Symmetry in 3D Geometry: Extraction and Applications

TL;DR: This report surveys and classifies recent developments in symmetry detection, elucidating the key similarities and differences between existing methods to gain a better understanding of a fundamental problem in digital geometry processing and shape understanding in general.

Abstract: The concept of symmetry has received significant attention in computer graphics and computer vision research in recent years. Numerous methods have been proposed to find, extract, encode and exploit geometric symmetries and high-level structural information for a wide variety of geometry processing tasks. This report surveys and classifies recent developments in symmetry detection. We focus on elucidating the key similarities and differences between existing methods to gain a better understanding of a fundamental problem in digital geometry processing and shape understanding in general. We discuss a variety of applications in computer graphics and geometry processing that benefit from symmetry information for more effective processing. An analysis of the strengths and limitations of existing algorithms highlights the plenitude of opportunities for future research both in terms of theory and applications.

Summary

1. Introduction

• The mathematical sciences particularly exhibit order, symmetry, and limitation; and these are the greatest forms of the beautiful.
• This abundance of symmetry in the natural world has inspired mankind from its origins to incorporate symmetry in the design of tools, buildings, artwork, or even music.
• Geometric data, acquired via scanning or modeled from scratch, is traditionally represented as a collection of lowlevel primitives, e.g., point clouds, polygon meshes, NURBS patches, etc., without explicit encoding of their high-level structure.
• The authors first introduce basic mathematical terms and present various high-level criteria to organize existing work into a set of categories, emphasizing their similarities and differences (see Table 1).
• The main part of this paper surveys different algorithms for symmetry detection (Sections 4-7).

2. Classical Theory

• Symmetry is a general concept in mathematics [Wey52]; broadly speaking, a symmetry preserves a certain property (e.g., geometric similarity) of an object under some operation applied to the object.
• In the context of geometry, the authors will consider geometric transformations as the symmetry operations, such as reflections, translations, rotations, or combinations thereof.
• It follows that all integer multiples of this rotation are also symmetries of the triangle.
• Together, these transformations form the dihedral group D3 consisting of six elements, three rotations (of which one is the identity transformation) and three reflections.
• If rep- etition occurs in two different directions, seventeen distinct groups are possible, denoted as wallpaper groups.

3. Classification

• This survey investigates algorithmic paradigms for extracting symmetries and relations in 3D geometric data.
• The authors first formalize the problem of symmetry detection and present several classification categories to highlight the similarities and differences of existing symmetry detection algorithms.
• While a number of common shapes exhibit global symmetries , self-similarities often occur only on parts of a shape.
• For this purpose the authors define a distance function d(M , T (M)) that measures the distance between the two shapes M and T (M).
• For a partial and approximate symmetry to be meaningful, the size of the symmetric subset needs to be sufficiently large, and the approximation threshold sufficiently small.

4. Complexity

• The authors consider a simple “brute-force” baseline algorithm in order to understand the complexity of the symmetry detection problem and to motivate the need for more sophisticated algorithms discussed in the subsequent sections.
• On the other hand, since the space of transformations is high-dimensional (e.g., 6-dimensional for rigid transforms), naïvely sampling the space can be highly inefficient.
• For each correspondence assignment, the authors verify if M is globally symmetric under the action of the corresponding transformation T , i.e., they evaluate if d(M,T (M))< ε. The verification test requires O(n) distance computations, with each distance computation taking O(1) (for example using an ε-grid as search data structure, and excluding degenerate cases).
• If reliable surface normal information is available, two corresponding point pairs are sufficient to fix a rigid mapping that matches the local surface orientation.
• The O(n) cost for verification can be addressed by random sampling:.

5. Global Symmetry Detection

• The main focus of this survey is on partial symmetry detection methods (see Section 6).
• Theoretical characterization of symmetry detection algorithms has been a topic of interest in computational geometry.
• Atallah et al. [Ata85] propose an O(n logn) optimal algorithm for enumerating all reflective symmetries of a planar figure consisting of segments, circles, and points.
• Since this condition on the gradient can produce false positives, they verify the candidate solutions in the last step of the algorithm.
• An alternative strategy has been proposed by Ovsjanikov et al. [OSG08], who use eigenfunctions of Laplace Beltrami operators to transform intrinsic symmetries of a shape into Euclidean symmetries in a signature space.

6. Partial Symmetry Detection

• The authors now discuss five main approaches for partial symmetry detection:.
• At an abstract level, these methods share many similarities, even though the individual algorithmic components are different.
• Feature selection typically uses local shape descriptors to significantly reduce the search space by considering geometric properties of the shape that are invariant under the considered symmetry transformations.
• Under intrinsically isometric transformations, Gaussian curvatures are preserved at the surface point and hence are commonly used as local features.
• The extracted symmetry candidates are finally verified and refined in the spatial domain.

