More Single View Geometry

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More Single View Geometry

• 15-463 Rendering and Image Processing
• Alexei Efros

with a lot of slides stolen from Steve Seitz
and Antonio Criminisi
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Quiz!
Image B
Image A
Image C
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How can we model this scene?

• Find world coordinates (X,Y,Z) for a few points
• Connect the points with planes to model geometry
• Texture map the planes

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Finding world coordinates (X,Y,Z)

• Define the ground plane (Z0)
• Compute points (X,Y,0) on that plane
• Compute the heights Z of all other points

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Measurements on planes
Approach unwarp, then measure
What kind of warp is this?
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Unwarp ground plane
p
p
Our old friend the homography Need 4 reference
points with world coordinates p (x,y) p
(X,Y,0)
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Holbeins Ambassadors
Can you see something weird?
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Finding world coordinates (X,Y,Z)

• Define the ground plane (Z0)
• Compute points (X,Y,0) on that plane
• Compute the heights Z of all other points

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Comparing heights
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Perspective cues
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Perspective cues
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Comparing heights
Vanishing Point
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Measuring height
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Computing vanishing points (from lines)
v
q2
q1
p2
p1

• Intersect p1q1 with p2q2

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Criminisi 99
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Measuring height without a ruler
Z
C
ground plane

• Compute Z from image measurements
• Need more than vanishing points to do this

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Measuring height
vz
r
vanishing line (horizon)
t0
vx
vy
R
H
b0
b
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Measuring height
vz
r
t0
vanishing line (horizon)
t0
vx
vy
m0
b

• What if the point on the ground plane b0 is not
  known?
• Here the guy is standing on the box
• Use one side of the box to help find b0 as shown
  above

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What if vz is not infinity?
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(No Transcript)
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The cross ratio

• A Projective Invariant
• Something that does not change under projective
  transformations (including perspective projection)

The cross-ratio of 4 collinear points
P4
P3
P2
P1

• Can permute the point ordering
• 4! 24 different orders (but only 6 distinct
  values)
• This is the fundamental invariant of projective
  geometry

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Measuring height
?
T (top of object)
R (reference point)
H
R
B (bottom of object)
ground plane
scene points represented as
image points as
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Measuring height
vz
r
vanishing line (horizon)
t0
vx
vy
R
H
b0
b
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Measuring heights of people
Here we go !
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Forensic Science measuring heights of suspects
Vanishing line
Reference height
Reference height
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Assessing geometric accuracy
Are the heights of the 2 groups of people
consistent with each other?
Flagellation, Piero della Francesca
Estimated relative heights
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Assessing geometric accuracy
The Marriage of the Virgin, Raphael
Estimated relative heights
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Criminisi et al., ICCV 99

• Complete approach
• Load in an image
• Click on lines parallel to X axis
• repeat for Y, Z axes
• Compute vanishing points
• Specify 3D and 2D positions of 4 points on
  reference plane
• Compute homography H
• Specify a reference height
• Compute 3D positions of several points
• Create a 3D model from these points
• Extract texture maps
• Cut out objects
• Fill in holes
• Output a VRML model

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Interactive silhouette cut-out
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Occlusion filling

• Geometric filling by exploiting
• symmetries
• repeated regular patterns
• Texture synthesis
• repeated stochastic patterns

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Occlusion Filling texture synthesis.
Non parametric texture synthesis to fill in
removed areas.
My son cannot walk but he can fly?
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Complete 3D reconstruction
Single View algorithms
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Reconstruction from single photographs
Reconstruction of the garden Hut from a single
image
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Virtual Museum DEMO