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Projective Geometry and Camera Models
Computer VisionCS 543 / ECE 549
University of Illinois
Derek Hoiem
01/21/10
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Note about HW1
• Out before next Tues
• Prob1: covered today, Tues
• Prob2: covered next Thurs
• Prob3: covered following week
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Last class: intro
• Overview of vision, examples of state of art
• Logistics
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Next two classes: Single‐view Geometry
How tall is this woman?
Which ball is closer?
How high is the camera?
What is the camera rotation?
What is the focal length of the camera?
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Today’s class
Mapping between image and world coordinates
– Pinhole camera model
– Projective geometry• homogeneous coordinates and vanishing lines
– Camera matrix
– Other camera parameters
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Image formation
Let’s design a camera– Idea 1: put a piece of film in front of an object– Do we get a reasonable image?
Slide source: Seitz
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Pinhole camera
Idea 2: add a barrier to block off most of the rays– This reduces blurring
– The opening known as the aperture
Slide source: Seitz
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Pinhole camera
Figure from Forsyth
f
f = focal lengthc = center of the camera
c
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Camera obscura: the pre‐camera
• First idea: Mo‐Ti, China (470BC to 390BC)
• First built: Alhacen, Iraq/Egypt (965 to 1039AD)
Illustration of Camera Obscura Freestanding camera obscura at UNC Chapel Hill
Photo by Seth Ilys
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First Photograph
First photograph– Took 8 hours on pewter plate
Joseph Niepce, 1826
Photograph of the first photograph
Stored at UT Austin
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Point of observation
Figures © Stephen E. Palmer, 2002
Dimensionality Reduction Machine (3D to 2D)
3D world 2D image
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Projection can be tricky…Slide source: Seitz
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Projection can be tricky…Slide source: Seitz
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Projective Geometry
What is lost?• Length
Which is closer?
Who is taller?
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Length is not preserved
Figure by David Forsyth
B’
C’
A’
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Projective Geometry
What is lost?• Length
• Angles
Perpendicular?
Parallel?
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Projective Geometry
What is preserved?• Straight lines are still straight
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Vanishing points and lines
Parallel lines in the world intersect in the image at a “vanishing point”
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Vanishing points and lines
oVanishing Point oVanishing Point
Vanishing Line
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Vanishing points and lines
Vanishingpoint
Vanishingline
Vanishingpoint
Vertical vanishingpoint
(at infinity)
Slide from Efros, Photo from Criminisi
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Vanishing points and lines
Photo from online Tate collection
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Note on estimating vanishing points
Use multiple lines for better accuracy… but lines will not intersect at exactly the same point in practice
One solution: take mean of intersecting pairs… bad idea!
Instead, minimize angular differences
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(more vanishing points on board)
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Vanishing objects
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Projection: world coordinates image coordinates
(work on board)
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Homogeneous coordinates
Conversion
Converting to homogeneous coordinates
homogeneous image coordinates
homogeneous scene coordinates
Converting from homogeneous coordinates
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Homogeneous coordinates
Invariant to scaling
Point in Euclidean is ray in Homogeneous
⎥⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡⇒
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
wywx
kwkykwkx
kwkykx
wyx
k
Homogeneous Coordinates
Euclidean Coordinates
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Basic geometry in homogeneous coordinates
• Line equation: ax + by + c = 0
• Append 1 to pixel coordinate to get homogeneous coordinate
• Line given by cross product of two points
• Intersection of two lines given by cross product of the lines
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=1i
i
i vu
p
jiij ppline ×=
jiij linelineq ×=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
i
i
i
i
cba
line
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Another problem solved by homogeneous coordinates
Euclidean: (Inf, Inf)Homogeneous: (1, 1, 0)
Intersection of parallel linesEuclidean: (Inf, Inf) Homogeneous: (1, 2, 0)
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Slide Credit: Saverese
Projection matrix
[ ]XtRKx = x: Image Coordinates: (u,v,1)K: Intrinsic Matrix (3x3)R: Rotation (3x3) t: Translation (3x1)X: World Coordinates: (X,Y,Z,1)
Ow
iw
kw
jwR,T
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Interlude: when have I used this stuff?
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When have I used this stuff?
Object Recognition (CVPR 2006)
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When have I used this stuff?
Single‐view reconstruction (SIGGRAPH 2005)
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When have I used this stuff?
Getting spatial layout in indoor scenes (ICCV 2009)
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When have I used this stuff?
Inserting photographed objects into images (SIGGRAPH 2007)
Original Created
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When have I used this stuff?