6.1. Geometric Hashing

• A fundamental technique often employed for indexing is geometric hashing.
• Curvature derivatives at vertices are then computed using the coefficients of the fitted quadratic patches and the mesh vertices are sorted according to the magnitude of the Gaussian curvature, i.e., the product of principal curvature values.
• They aggregate transformations based on the following observation: several transformation families have only a few degrees of freedom and are uniquely determined by a small number of correct corresponding point pairs.
• In the geometric hashing step, they first bring each query patch to a canonical position by indexing a small number of rotation-invariant features.
• Bi = {bij} that vote for the current cell.

6.2. Transformation Space Voting

• Mitra et al. [MGP06] propose a method for computing pairwise partial and approximate symmetries based on accumulating local symmetry votes in a symmetry transformation space.
• The reflective plane can be represented as a point in a 3D space consisting of two angles that define the normal vector and the distance of the plane from the origin.
• Rections, they are left out of the point-pairing.
• In the case of rigid transformations each point-pair along with the respective intrinsic coordinate frame produces a rigid transformation parameterized by a translation vector and three Euler angles, i.e., a point in a 6D transformation space.
• Each cluster corresponds to a potential pairwise partial symmetry of the shape and the extracted cluster centers act as symmetry transformation candidates that are subsequently validated and refined.

6.3. Planar Reflective Symmetry Transform

• Motivated by the notion of continuous symmetry introduced by Zabrodsky et al. [ZPA95], Podolak et al. [PSG∗06] investigate the notion of a symmetry transform under reflective transformations.
• They propose the planar reflective symmetry transform (PRST) to encode a continuous notion of symmetry of an object about any reflective line in 2D, or about any reflective plane in 3D.
• Observe that functions arising from rasterized surfaces are typically sparse over the volume grids, and propose a Monte Carlo algorithm to efficiently compute the PRST.
• The definition has subsequently been extended by Rustamov [Rus07a] to incorporate local neighborhood of points or contexts using a scale factor for controlling the neighborhood range.
• The PRST values are computed over a medium resolution volume grid, and then candidate maxima are identified via a thresholding step.

6.4. Graph-Based Symmetry Detection

• Instead of operating at the level of sample points, it is sometimes more practical to work at the level of feature curves, in particular for data sets where these feature curves can be extracted robustly.
• Feature lines are then used to define local coordinates or bases.
• The algorithm considers rigid motions as potential symmetry transformations and detects lines by slippage analysis [GG04].
• For efficiency reasons, not all pairs of bases are checked but instead random sampling is applied.
• Later, Bokeloh et al. [BWS10] introduce a similar algorithm that lifts this restriction but outputs overlapping symmetric areas.

6.5. Symmetry Factored Embedding

• Instead of working in the transformation space, one can also work directly in the space of correspondences.
• They extract connectedness of the graph using spectral methods.
• A popular solution to this problem is the Iterated Closest Point (ICP) algorithm [BM92, CM92].
• After extracting potential symmetry transformations and then refining the coarse estimates, the last step of most symmetry detection algorithms involves extracting patches of the mesh that are symmetric under the detected symmetry transformations.
• The challenge is to simultaneously determine {Ri} and {Ti}.

7. Intrinsic Symmetries

• While the geometry of the dragon in Figure 15(a) does not exhibit any global symmetries, the pose of Figure 15(b) exhibits a global mirror symmetry.
• Each such Möbius transform can then be used to vote for correspondence between shapes M1 and M2.
• The algorithm successfully finds correspondence across model pairs even under isometric deformations resulting in large Euclidean deformations.
• Earlier, Raviv et al. [RBBK07] employ generalized multi- dimensional scaling to compute a new embedding that best preserves the original geodesic distances on the object in the form of corresponding Euclidean distances in the new space.

8. Encoding Extracted Symmetries

• Independent of the specific symmetry detection algorithm, a number of different representations of the extracted symmetries are possible with the preferred representation depending on the target application.
• In particular, this allows improved segmentation results that take consistent groups of symmetric mappings into account.
• Finding an optimal hierarchy with respect to compression (minimal coding costs) is NP-hard; a two-level hierarchy already reduces to the NP-hard set-cover problem.
• Continuous symmetries with respect to rigid motion have been studied by Gelfand et al. [GG04].
• In summary, while different models are possible to structure the space of pairwise symmetries, the choice often depends on the target application.

9. Applications

• Extracted symmetries, global or partial, exact or approximate, essentially provide relations across different parts of a shape, as discussed in the last section.
• Further, in regions of missing data, instead of looking for points qi, surface information is propagated using symmetry information Ti(p) for data completion [PMG 05, TW05, XZT 09] .
• Detected symmetries, if organized by explicitly storing the symmetry transforms, factor out model redundancies and thus produce a compressed representation, e.g., if geometry is organized as a tree-structure [MGP06, SKS06] or in a hierarchy [WXL 11] .
• Symmetry is ubiquitous in natural objects, man-made shapes, and architectural forms.
• This exchange of parts encapsulated in symmetry defines a rewriting system for creating shape modifications, which can be transformed into a constructive grammar towards creating a rich set of shape variations.

