Transcript PPT

Perspective & Renaissance
North Doors (1424)
Lorenzo
Ghiberti
(1378-1455)
East Doors (1452)
Imaging Process
Unsolvable Problem
Recovering 3D geometry from
single 2D projection
Infinite number of possible
solutions!
from [Sinha and Adelson
1993]
Dimensionality Reduction Machine (3D to 2D)
3D world
2D image
Point of observation
What have we lost?
• Angles
• Distances (lengths)
Figures © Stephen E. Palmer, 2002
Funny things happen…
Parallel lines aren’t…
Figure by David Forsyth
Lengths can’t be trusted...
A’
C’
B’
Figure by David Forsyth
Understanding Scenes
Hock, Romanski, Galie, & Williams 1978
• Biederman’s Relations among Objects in a WellFormed Scene (1981):
– Support
– Size
– Position
– Interposition
– Likelihood of
Appearance
Support
Rene Magritte, Golconde
Object Scale /
Camera Viewpoint
Image
Image Coordinates
World Coordinates
Size
Rene Magritte, The Listening Room
Interposition
Rene Magritte, Black Check
Position, Probability, Size
Rene Magritte, Personal Values
Size Constancy
16.24
When we move closer, an object’s projected
image gets bigger, but its perceived size does not.
© Stephen E. Palmer, 2002
Size Constancy
16.25
Size-Distance Relation
The size of an object (h) can be computed from its
retinal size (a) and its distance (d) from the eye.
a
h
d
h = 2d * (tan (a/2))
© Stephen E. Palmer, 2002
Size Constancy
16.26
Hallway Illusion
© Stephen E. Palmer, 2002
Size Constancy
16.27
Hallway Illusion
© Stephen E. Palmer, 2002
Size Constancy
16.28
Relative Size
© Stephen E. Palmer, 2002
Size Constancy
16.29
Moon Illusion
Zenith Moon
dz
Flattened
dz
Sky
Horizon
Moon
Ground Plane
© Stephen E. Palmer, 2002
Size Constancy
16.30
Ponzo Illusion
© Stephen E. Palmer, 2002
Size Constancy
16.31
Ponzo Illusion
© Stephen E. Palmer, 2002
Shape Constancy
17.3
The ability to perceive the constant shape of an
object despite changes in retinal shape due to
viewing perspective.
© Stephen E. Palmer, 2002
Shape Constancy
17.4
3-D Shape constancy
Original view
Good constancy
Poor constancy
© Stephen E. Palmer, 2002
Shape Constancy
17.5
The Ames Room Illusion
2D Projection
Viewing Geometry
© Stephen E. Palmer, 2002
Shape Constancy
17.6
Bias toward “better” shape
Principle of Prägnanz: The percept will be as
“good” as the prevailing conditions allow
© Stephen E. Palmer, 2002
Shape Constancy
17.7
The Ames Room Illusion
2D Projection
Viewing Geometry
© Stephen E. Palmer, 2002
Shape Constancy
17.8
Shepard Illusion
Standard
Comparison A
Comparison B
© Stephen E. Palmer, 2002
Field of View (Zoom)
Field of View (Zoom)
Field of View (Zoom) =
Cropping
FOV depends of Focal Length
f
Smaller FOV = larger Focal Length
From Zisserman & Hartley
Field of View / Focal Length
Large FOV, small f
Camera close to car
Small FOV, large f
Camera far from the car
Fun with Focal Length (Jim
Sherwood)
http://www.hash.com/users/jsherwood/tutes/focal/Zoomin.mov
Vanishing points
image plane
vanishing point
camera
center
ground plane
Vanishing point
• projection of a point at infinity
• Caused by ideal line
Vanishing points (2D)
image plane
vanishing point
camera
center
line on ground plane
Vanishing points
image plane
vanishing point V
camera
center
C
line on ground plane
line on ground plane
Properties
• Any two parallel lines have the same vanishing point v
• The ray from C through v is parallel to the lines
• An image may have more than one vanishing point
Vanishing lines
v1
v2
Multiple Vanishing Points
• Any set of parallel lines on the plane define a vanishing point
• The union of all of these vanishing points is the horizon line
– also called vanishing line
• Note that different planes define different vanishing lines
Vanishing lines
Multiple Vanishing Points
• Any set of parallel lines on the plane define a vanishing point
• The union of all of these vanishing points is the horizon line
– also called vanishing line
• Note that different planes define different vanishing lines
vanishing lines
C
l
ground plane
Properties
• l is intersection of horizontal plane through C with image plane
• Compute l from two sets of parallel lines on ground plane
• All points at same height as C project to l
– points higher than C project above l
• Provides way of comparing height of objects in the scene
Fun with vanishing points
“Tour into the Picture” (SIGGRAPH ’97)
Create a 3D “theatre stage” of
five billboards
Specify foreground objects
through bounding polygons
Use camera transformations to
navigate through the scene
The idea
Many scenes (especially paintings), can be represented
as an axis-aligned box volume (i.e. a stage)
Key assumptions:
• All walls of volume are orthogonal
• Camera view plane is parallel to back of volume
• Camera up is normal to volume bottom
How many vanishing points does the box have?
• Three, but two at infinity
• Single-point perspective
Can use the vanishing point
to fit the box to the particular
Scene!
Fitting the box volume
User controls the inner box and the vanishing point
placement (# of DOF???)
Q: What’s the significance of the vanishing point
location?
A: It’s at eye level: ray from COP to VP is perpendicular
to image plane.
Example of user input: vanishing point and back face of
view volume are defined
High
Camera
Example of user input: vanishing point and back face of
view volume are defined
High
Camera
Example of user input: vanishing point and back face of
view volume are defined
Low
Camera
Example of user input: vanishing point and back face of
view volume are defined
Low
Camera
Comparison of how image is subdivided based on two
different camera positions. You should see how moving
the box corresponds to moving the eyepoint in the 3D
world.
High Camera
Low Camera
Another example of user input: vanishing point and back
face of view volume are defined
Left
Camera
Another example of user input: vanishing point and back
face of view volume are defined
Left
Camera
Another example of user input: vanishing point and back
face of view volume are defined
Right
Camera
Another example of user input: vanishing point and back
face of view volume are defined
Right
Camera
Comparison of two camera placements – left and right.
Corresponding subdivisions match view you would see if
you looked down a hallway.
Left Camera
Right Camera
2D to 3D conversion
First, we can get ratios
left
right
top
vanishing
point
back
plane
bottom
2D to 3D conversion
• Size of user-defined back plane must equal
size of camera plane (orthogonal sides)
• Use top versus side ratio
to determine relative
left
right
height and width
top
dimensions of box
• Left/right and top/bot
camera
ratios determine part of
bottom
pos
3D camera placement
Depth of the box
Can compute by similar triangles (CVA vs. CV’A’)
Need to know focal length f (or FOV)
Note: can compute position on any object on the ground
• Simple unprojection
• What about things off the ground?
DEMO
Now, we know the 3D geometry of the box
We can texture-map the box walls with texture from the
image
Foreground Objects
Use separate billboard for
each
For this to work, three
separate images used:
• Original image.
• Mask to isolate desired
foreground images.
• Background with objects
removed
Foreground Objects
Add vertical
rectangles for
each foreground
object
Can compute 3D
coordinates P0,
P1 since they are
on known plane.
P2, P3 can be
computed as
before (similar
triangles)
Foreground DEMO