Outline
Moebius Transformations and
Omnidirectional Images
Luiz Velho
IMPA
• Moebius Tranformations
- Mathematical Fundamentals
• Omnidirectional Images
- Basic Concepts
- 360 Panoramas • Applications
- Wide Field of View
Möbius Transformations
• Complex Map
M : C 7! C
!
Moebius
Transformations
!
• Definition:
M (z) =
az + b
cz + d
with
(ad
Anatomy of M
• Decomposition into Sequence
m4 m3 m2 m1 (z)
Fixing the Inversion
• Point at Infinity
1
!
!
m1 (z) = z +
d
c
translation
m2 (z) =
1
z
inversion
m3 (z) =
(bc ad)
z
c2
scaling and rotation
m4 (z) = z +
a
c
translation
bc) 6= 0
1
1
=0
1
0
!
• Extended Complex Plane
b = C [ {1}
C
=1
Riemann Sphere
Complex Projective Space
• Stereographic Projection
• Isomorphism
!
N0
point at infinity
!
z 7! w = M (z)
in
Ĉ
induces
!
!
ẑ 7! ŵ
• Geometry
ẑ = (✓, ) 7! z = cot( /2)ei✓
(Lie Group)
!
2
!
•
= [ 1,
!
2
2
6= 0
(b/c),
(c/d)
az+b
cz+d
= M (z) =
kaz+kb
kcz+kd
!
• Normalization
(ad
bc) = 1
Cross Ratio
• The unique
2 ] 6= [0, 0]
(w
(w
Two Cases
(a/b),
• Ratios and Uniqueness
Homogeneous Coordinates
• Ratio of 2 complex numbers
1
Complex
Plane
• Images of 3 points (e.g)
• Preservation of:
- Circles (lines to circles)
- Angles (conformal)
- Symmetry (w.r.t. circles)
z=
and Algebra
Defining M
P
GL(2, C)
!
!
⌃
Riemann Sphere
Properties of M
• Projective Linear Group
in
z!
7 w = M (z)
sending q, r, s 7! q̃, r̃, s̃
!
q̃)(r̃
!
s̃)(r̃
(z
s̃)
= [w, q̃, r̃, s̃] = [z, q, r, s] =
q̃)
(z
q)(r
s)(r
!
z=
1/
=0
z=1
2
• Theorem:
If M maps 4 points
p, q, r, s 7! p̃, q̃, r̃, s̃
then, the cross-ratio is invariant.
s)
q)
Orientation Properties
• Maps Oriented Circles to Oriented Circles
- s.t. Regions are mapped accordingly
Fixed Points
• Solution of
!
z = M (z)
• M has at most two fixed points
- except for Id.
• For M Normalizedp
⇠± =
M - Classification
• Fixed Point at Infinity :
!
c=0
M (z) = Az + B
!
•
(a+d)2 4
2c
Elliptic Transform
• Rotation
!
!
Basic Types
-
(a d)±
Elliptic
Hyperbolic
Parabolic
Loxodromic
z! 7! ei↵ z
!
• two fixed points
(0, 1)
Hyperbolic Transform
Loxodromic Transform
• Scaling
• Rotation and Scaling
!
!
z 7! ⇢z
!
z! 7! ⇢ei↵ z
!
!
!
• two fixed points
(0, 1)
!
• two fixed points
(combination of elliptic and hyperbolic)
Parabolic Transform
• Translation
!
Omnidirectional Images
!
z 7! z + b
!
!
• one fixed point at 1
Basic Concepts
Plenoptic Function
Complete description of Visual Information
in a 3D environment
• Plenoptic Function
• Capturing Light Fields
• 360 Panoramas
• Parametrization and Projections
•
I = P (x, y, z, ✓, , t)
Holographic Image
•
P : R3 ⇥ S2 ⇥ R 7! E
t
6D Phase Space
x,y,z
✓,
Light Field
Omnindirectional Image
A Slice of the Plenoptic Function
The Set of All Rays incident at a point (x,y,z)
• Structured Sampling of P
- example: Camera
• Spherical Light Field
= 360 degrees
Image
x,y,z
x,y,z fixed
Ray Space
Representation of Choice
Capturing Point Light Fields
Catadioptric Cameras
• Mirror-Based
• Omnidirectional Cameras
- Catadioptric
- Dioptric
- Multi-Camera
camera
Dioptric Cameras
(parabolic or hyperbolic)
mirror
Multi-Camera Systems
• Point Grey's Ladybug
• Fish Eye Lenses
image
(6 Perspective Cameras)
Cam 5
Cam 3
Cam 2
Cam 4
Cam 1
Cam 0
Cam 0
camera
lens
Cam 1
Cam 2
Cam 3
Cam 4
Cam 5
image
Practical Option
Panoramic Surfaces
Parametrizations
Generalized Support for Visual Information
Maps 2D Surface to Planar Domain
• Data Representation
- example: Cilindrical Panorama
• Coordinate Systems
- example: Cilindrical Mapping
7! h
✓
360˚ Image Formats
Omnidirectional Panoramas
Equirectangular Projection
• Latitude-Longitude Mapping
(e.g., Flickr)
!
• Parametrizations of the Sphere
- Lat-Long
- Cube Map
- Azimuthal
- Stereographic (*)
✓
natural coordinate system
distortion toward poles
Most Convenient Format
Cube Mapping
• 6 Perspective Projections
suitable for CG rendering
Azimuthal Projection
• Hemispherical Mapping
Dome Master standard
Exhibition
• Viewing Scenarios
Applications
to
360 Cinema
conventional theater
full dome
Field of View
Film Language
• Reference to Observer
- 30 to 90 degrees
• Conventional Cinema
- HD Television
- Theater Panavision
• 360 Degrees Dome
- Omnimax
- Dome Master
Conventional Cinema
• Camera Moves
Track
360 Camera
• Camera Moves
Pan / Tilt
Zoom
Authoring Issues
Track
Pan / Tilt
yes
maybe
Zoom
?
360˚ Image Transforms
OBS: Post-Production
• Passive
- Movies
• Interactive
- Google Street View
• Immersive
- AR Cinema
Moebius Transformations for Manipulation and
Visualization of Spherical Panoramas
Emerging Technologies
• Current Research at VISGRAF Lab
• Collaboration with
- Leonardo Koller Sacht
- Luis Penaranda
Math of Camera Moves
Transformation Pipeline
• Omnidirectional Images and
• Möbius Mapping
Moebius Transformations -
Pan / Tilt
Elliptic Transform Zoom
Hyperbolic Transform M
S(i)
S
1
(i)
!
-
Perspective
Parabolic Transform ?
input
equirectangular image
Example
complex
plane
representation
scaled
complex
image
final
equirectangular image
Comparison
• Alternative Projections
• Extreme Zoom
input panorama
equirectangular
projective
mercator
equi-rectangular
Current Work
• Preserving Lines
!
• Perspective Control
Control of
Perspective
möbius
Questions?