10. Future Directions

• Arguably, it takes us one step closer to the ultimate goal of a computationally understanding threedimensional objects.
• The authors structure these into three areas: (i) improving de- c The Eurographics Association 2012.
• Intrinsic approaches are particularly troubled – currently, none of the techniques could handle objects with large scale topological noise, such as acquisition holes and contacts in a raw 3D scan.
• Statistical representation of potential symmetry is a direction that could possibly contribute to addressing such challenges.

11. Conclusion

• Automatic detection of symmetry depends on the type of symmetries that are of interest, i.e., whether exact or approximate, global or partial, and Euclidean or intrinsic.
• Symmetry-aided processing can be applied in the shape acquisition phase, in the geometry manipulation phase, and also can be used for categorizing and organizing the captured 3D geometry of shapes.
• A grand goal, however, is to infer meaningful shape semantics via computational shape analysis.
• His research interests include geometry processing, architectural geometry and design, computer graphics and animation.
• He received his Ph.D. degree from Tübingen University in 2004 and his Diploma in Computer Science from Paderborn University in 2000.

Figures

Figure 6: A rigid correspondence is determined by a surface point and a direction. In practice, we expect that an object encoded by n points has no more than O(n1.5) candidate transformations that a brute-force search algorithm would need to test.

Figure 3: Frieze groups are composed of translation, rotation by 180 degrees, glide reflection, reflection about a horizontal line, or reflection about a vertical line.

Figure 11: Planar Reflective Symmetry Transforms (PRST) for various curves. For visualization, each pixel is colored by the score of the plane with maximum symmetry score among all the planes passing passing through the pixel. (Images courtesy of Podolak and et al. [PSG∗06]).

Figure 19: Establishing a hierarchical structure is ambiguous; there are usually several equivalent solutions that all explain the structure compactly.

Figure 2: The dihedral group D3 represents the symmetries of the equilateral triangle (the colored flags are added to illustrate the transformation), while D5 is the symmetry group of the five-sided star. The triskelion has three rotational symmetry but no reflectional symmetries and is represented by the cyclic group C3. All of these finite point groups are subsets of the isometry group O(2) that represents the symmetries of the circle.

Figure 10: Regular structures discovered using transformation domain grid-detection [PMW∗08] on an amphitheater, a nautilus shell, a laser-scan of a building, and a procedurally generated helical segment.

Figure 9: To detect symmetries in geometric models, the boundary of the shape is uniformly sampled (left) [MGP06]. Every pair of samples with compatible local surface geometry provides local evidence for a symmetry transformation (center). In this example we consider reflections that are parameterized by an angle φ and the distance d to the origin. Accumulating such evidence using a clustering approach yields the dominant symmetries of the model (right).

Figure 16: Isometric point-to-point correspondence between two meshes M1 and M2. The meshes are first conformally flattened to the plane. Any pair of triplet of correspondences, then uniquely specifies a Möbius transform between the flattened meshes. Each such Möbius transform is ranked based on the votes from other mesh vertices, and the highest rated transform is retained as the isometric deformation linking M1 and M2. Intuitively, a correct pair of correspondence triplets (like shown in red, green, blue) brings the flattened meshes to alignment (bottom row). (Images courtesy of Lipman and Funkhouser [LF09].)

Figure 17: Intrinsic symmetries detected on 3D geometry. (Images courtesy of Ovsjanikov et al. [OSG08], Kim et al. [KLCF10] and Lasowski et al. [LTSW09], respectively.)

Figure 18: Intrinsic symmetries detected on 3D geometry. (Image courtesy of Raviv et al. [RBBK07] and Xu and et al. [XZT 09], respectively.)

Figure 4: The symmetries of the tiling patterns of the Alhambra can be described by wallpaper groups of which 17 distinct types exist.

Figure 23: Inverse procedural modeling from symmetry: By attaching building blocks derived from boundaries between non-symmetric area (docking sides), shape variations can be created automatically (top; original in red). Sliding dockers (bottom, orange) generalize the idea to partial patterns.

Figure 24: (Top panel) Preservation of analyzed symmetry relations in iWires a shape manipulation framework [GSMCO09], (bottom panel) distilling and highlighting of symmetry relations for creating abstractions of manmade shapes [MZL 09]. (Images courtesy of Gal et al. and Mehra et al. respectively.)

Figure 14: Under the action of transformation Ti, the input shape M gets mapped to Ti(M). The overlap regions, shown in black, between M and Ti(M) are the parts that are symmetric under the action of transformation Ti. Overlap is measured with respect to a tolerance ε as specified by the approximation level.

