Amir%20Pelled

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3D-Symmetry

• Amir Pelled

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Topics

• What is 3D-Symmetry ?
• Basic 3d-Symmetry Definitions
• Mathematical Definitions for 3D-Space
• Symmetry Detection Algorithms
• Applications
• Bibliography

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What is 3D-Symmetry ?

• Does this building possess 3D-Symmetry ?

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What is 3D-Symmetry ?

• A basic misconception
• We can only see the 2D-projection that is facing
  us.
• 3D-symmetry has to consider the whole of the
  object

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Basic Definitions

• Topics
• 3d-reflection symmetry
• 3d-rotation symmetry

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Basic Definitions3D-Reflection-Symmetry

• A reflection symmetry has
• N mirror planes each passing through the objects
  mass center ( in contrast to mirror axes in 2D)
• A rotational symmetry has a center of rotation
• an axis (a line) in a 3D object ( in contrast to
  a point in 2D)
• A number of times N by which shapes are repeated
  about that axis

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Basic Definitions3D-Reflection-Symmetry

• Reflection planes of a water molecule

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Basic Definitions3D-Reflection-Symmetry
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Basic Definitions3D-Reflection-Symmetry

• How many 3D-reflection symmetries does a cube
  have ?

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Basic Definitions3D-Rotation-Symmetry

• A cube has these axes of rotational symmetry

4 axes of 3-fold each passing through the
centers of two opposite vertices
3 axes of 4-fold each passing through the
centers of two opposite faces
6 axes of 2-fold each passing through the
centers of two opposite edges
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Basic Definitions3D-Rotation-Symmetry

• Table of rotational symmetries for other platonic
  objects

Symmetry Basis
axes through vertices
Axes through face centers
Axes through edge centers
Icosahedron
6 5-fold
10 3-fold
15 2-fold
Dodecahedron
10 3-fold
6 5-fold
15 2-fold
Octahedron
3 4-fold
4 3-fold
6 2-fold
Hexahedron
4 3-fold
3 4-fold
6 2-fold

Tetrahedron
4 3-fold
4 3-fold
3 2-fold
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Mathematical Definitions for 3D-Space

• We need one standard method which could represent
  an objects characteristics in 3D space
• Shape
• Size
• Position
• Orientation

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Mathematical Definitions for 3D-Space

• Inertia Matrix
• Eigenvalues Eigenvectors
• Principal Axes

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Mathematical Definitions for 3D-Space (inertia
matrix)

• The inertia matrix of an object, is a
  mathematical descriptor of the dispersion of the
  objects mass, around its center.
• If B is an object and
• O is the center of mass for B
• We sample points at coordinates (x,y,z) where x
  is the distance from O on the yz plane
• Mi is the mass of the object located at point i

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Mathematical Definitions for 3D-Space (inertia
matrix)
Notice that matrix I is symmetrical
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Mathematical Definitions for 3D-Space
(EigenValues EigenVectors)

• Scalar ? is called an eigenvalue of a linear
  mapping A if there exists a non-zero vector x
  such that Ax ?x
• Vector x is called an eigenvector

Example
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Mathematical Definitions for 3D-Space
(EigenValues EigenVectors)

• The dominant eigenvector of a matrix is an
  eigenvector corresponding to the eigenvalue of
  largest magnitude (for real numbers, largest
  absolute value) of that matrix
• EigenVectors of a symmetrical matrix are mutually
  orthogonal

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Mathematical Definitions for 3D-Space
(EigenValues EigenVectors)
x-gt
lt-x
o
The eigenvector in the direction of x, is more
dominant in describing the dispersion of mass
around O than the other two. As such, the
eigenvalue associated with it will be the largest.
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Mathematical Definitions for 3D-Space (principle
axes)

• Principle axes can be defined as a set of N
  vectors that satisfy the following conditions
• They have a common intersection point at the
  objects centroid
• They are mutually orthogonal
• They point in the direction of the maximum
  variance of data

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Mathematical Definitions for 3D-Space (principle
axes cont)

• Principal Axis Theorem
• Any plane of reflectional symmetry of a body is
  perpendicular to a principal axis
• Any axis of rotational symmetry of a body is a
  principal axis
• Principal axes are eigenvectors of the objects
  inertia matrix ( and its covariance matrix as
  well)

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Symmetry Detection Algorithms

• There are numerous algorithms for 3D-symmetry
  detection but they have flaws
• Computational problems
• 2D based algorithms could detect symmetry only in
  specific directions
• Noise
• Standard of object/symmetry representation
• Unable to work with complex objects

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Symmetry Detection Algorithms

• We would like to introduce two improved
  algorithms
• An Oct-Tree based algorithm
• An Extended-Gaussian-Image algorithm

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Symmetry Detection Algorithms (Oct-Tree)

