design and drafting , is the use of computer technology for the process of design and design-documentation. Computer Aided Drafting describes the process of drafting with a computer...
), by scanning (3D scanner
3D scanner
A 3D scanner is a device that analyzes a real-world object or environment to collect data on its shape and possibly its appearance . The collected data can then be used to construct digital, three dimensional models....
) 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
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 by Blum as a tool for biological shape recognition....
transform).
Application fields
Shape analysis is used in many application fields:- archeology for example, to find similar objects or missing parts
- architectureArchitectureArchitecture is both the process and product of planning, designing and construction. Architectural works, in the material form of buildings, are often perceived as cultural and political symbols and as works of art...
for example, to identify objects that spatially fit into a specific space - medical imagingMedical imagingMedical imaging is the technique and process used to create images of the human body for clinical purposes or medical science...
to understand shape changes related to illness or aid surgical planningSurgical planningThe 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.... - virtual environmentsVirtual realityVirtual reality , also known as virtuality, is a term that applies to computer-simulated environments that can simulate physical presence in places in the real world, as well as in imaginary worlds...
or on the 3D model market to identify objects for copyright purposes - security applications such as face recognition
- entertainment industry (movies, games) to construct and process geometric models or animations
- computer-aided designComputer-aided designComputer-aided design , also known as computer-aided design and drafting , is the use of computer technology for the process of design and design-documentation. Computer Aided Drafting describes the process of drafting with a computer...
and computer-aided manufacturingComputer-aided manufacturingComputer-aided manufacturing is the use of computer software to control machine tools and related machinery in the manufacturing of workpieces. This is not the only definition for CAM, but it is the most common; CAM may also refer to the use of a computer to assist in all operations of a...
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 momentMoment (mathematics)
In mathematics, a moment is, loosely speaking, a quantitative measure of the shape of a set of points. The "second moment", for example, is widely used and measures the "width" of a set of points in one dimension or in higher dimensions measures the shape of a cloud of points as it could be fit by...
or spherical harmonic
Spherical Harmonic
Spherical Harmonic is a science fiction novel from the Saga of the Skolian Empire by Catherine Asaro. It tells the story of Dyhianna Selei , the Ruby Pharaoh of the Skolian Imperialate, as she strives to reform her government and reunite her family in the aftermath of a devastating interstellar...
based descriptors or Procrustes analysis
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 stretching their limbs or cutting them off.To compare the shape of...
operating on point clouds).
Another class of shape descriptors (called intrinsic shape descriptors) is invariant with respect to isometry
Isometry
In mathematics, an isometry is a distance-preserving map between metric spaces. Geometric figures which can be related by an isometry are called congruent.Isometries are often used in constructions where one space is embedded in another space...
. 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
Laplace-Beltrami operator
In differential geometry, the Laplace operator, named after Pierre-Simon Laplace, can be generalized to operate on functions defined on surfaces in Euclidean space and, more generally, on Riemannian and pseudo-Riemannian manifolds. This more general operator goes by the name Laplace–Beltrami...
spectrum
Spectrum (functional analysis)
In functional analysis, the concept of the spectrum of a bounded operator is a generalisation of the concept of eigenvalues for matrices. Specifically, a complex number λ is said to be in the spectrum of a bounded linear operator T if λI − T is not invertible, where I is the...
.
There are other shape descriptors, such as graph-based descriptors like the medial axis
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 by Blum as a tool for biological shape recognition....
or the Reeb graph
Reeb graph
In Morse theory, a branch of mathematics, a Reeb graph of a scalar function describes the connectivity of its level sets.Reeb graphs are named after Georges Reeb....
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
- Discrete Morse theory
- Discrete differential geometryDiscrete differential geometryDiscrete 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...
- Topological data analysisTopological data analysisTopological data analysis is a new area of study aimed at having applications in areas such as data mining and computer vision.The main problems are:# how one infers high-dimensional structure from low-dimensional representations; and...
- EquidimensionalEquidimensionalEquidimensional is an adjective applied to objects that have nearly the same size or spread in multiple directions. As a mathematical concept, it may be applied to objects that extend across any number of dimensions...