Figure 15: Changes of pose cause a change in extrinsic symmetries. While the pose in (a) does not exhibit a global mirror symmetry, a (nearly) isometric deformation exposes the global reflection (images from [MGP07]).

Figure 7: The suction cup on the tentacle of the octopus is identified as a salient feature and its similar occurences are detected [GCO06]. (Images courtesy of Gal et al.)

Table 1: The table classifies the related work on symmetry detection according to the method used, type of input (meshes, point sets, volume data, and images), and type of output (global vs. partial, discrete vs. continuous measure, extrinsic vs. intrinsic, structure of the symmetries, and the class of transformations, and features used.)

Figure 25: (Top panel) Symmetry descriptors are used to query a database for shape retrieval [KFR04]. (Bottom panel) Good viewpoints are automatically selected to minimize the symmetry in the scene [PSG∗06] and a symmetryinduced hierarchical organization of the parts of a model is presented[WXL∗11]. (Images courtesy of Kazhdan et al., Podolak et al., and Wang et al. respectively.)

Figure 22: Detecting regularity in shapes specially building facades and similar man-made structures, allow us to infer plausible procedural rules [PM01, MWH 06, MZWG07] to recreate the models. This allows smart geometric modeling to perform global geometric edits by inserting and deleting geometric patches, or replacing the base geometric patches, while conferring to the inferred procedural rules [PMW 08, MP08].

Figure 12: Symmetry detection result using line features and sub-graph matching [BBW∗09b]. Detected symmetric or repeating parts are highlighted.

Figure 13: Symmetry factored distance and the symmetry orbits detected. (Left) Symmetry factored distance from the marked point (with arrow) to all other points on the model, with blue denoting small distance and red denoting large distance. Note that symmetric points are at similar distances. (Right) Segmentation in the symmetry-factored embedding space produces symmetry-aware segmentation. (Images courtesy of Lipman et al. [LCDF10].)