• Why use an Oct-Tree ?
• We need an object representation that allows
• Computing moment ( in order to form the inertia
  matrix)
• Linear transformation ( translation and rotation)
• Cross-section extraction
• Symmetry evaluation
• An Oct-Tree is a data structure that allows these
  operations

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Symmetry Detection Algorithms (Oct-Tree cont)

• A cross-section represented by a quad-tree can be
  found simply by an oct-tree traversal.
• Symmetry evaluation of rotation and reflection
  can be obtained by traversing and comparing
  appropriate tree nodes. ( for rotation of nlt4.
  For ngt5 a more complex evaluation is required)
• Linear transformation can be achieved by moving
  the object from one group of cells, to another
  correlating group in the oct-tree. (the
  transformations are not immune to quantization
  errors, but this does not significantly affect
  their performance

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Symmetry Detection Algorithms (Oct-Tree cont)

• Building the tree
• Starting with one cube enclosing the entire
  object
• Mark all unmarked boxes White and Black
  (representing the absence, presence on the
  object)
• Mark all boxes containing a border of the object
  Grey and split them
• Continue until no more unmarked boxes exist, or
  reached requested resolution.

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Symmetry Detection Algorithms (Oct-Tree cont)
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Symmetry Detection Algorithms (Oct-Tree cont)

• Using the tree
• Step 1
• Compute the eigenvalues associated with the
  inertia matrix
• Compute the principal axis transform
• For eigenvectors not uniquely determined, choose
  any orthogonal vectors (to the ones that are)

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Symmetry Detection Algorithms (Oct-Tree cont)
Computing the inertia matrix with an oct-tree
The inertia matrix is composed of central moments
that are related to the ordinary moments Mpqr of
order (pqr p,q,r ?N0)
Where S is the oct-trees space and ? is the
density function
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Symmetry Detection Algorithms (Oct-Tree cont)

• Step 2
• Perform the principal axis transform of the
  input tree in order to build the principal
  oct-tree
• The result of this step is that the object is
  centered at the origin, and its principal axes
  are aligned with the coordinate axes

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Symmetry Detection Algorithms (Oct-Tree cont)

• Step 3
• According to the eigenvalues acquired in step 1
  evaluate the types of symmetries we need to
  search ( that there is a potential of )
• Case 1 If all eigenvalues are distinct, check
  for
• reflectional symmetry using the coordinate planes
• axial symmetry using the coordinate axes
• central symmetry, using the centroid
• Case 2 If two eigen values are equal also check
  for
• Rotational symmetry using a cross-section of
  coordinate plane perpendicular to the unique
  eigenvector
• Case 3 If all eigenvalues are equal also check
  for
• central symmetry with respect to the origin
• Spherical symmetry (if potential axes of symmetry
  are given)

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Symmetry Detection Algorithms (Oct-Tree cont)
If all eigenvalues are distinct, search for
symmetry along the dominant eigenvector or
perpendicular to it
If two eigen values are the same, we need to look
for symmetry along the third eigenvector or
perpendicular to it
If all three eigenvalues are equal, there is a
possibility for spherical symmetry
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Symmetry Detection Algorithms (symmetry detection
using Oct-Tree)
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Symmetry Detection Algorithms (Extended Gaussian
Image)

• The Oct-Tree based algorithm operates well on
  synthetic objects, but when it comes to real
  images, it has its short-comings.
• The EGI algorithm solves this problem and as
  such, is much more flexible than the Oct-Tree
  solution.

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Symmetry Detection Algorithms (Extended Gaussian
Image cont)

• The goals of the EGI algorithm
• Determine the position and orientation of the
  plane of reflection symmetry
• Determine the axis and order of rotational
  symmetry for images in a variety of formats

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Symmetry Detection Algorithms (Extended Gaussian
Image cont)

• It is not an objective of the algorithm to
  determine whether or not an image is symmetric,
  but assumes that the image has some degree of
  symmetry.
• Not all symmetries are required to be found, only
  the strongest. For rotational symmetry, the
  highest order is sought.

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Symmetry Detection Algorithms (Extended Gaussian
Image cont)

• The Gaussian Image
• Take a 3D object. Declare a sphere that encloses
  the object whose center of mass is the center of
  the object
• From every surface of the object, draw a normal
  vector whose tail lies at the center of the
  sphere, and head lies on a point on the sphere
  appropriate to the particular face

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Symmetry Detection Algorithms (Extended Gaussian
Image cont)
Block object
Gi of a block object
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Symmetry Detection Algorithms (Extended Gaussian
Image cont)

• The Extended Gaussian Image
• Create the Gaussian Image, but assign each vector
  a weight (scalar) which is equal to the area of
  the surface the vector is mapping