Inserting synthetic objects into images (submitted)
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[ ]X0IKx =⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
10100000000
1zyx
ff
vu
k
K
Slide Credit: Saverese
Projection matrix
Intrinsic Assumptions• Unit aspect ratio• Optical center at (0,0)• No skew
Extrinsic Assumptions• No rotation• Camera at (0,0,0)
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Remove assumption: known optical center
[ ]X0IKx =⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
101000000
10
0
zyx
vfuf
vu
k
Intrinsic Assumptions• Unit aspect ratio• No skew
Extrinsic Assumptions• No rotation• Camera at (0,0,0)
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Remove assumption: square pixels
[ ]X0IKx =⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
101000000
10
0
zyx
vu
vu
k βα
Intrinsic Assumptions• No skew
Extrinsic Assumptions• No rotation• Camera at (0,0,0)
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Remove assumption: non‐skewed pixels
[ ]X0IKx =⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
10100000
10
0
zyx
vus
vu
k βα
Intrinsic Assumptions Extrinsic Assumptions• No rotation• Camera at (0,0,0)
Note: different books use different notation for parameters
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Oriented and Translated Camera
Ow
iw
kw
jw
t
R
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Allow camera translation
[ ]XtIKx =⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
1100010001
1000
0
10
0
zyx
ttt
vu
vu
z
y
x
βα
Intrinsic Assumptions Extrinsic Assumptions• No rotation
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3D Rotation of Points
Rotation around the coordinate axes, counter-clockwise:
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡ −=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−=
1000cossin0sincos
)(
cos0sin010
sin0cos)(
cossin0sincos0001
)(
γγγγ
γ
ββ
βββ
ααααα
z
y
x
R
R
R
p
p’
γ
y
z
Slide Credit: Saverese
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Allow camera rotation
[ ]XtRKx =
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
1100
01 333231
232221
131211
0
0
zyx
trrrtrrrtrrr
vus
vu
z
y
x
βα
![Page 45: Projective Geometry and Camera Models - University Of Illinoisdhoiem.cs.illinois.edu/courses/vision_spring10/lectures/Lecture2... · Projective Geometry and Camera Models Computer](https://reader034.vdocument.in/reader034/viewer/2022042215/5ebbee0b13d71227b768637d/html5/thumbnails/45.jpg)
Degrees of freedom
[ ]XtRKx =
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
1100
01 333231
232221
131211
0
0
zyx
trrrtrrrtrrr
vus
vu
k
z
y
x
βα
5 6
![Page 46: Projective Geometry and Camera Models - University Of Illinoisdhoiem.cs.illinois.edu/courses/vision_spring10/lectures/Lecture2... · Projective Geometry and Camera Models Computer](https://reader034.vdocument.in/reader034/viewer/2022042215/5ebbee0b13d71227b768637d/html5/thumbnails/46.jpg)
Scaled Orthographic Projection
• Special case of perspective projection– Distance from the COP to the PP is infinite
– Also called “parallel projection”
– What’s the projection matrix?
Image World
Slide by Steve Seitz
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
1100000100001
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Other things to be aware of
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Radial Distortion
Image from Martin Habbecke
Corrected Barrel Distortion
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Focal length, aperture, depth of field
A lens focuses parallel rays onto a single focal point– focal point at a distance f beyond the plane of the lens
– Aperture of diameter D restricts the range of rays
focal point
F
optical center(Center Of Projection)
Slide source: Seitz
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Depth of field
Changing the aperture size or focal length affects depth of field
f / 5.6
f / 32
Flower images from Wikipedia http://en.wikipedia.org/wiki/Depth_of_field
Slide source: Seitz
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Varying the aperture
Large apeture = small DOF Small apeture = large DOF
Slide from Efros
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Shrinking the aperture
• Why not make the aperture as small as possible?– Less light gets through– Diffraction effects
Slide by Steve Seitz
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Shrinking the aperture
Slide by Steve Seitz
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Relation between field of view and focal length
Dolly zoom (or “Vertigo effect”)http://www.youtube.com/watch?v=Y48R6‐iIYHs
Field of view (angle width) Film/Sensor Width
Focal lengthfdfov 2tan 1−=
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Things to remember
• Vanishing points and vanishing lines
• Pinhole camera model and camera projection matrix
• Homogeneous coordinates
Vanishingpoint
Vanishingline
Vanishingpoint
Vertical vanishingpoint
(at infinity)
[ ]XtRKx =
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Next class
• Applications of camera model and projective geometry– Recovering the camera intrinsic and extrinsic parameters from an image
– Recovering size in the world
– Projecting from one plane to another (if time allows)
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Questions