Full text

EUROGRAPHICS 2012/ M.-P. Cani, F. Ganovelli STAR – State of The Art Report
Symmetry in 3D Geometry: Extraction and Applications
Niloy J. Mitra Mark Pauly Michael Wand Duygu Ceylan
UCL EPFL MPII EPFL
Abstract
The concept of symmetry has received significant attention in computer graphics and computer vision research
in recent years. Numerous methods have been proposed to find and extract geometric symmetries and exploit
such high-level structural information for a wide variety of geometry processing tasks. This report surveys and
classifies recent developments in symmetry detection. We focus on elucidating the similarities and differences
between existing methods to gain a better understanding of a fundamental problem in digital geometry processing
and shape understanding in general. We discuss a variety of applications in computer graphics and geometry
that benefit from symmetry information for more effective processing. An analysis of the strengths and limitations
of existing algorithms highlights the plenitude of opportunities for future research both in terms of theory and
applications.
1. Introduction
The mathematical sciences particularly exhibit order, sym-
metry, and limitation; and these are the greatest forms of the
beautiful. - Aristotle
Symmetry is a universal concept in nature, science, and
art (see Figure 1). In the physical world, geometric symme-
tries and structural regularity occur at all scales, from crystal
lattices and carbon nano-structures to the human body, archi-
tectural artifacts, and the formation of galaxies. Many bio-
chemical processes are governed by symmetry and as a re-
sult we experience a wealth of biological structures that ex-
hibit strong regularity patterns. This abundance of symmetry
in the natural world has inspired mankind from its origins to
incorporate symmetry in the design of tools, buildings, art-
work, or even music. Besides aesthetic considerations, phys-
ical optimality principles and manufacturing efficiency often
lead to symmetric designs in engineering and architecture.
Geometric data, acquired via scanning or modeled from
scratch, is traditionally represented as a collection of low-
level primitives, e.g., point clouds, polygon meshes, NURBS
patches, etc., without explicit encoding of their high-level
structure. Finding symmetries in such geometric data is thus
an important problem in geometry processing that has re-
ceived significant attention in recent years. Numerous ap-
plications immediately benefit from extracted symmetry in-
formation, e.g., shape matching, segmentation, retrieval, ge-
Human brain
Silicon nanostructures
Steam turbine
Starfish
Nautilus shell
Persian carpet
Taj Mahal Vitruvian Man
Simian virus
Insect eye
Geodesic dome
Spiral galaxy
Figure 1: Examples of symmetry in nature, engineering, ar-
chitecture, and art.
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Mitra, Pauly, Wand, Ceylan / Symmetry in 3D Geometry
ometry completion, beautification, meshing, or procedural
modeling.
This survey reviews the state-of-the-art in symmetry de-
tection methods for geometric data sets. We first introduce
basic mathematical terms and present various high-level cri-
teria to organize existing work into a set of categories, em-
phasizing their similarities and differences (see Table 1). We
hope that this comparative survey will help readers navigate
through the constantly expanding literature on symmetry de-
tection and inspire researchers to contribute to this emerging
field in the future.
The rest of the survey is structured as follows: In Sec-
tion 2 we start with a discussion of the classical mathemat-
ical theory of symmetry groups that characterizes the struc-
ture of globally and exactly symmetric objects. In Section 3
we look at more general cases including partial and approx-
imate symmetry, which are particularly relevant in practical
applications in computer graphics and vision. The main part
of this paper surveys different algorithms for symmetry de-
tection (Sections 4-7). We then examine what types of ge-
ometrical structures these different algorithms discover and
how symmetry information is encoded (Section 8). Finally,
we discuss various applications of symmetry detection (Sec-
tion 9) and conclude with thoughts on future challenges in
the field (Sections 10 and 11).
2. Classical Theory
Symmetry is a general concept in mathematics [Wey52];
broadly speaking, a symmetry preserves a certain property
(e.g., geometric similarity) of an object under some opera-
tion applied to the object. This notion of invariance is for-
malized in an elegant branch of mathematics called group
theory [Rot94]. In the context of geometry, we will con-
sider geometric transformations as the symmetry operations,
such as reflections, translations, rotations, or combinations
thereof.
We say that a geometric object M is symmetric with re-
spect to a transformation T , if M = T (M), i.e., M is in-
variant under the action of transformation T . The set S of
all symmetry transformations of a shape has a very specific
structure, namely that of a group. A symmetry group is a set
of transformations that satisfies the following group axioms
with composition as the group operation:
• Closure: If M is symmetric with respect to two transfor-
mations T
1
and T
2
, then it will also be symmetric with
respect to the composition T
1
T
2
. Thus if T
1
,T
2
∈ S, it fol-
lows that (T
1
T
2
) ∈ S also.
• Identity element: The identity transform I ∈ S is always
a symmetry transformation, since it trivially leaves any
object unchanged, i.e., I(M) = M.
• Inverse element: For each symmetry transformation T ∈
S there exists an inverse element T
−1
∈ S, such that
T
−1
T = T T
−1
= I.
120°
240°
I
dihedral group D
3
dihedral group D
5
cyclic group C
3
infinite group O(2)
(a)
(b) (c) (d)
Figure 2: The dihedral group D
3
represents the symmetries
of the equilateral triangle (the colored flags are added to il-
lustrate the transformation), while D
5
is the symmetry group
of the five-sided star. The triskelion has three rotational sym-
metry but no reflectional symmetries and is represented by
the cyclic group C
3
. All of these finite point groups are sub-
sets of the isometry group O(2) that represents the symme-
tries of the circle.
• Associativity: The compositions of multiple transforma-
tions is independent of the priority of composition, i.e.,
(T
1
T
2
)T
3
= T
1
(T
2
T
3
) ∀ T
1
,T
2
,T
3
∈ S.
Note that while the priority of composition is irrelevant,
the order of transformations can be important. For example,
composing two rotations in 3D about different axes in gen-
eral leads to a different transformation when switching the
order of application. Groups for which the relation T
1
T
2
=
T
2
T
1
holds ∀T
1
,T
2
∈ S are called commutative or Abelian
groups.
The notion of symmetry as invariance under transforma-
tions is a powerful concept that has been advocated promi-
nently by Felix Klein in his Erlanger Programm [Kle93].
Klein proposed to characterize different classes of geometry,
such as projective geometry or Euclidean geometry, based
on the underlying symmetry groups. For example, distances
and angles are invariants in Euclidean geometry. These prop-
erties are preserved under transformations of the Euclidean
group, the group of all isometries with respect to the Eu-
clidean metric. This notion of classifying geometries based
on symmetry groups can be transferred to geometric objects
as well.
Let us consider the example of a 2D equilateral triangle
c
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Mitra, Pauly, Wand, Ceylan / Symmetry in 3D Geometry
translation (T)
T+ glide reflection (GR)
T+ GR +
horizontal reflection (HR)
T+ vertical reflection (VR)
T+ 180° rotation (R)
T + R + VR + GR
T + R + HR + VR + GR