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8
2
2
8
8
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Symmetry Detection Algorithms (Extended Gaussian
Image cont)

• The Orientation Histogram
• We need a data structure that stores the EGI, in
  a fashion that a computer can use

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Symmetry Detection Algorithms (Extended Gaussian
Image cont)

• Tessellating the Gaussian Sphere
• Place an Icosahedron in the gaussian sphere, so
  that they both have the same center, and the
  vertices are touching the sphere
• Divide each triangle face into four triangle by
  connecting the mid-points of the three edges
• Normalize the new triangles onto the unit sphere
• Repeat until desired resolution is achieved
• Convert the triangular faces into hexagonal faces

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Symmetry Detection Algorithms (Extended Gaussian
Image cont)

• Properties of Tessellation
• All the cells have the same area
• All cells are of the same shape
• All cells have compact regular shapes
• The division should be fine enough to allow
  curvature
• Each bin (hexagon) of the tessellation sphere is
  enumerated

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Symmetry Detection Algorithms (Extended Gaussian
Image cont)
object
Tessellation sphere
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Symmetry Detection Algorithms (Extended Gaussian
Image cont)

• Creating the orientation histogram
• Using the Gaussian Image, assign each normal to
  the appropriate bin (hexagon) in the Tessellation
  Sphere
• For every bin, mark the number of normal that are
  assigned to it
• Create the histogram F bin -gt num(normals)

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Symmetry Detection Algorithms (Extended Gaussian
Image cont)

• Symmetry detection using the orientation
  histogram
• Reflectional symmetry
• The degree of reflectional symmetry in a plane is
  measured by the correlation of the histogram with
  itself after reflection in the plane
• The correlation is formed by visiting each
  histogram bin and multiplying its value by the
  value in the bin which mirrors it in the plane
• Accumulate the products
• The candidates for the symmetry plane normal are
  the three principal axis directions and their
  neighbors (5-6)
• The plane with the highest histogram correlation
  is the plane with the strongest symmetry

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Symmetry Detection Algorithms (reflection
detection using EGI)
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Symmetry Detection Algorithms (Extended Gaussian
Image cont)

• Symmetry detection using the orientation
  histogram
• Rotation symmetry
• We denote the rotation around vector n through an
  angle ?
• For each candidate rotation degree, the histogram
  is rotated by the corresponding angle, and the
  product is correlated with the original

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Symmetry Detection Algorithms (Extended Gaussian
Image cont)

• This is performed for all candidate axes for
  rotational symmetry ( the eigenvectors )
• The order which results in a peak in the
  correlation is the strongest order of symmetry
  for the given axis
• The largest correlation from these axes indicates
  the axis of rotational symmetry

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Symmetry Detection Algorithms (rotation detection
using EGI)
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Symmetry Detection Algorithms (rotation detection
using EGI)
                                                
                                                  
  (a) (b) (c)
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Symmetry Detection Algorithms (rotation detection
using EGI)
This object also has an axis of 2-fold rotational
symmetry, but the axis with the higher order will
be the one identified by the algorithm
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Symmetry Detection Algorithms (Extended Gaussian
Image cont)

• Advantages of using the EGI
• Reduce search space
• We only need to search one hemisphere of the
  histogram
• For high resolution tessellations, we can further
  reduce the search by focusing on the principal
  axes of the object and their neighbors ( because
  of the correlation between the principal axes and
  the planes/axes of symmetry)

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Symmetry Detection Algorithms (Extended Gaussian
Image cont)

• Advantages of using the EGI
• The corresponding EGI for each stored object
  model can be saved in the model database
• Model stored as a surface normal vector histogram
• Surface normals available for extraction
• Once a match is found ( by compairing EGI
  histograms), both the identity and orientation of
  the object may be calculated

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Symmetry Detection Algorithms (Extended Gaussian
Image cont)

• Disadvantages of using the EGI
• EGIs only uniquely define convex objects
• An infinite number of non-convex objects can
  possess the same EGI
• The following objects have the same EGI

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Applications

• Object descriptor
• Use the object symmetry axes as identifiers
• Computer vision
• Identify object
• Compression
• Tessellation and Crystalography
• Medical Technology (example advantages of MRI
  over mammography because of MRI symmetry)

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Bibliography

• 3D Symmetry Detection Using the EGI Changming
  Sun and Jamie Sherrah
• Symmetry Identification of a 3-D Object
  Represented by Octree Predrag Minovic, Sciji
  Ishikawa and Kiyoshi Kato
• Answers.com
• wolfram research
• Mathworld.wolfram.com
• Scienceworld.wolfram.com
• EG1527 Dynamics Note 10 Moments of Inertia
• City University London DOC Lecture 7
  Polygons and Polyhedrons
• Extended Gaussian Images Berthold K.P.Horn

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The End
or is it ?