Lecture 16
Geometry and Reconstruction from
Symmetry
Allen Y. Yang
October 23rd, 2006
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Geometry in single views
 We have learned reconstruction of 3-D objects in multiple views.
 Today: study multiple-view geometry in single views.
Humans have the ability to infer 3-D structures from a single view!
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Symmetry plays a role
 Humans rely on symmetry information to reconstruct the space from
single views.
 Illusions take advantage of symmetry to deceive human vision
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History of Symmetry: 2-D Patterns
 There are 7 patterns that can be expand infinitely in one
dimension: Frieze groups.
There are 17 patterns in two dimensions: Wallpaper groups.
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History of Symmetry: 3-D Solids

Five Plato solids had been known
to Neolithic Scotland people.

Theaetetus (417 BC-369 BC) may
be the first to prove that there
exist only five regular solids.
 Dual properties of the five solids:
1. Tetrahedron is self-dual
2. Cube and octahedron are a dual
3. Dodecahedron and icosahedron
are a dual
 Three symmetry groups:
Tetrahedron group, Octahedron
group, and Icosahedron group.
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Structure from Symmetry
Recall that a single perspective image may be projected from infinitely
many 3-D structures.
1. If additional symmetry constraint is applied, no 3-D information is
lost in the perspective projection (up to a scale factor).
2. Detection of a symmetric object relies on the consistency of
symmetric relations on itself together with its neighboring objects.
3. Planar symmetric patterns can be used (after to reconstruction) to
align multiple views of the same object.
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Geometry of A Planar Symmetric Structure

3 types of symmetry: reflective, rotational, translational.
reflective
translational
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rotational
Representation of Symmetry in Space
 Group structure of reflective and rotational symmetry in space:
 Example
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Homography Group in Perspective Images
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Relation between g0 and g’
Goal: Given a symmetry assumption (R, T), recovering R0
and T0 between the object and camera.
 Proposal:
1. Use homography relation between the real vantage
point and the “hidden” one to recover R’ and T’.
2. Solve for R0 and T0 from the above constraint.

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Relation between H0 and H’
 Since S is a planar structure:
 H represents symmetric transformation in space:
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Recover H’: Reflective Symmetry
With two solutions from the homography decomposition, only one admits
symmetric pattern, in general.
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Recover H’: Rotational Symmetry
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Recover R0 and T0 for Ref and Rot
 Given R’ and R known, and constraint
We are solving for R0 from the following equation:
R’R0 - R0R = 0
 This equation is called Lyapunov map
from R0  R’R0 - R0R.
 R0 is solved by finding the kernel of
the Lyapunov map.
 T0 is recovered by taking reprojected
center of the object from R’ and T’.
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Recover Translational Symmetry
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Symmetry Detection
We use structure-from-symmetry to detect symmetric objects:
Local constraint:
Global constraint:
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Implementation
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More Results
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Matching Single Symmetry Cells
 We deal with multiple views of a symmetric object.
 Each camera can be reconstructed up to a scale.
 Goal for matching: Finding a global scale factor across multiple
views.
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Matching Multiple Symmetry Cells
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Matching Multiple Symmetry Cells
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3-D Rendering from Symmetry Cells
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