Figure 3: Frieze groups are composed of translation, rota-
tion by 180 degrees, glide reflection, reflection about a hori-
zontal line, or reflection about a vertical line.
shown in Figure 2(a). We observe that a rotation of 120
◦
around the triangle center maps the triangle onto itself. It fol-
lows that all integer multiples of this rotation are also sym-
metries of the triangle. However, only three of these rotations
are unique, since a rotation by 360
◦
is equal to the identity
transformation. We also see that the triangle has three reflec-
tional symmetries across the lines from each vertex through
the triangle center. Together, these transformations form the
dihedral group D
3
consisting of six elements, three rotations
(of which one is the identity transformation) and three reflec-
tions. In general, the dihedral group D
n
represents the sym-
metries of a regular n-gon. These symmetries can be rep-
resented as finite combinations of two generating transfor-
mations. For example, repeated application of a 72
◦
rotation
and a reflection can generate all elements of a dihedral group
D
5
(see Figure 2(b)). Shapes that have rotational symmetries
but no reflectional symmetries, such as the triskelion shown
in Figure 2(c) can be characterized by a cyclic group C
n
that
is generated by a rotation of 360
◦
/n. The cyclic and dihe-
dral groups are finite point groups. In two dimensions, they
are subgroups of the orthogonal group O(2), the group of all
Euclidean isometries that leave the origin fixed. This infinite
group is the symmetry group of the circle, which is sym-
metric with respect to rotations of arbitrary angle around its
center and reflections across arbitrary lines through the cen-
ter (Figure 2(d)).
Symmetry groups have been used extensively in the study
of decorative art and structural ornaments. The symmetries
of a two-dimensional surface that is repetitive in one direc-
tion and extends to infinity along that direction can be classi-
fied by one of exactly seven Frieze groups (Figure 3). If rep-
Figure 4: The symmetries of the tiling patterns of the Al-
hambra can be described by wallpaper groups of which 17
distinct types exist.
etition occurs in two different directions, seventeen distinct
groups are possible, denoted as wallpaper groups. These
groups combine reflections, rotations, and translations so
that all these transformations and all combinations of them
leave the entire grid unchanged. This leads to a wealth of
repetitive patterns that can, for example, be observed in the
beautiful tiling patterns of the Alhambra in Granada, Spain
(Figure 4).
In summary, the classical theory of symmetry groups de-
scribes the structure of transformations that map objects to
themselves exactly. Such exact, global symmetry leads to a
group structure because after applying a transformation, we
end up with the same situation as before, creating a closed
algebraic structure.
In computer graphics applications, we often face a more
general problem, where symmetry is approximate and par-
tial. For example, for a simple building facade with win-
dows related by translational symmetry, the closure property
is violated since the facades have finite extent. Further, we
have to handle different classes of transformations. Finally,
we need efficient algorithms to compute such symmetries. In
the following subsection, we discuss these issues.
3. Classification
This survey investigates algorithmic paradigms for extract-
ing symmetries and relations in 3D geometric data. We first
formalize the problem of symmetry detection and present
several classification categories to highlight the similarities
and differences of existing symmetry detection algorithms.
In contrast to the classical theory discussed before, we now
take practically important aspects such as approximate sym-
metry and partiality into account.
c
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Mitra, Pauly, Wand, Ceylan / Symmetry in 3D Geometry
Input Output
Reference Method
mesh
points
volume
image
global
partial
disc.
cont.
extrin.
intrin.
structure class of
transform.
Type of
Features
Alt et al. 1988 [AMWW88] object-space graph isomorph. X X X X pairwise rigid
Atallah et al. 1985 [Ata85] string pattern matching X X X X pairwise reflections
Bermanis et al. 2010 [BAK10] spectral analysis X X X X pairwise rotations,
reflections
angular difference
functions
Berner et al. 2008 [BBW
∗
08] feature-graph matching X X X X X X segmentation rigid slippage features
Berner et al. 2009 [BBW
∗
09a] feature-graph matching X X X X X X segmentation (relaxed)
isometries
Gaussian
curvature,
curvature lines
Bokeloh et al. 2009 [BBW
∗
09b] feature graph matching X X X X X segmentation rigid lines
Ben-Chen et al. 2010 [BCBSG10] flow of killing vector fields X X X X continuous continuous
isometries
Killing vector
fields
Bokeloh et al. 2011 [BWKS11] RANSAC-based transformation
verification
X X X X X symmetry
groups,
continuous
translations sharp creases
Berner et al. 2011 [BWM
∗
11] feature-graph matching X X X X X segmentation subspace
symmetries
lines of large
principle curvature
Chertok et al. 2010 [CK10] spectral analysis X X X X X pairwise rotations,
reflections
local image
features
Gal et al. 2006 [GCO06] geometric hashing X X X X segmentation similarity
transform.
curvature based
salient features
Gelfand et al. 2004 [GG04] slippage analysis X X X X X continuous rigid local slippage
signatures
Kazhdan et al. 2003 [KCD
∗
03] descriptor computation with
Fourier methods
X X X X pairwise reflections
Kazhdan et al. 2004 [KFR04] descriptor computation with
Fourier methods
X X X X pairwise rotations,
reflections
Kim et al. 2010 [KLCF10] search in möbius
transformations
X X X X pairwise isometries average geodesic
distance
Lipman et al. 2010 [LCDF10] spectral analysis in
correspondence space
X X X X X X symmetry
aware
embedding
rigid
Lasowski et al. 2009 [LTSW09] belief propagation X X X X X segmentation isometries
Mitra et al. 2010 [MBB10] multi-dimensional scaling X X X X symmetry
groups
isometries discrete
Laplacians
Mitra et al. 2006 [MGP06] transformation voting X X X X X segmentation,
hierarchy
similarity
transform.
curvature
Martinet et al. 2006 [MSHS06] generalized moment functions X X X X X pairwise,
hierarchy
rotations,
reflections
moments
Ovsjanikov et al. 2008 [OSG08] search in signature embedding X X X X pairwise isometries global point
signatures
Pauly et al. 2008 [PMW
∗
08] transformation voting X X X X X symmetry
groups
similarity
transform.
curvature
Podolak et al. 2006 [PSG
∗
06] symmetry transform
computation
X X X X X pairwise reflections
Raviv et al. 2007 [RBBK07] generalized multi-dimensional
scaling
X X X X pairwise isometries geodesic distances
Raviv et al. 2009 [RBBK09] generalized multi-dimensional
scaling
X X X X X pairwise isometries geodesic distances
Raviv et al. 2010 [RBS
∗
10] matching of distance
histograms
X X X X pairwise isometries diffusion distances
Simari et al. 2006 [SKS06] reweighted least squares auto
alignment
X X X X hierarchy reflections PCA axes
Sun et al. 1997 [SS97] search in orientation histograms X X X X X X pairwise reflections,
rotations
extended Gaussian
image
Thrun et al. 2005 [TW05] symmetry space scoring X X X X segmentation reflections,
rotations
Xu et al. 2009 [XZT
∗
09] voting for symmetry transforms X X X X segmentation reflections SDF
Zabrodsky et al. 1995 [ZPA95] symmetry distance computation X X X X X pairwise isometries
Table 1: The table classifies the related work on symmetry detection according to the method used, type of input (meshes, point
sets, volume data, and images), and type of output (global vs. partial, discrete vs. continuous measure, extrinsic vs. intrinsic,
structure of the symmetries, and the class of transformations, and features used.)
c
 The Eurographics Association 2012.

Mitra, Pauly, Wand, Ceylan / Symmetry in 3D Geometry
Correspondences as Building Blocks. The elementary
building block for symmetry detection is the identification of
matching geometry: Given a shape, the goal is to identify and
extract pairs of regions such that each pair of regions, under
an appropriate distance measure, is similar when the respec-
tive regions are aligned using an allowable transformation.
Specifically, given a geometric model M, the goal is to iden-
tify subsets M
1
 M
2
 M such that M
1
 T (M
2
), where T
denotes a transformation and  denotes equality under the
chosen distance measure. In case of partial symmetry detec-
tion (see later), the surface patches are often required to be
non-overlapping, i.e., M
1
M
2
= . In this survey, we focus
on shapes represented as surfaces, e.g., point cloud data, tri-
angle meshes, or NURBS surfaces, rather than 2D images or
volumetric data.
Research efforts target variations of the symmetry detec-
tion problem primarily based on the choice of (i) how the
shape is segmented, (ii) how distance between two surface
patches is measured, and (iii) what classes of transforma-
tions are allowed to bring surface patches into alignment (see
Table 1 for a classification of recent related work). The sym-
metry detection problem is challenging because we have to
simultaneously segment the shape and establish correspon-
dence across the resultant segments, while solving for the
respective aligning transforms. Note that even the decou-
pled versions of the problem are non-trivial with various
solution strategies: we refer the readers to respective sur-
veys on mesh segmentation [Sha08] and surface correspon-
dence [vKZHCO10]. Next we discuss some common vari-
ants of the problem.
Global vs. Partial Symmetries. For global symmetry de-
tection we seek transformations that map the whole object
to itself, i.e., M
1
= M
2
= M. Consequently, we do not have
to solve the segmentation problem, which greatly simplifies
the symmetry detection process.
For global symmetries of a finite object the centroid of
an object is a fixpoint, i.e., is invariant under the shape’s
symmetry transformations. Specifically, symmetry rotations
have the object centroid as rotation centers, while planes of
reflection must pass through the object centroid. Methods
for global symmetry detection exploit this property to sig-
nificantly reduce the search space.
While a number of common shapes exhibit global sym-
metries (see Figure 1), self-similarities often occur only on
parts of a shape. In order to capture these regularity pat-
terns and enable a fine-grain analysis of geometric objects,
we consider partial symmetries. There are two aspects to
partial symmetries. Symmetry can be restricted to a subset
M

 M as shown in Figure 5(a). If we consider M

as a
separate shape, then we can apply the notion of symmetry
groups as defined above. Symmetry detection thus amounts
to segmenting the shape into subsets that exhibit global sym-
metries represented by a transformation group.
(a) complete symmetry group on parts of a shape
(b) partial translational
symmetry
(c) partial rotational
symmetry
Figure 5: Partial symmetries commonly occur in geometric
data sets.
In many instances, however, we do not have a complete
symmetry as defined by a symmetry group. For example,
translational structures in bounded shapes are very common,
such as the repetitive patterns of the steps of the stairs shown
in Figure 5(b). For such a structure, we can find a transfor-
mation that maps, e.g., the three lower-most steps to the three
upper-most ones, but there is no self-similarity of the entire
set of steps, except for the identity transform. Specifically,
we say a shape M has a partial symmetry with respect to
a transformation T , if there exist two subsets M
1
 M
2
 M
such that T (M
1
)=M
2
. This definition coincides with the
definition of a global symmetry if M
1
= M
2
= M, thus global
symmetry is a special case of partial symmetry.
Partial symmetry allows handling a broader class of sym-
metries, but typically does not preserve the group structure.
However, we can classify partial symmetries by the small-
est group that contains the partial symmetry transformations.
Conceptually, we can compute the closure of the symme-
try set with respect to composition, which is analogous to
repeating the pattern to infinity (or until a full rotation is
achieved for a cyclic rotation group) as illustrated in Fig-
ure 5.
Exact vs. Approximate Symmetries. In another axis of
variation, we look at the notion of equivalence  under
transformations. Physical objects as shown in Figure 1 are
c
 The Eurographics Association 2012.

Frequently asked questions

Q1. What contributions have the authors mentioned in the paper "Symmetry in 3d geometry: extraction and applications" ?

This report surveys and classifies recent developments in symmetry detection. The authors discuss a variety of applications in computer graphics and geometry that benefit from symmetry information for more effective processing. 

Q2. What is the role of symmetry in the design of objects?

Symmetry – the redundancy within a geometric object and the structure of such redundancy – will continue to play an important role in model acquisition, understanding, manipulation, manufacturing, and also towards efficient, economic, aesthetic, and functional object designs. 

Q3. What can be used for the analysis of shapes?

Symmetry-aided processing can be applied in the shape acquisition phase, in the geometry manipulation phase, and also can be used for categorizing and organizing the captured 3D geometry of shapes. 

Q4. How is the graph of linear features built?

The graph of linear features is built using detected feature lines as nodes and using an edge set obtained by connecting k-nearest line segment neighbors. 

Q5. What can be done to characterize shapes?

the information that two regions M1 , M2 = T(M1) are coupled by a symmetry transformation T can be exploited to characterize shapes for recognition or for structure-preserving editing. 

Q6. What is the way to extract symmetric patches?

The extracted symmetric patches can be treated as alphabets and combined with the detected symmetry transforms in order to construct an inverseshape grammar [SG71] for the input shapes. 

Q7. What is the main difficulty in developing a symmetry detection algorithm?

The chief difficulty in developing such algorithms come from the global nature of symmetry that tightly couples the overall geometry making it difficult to work solely with local reasoning, without any prior knowledge or preprocessing of the data. 

Q8. What is the way to detect symmetries in 3D?

The state-of-the-art provides a large range of algorithms for detecting extrinsic and intrinsic symmetries in various types of data, ranging from clean meshes to point clouds from 3D scanners. 

Q9. What methods can be used to detect n-fold rotational symmetries?

Popular methods detect n-fold rotational symmetry based on the correlation of the extended Gaussian image [SS97], moment coefficients [TMS09], or spherical harmonic coefficients [KFR04]. 

Q10. What is the simplest way to aggregate transformations?

They aggregate transformations based on the following observation: several transformation families have only a few degrees of freedom and are uniquely determined by a small number of correct corresponding point pairs. 

Q11. What is the way to compute a new embedding?

Raviv et al. [RBBK07] employ generalized multi-dimensional scaling to compute a new embedding that best preserves the original geodesic distances on the object in the form of corresponding Euclidean distances in the new space. 

Q12. How many Frieze groups can be classified?

The symmetries of a two-dimensional surface that is repetitive in one direction and extends to infinity along that direction can be classified by one of exactly seven Frieze groups (Figure 3). 

Q13. What is the problem of finding a segmentation that leads to building blocks that can be connected?

The issue of finding a segmentation that leads to building blocks that can be connected to form new shapes has been considered by Bokeloh et al. [BWS10]: Instead of cutting pieces at Voronoi cells of a region growing algorithm, their approach cuts at the boundaries of partial symmetries. 

Q14. What is the definition of approximate symmetry?

In order to enable more practical symmetry detection algorithms, the authors need a mathematical definition of approximate symmetry that is suitable for computation. 

Q15. What is the symmetry factored distance between a mesh?

symmetry factored distance between any two points xi and x j on the mesh can be simply computed as the Euclidean distance in this embedded space, i.e., dt(xi,x j)2 = ‖Φt(xi)−Φt(x j)‖2 = ∑k λ2tk ‖ψk(xi)− ψk(x j)‖2 (see Figure 13).