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CS252A, Fall 2011 Computer Vision I
Cameras and Radiometry
Computer Vision I CSE 252A Lecture 6
CS252A, Fall 2011 Computer Vision I
Announcements • Assignment 1 Posted to the web page: Due
Tuesday, 10/18/10 • Reading Szeliski Chapter 9, particularly section
9.1
CS252A, Fall 2011 Computer Vision I
When we last met … and what Andrew covered.
CS252A, Fall 2011 Computer Vision I
Homogenous coordinates A way to represent points in a projective space
Use three numbers to represent a point on a projective plane
Add an extra coordinate e.g., (x,y) -> (x,y,1)
Impose equivalence relation (x,y,z) ≈ λ*(x,y,z) such that (λ not 0)
i.e., (x,y,1) ≈ (λx, λy, λ) X
Y (x,y)
(x,y,1)
1 Z
CS252A, Fall 2011 Computer Vision I
Conversion Euclidean -> Homogenous -> Euclidean In 2-D • Euclidean -> Homogenous:
(x, y) -> k (x,y,1)
• Homogenous -> Euclidean: (x, y, z) -> (x/z, y/z)
In 3-D • Euclidean -> Homogenous:
(x, y, z) -> k (x,y,z,1)
• Homogenous -> Euclidean: (x, y, z, w) -> (x/w, y/w, z/w)
X
Y (x,y)
(x,y,1)
1 Z
CS252A, Fall 2011 Computer Vision I
Points at infinity
X
Y
(x,y,1)
1 Z
Point at infinity – zero for last coordinate (x,y,0)
and equivalence relation (x,y,0) ≈ λ*(x,y,0)
No corresponding Euclidean point
(x,y,0)
Projective Plane
Affine Plane = + Line at
Infinity
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CS252A, Fall 2011 Computer Vision I
The equation of projection
Cartesian coordinates:
€
(x,y,z)→( f ' xz, f ' y
z)
€
UVW
⎛
⎝
⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟
=
1 0 0 00 1 0 00 0 1
f ' 0
⎛
⎝
⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟
XYZT
⎛
⎝
⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟
Homogenous Coordinates and Camera matrix
CS252A, Fall 2011 Computer Vision I
A convenient notation
Points: AP1 Leading superscript indicates the coordinate system that the coordinates are with respect to Subscript – an identifier
Rotation Matrices Lower left (Going from this system) Upper left (Going to this system)
To add vectors, coordinate systems must agree To rotate a vector, points coordinate system must agree with lower left of rotation matrix
CS252A, Fall 2011 Computer Vision I
SO(3) • RTR = I, det(R) = 1 -> R-1 = RT
• SO(3) is 3-D manifold • Parameterizations of SO(3)
– Roll-Pitch-Yaw – Euler Angles – Axis Angle (Rodrigues formula) – Cayley’s formula – Matrix Exponential – Quaternions (four parameters + one constraint)
• Not commutative CS252A, Fall 2011 Computer Vision I
Block Matrix Multiplication
What is AB ?
Homogeneous Representation of Rigid Transformations
€
BP1
⎡
⎣ ⎢
⎤
⎦ ⎥ = A
BR BOA
0T 1
⎡
⎣ ⎢
⎤
⎦ ⎥
AP1
⎡
⎣ ⎢
⎤
⎦ ⎥ = A
BR AP + BOA
1
⎡
⎣ ⎢
⎤
⎦ ⎥ = A
BTAP1
⎡
⎣ ⎢
⎤
⎦ ⎥
CS252A, Fall 2011 Computer Vision I
Camera parameters • Issue
– World units (e.g., cm), camera units (pixels) – camera may not be at the origin, looking down the z-axis
• extrinsic parameters – one unit in camera coordinates may not be the same as one
unit in world coordinates • intrinsic parameters - focal length, principal point, aspect ratio,
angle between axes, etc.
• Can be written as u = Mp where u=( u v w)T, p= (x,y,z,t)T
, and M is 3x4 camera matrix
3 x 3 4 x 4: Rigid transformation
CS252A, Fall 2011 Computer Vision I
, estimate intrinsic and extrinsic camera parameters
• See Text book for how to do it. • Camera Calibration Toolbox for Matlab (Bouguet) http://www.vision.caltech.edu/bouguetj/calib_doc/
Camera Calibration
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CS252A, Fall 2011 Computer Vision I
A Hierarchy of 2-D to 2D transformations
€
u1u2u3
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥
=
a11 a12 a13a21 a22 a23a31 a32 a33
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥
x1x2x3
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥
Projective (Homography)
Affine
Similarity
Rigid
€
u1u2u3
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥
=
a11 a12 a13a21 a22 a230 0 1
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥
x1x2x3
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥
€
u1u2u3
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥
=
a11 a12 a13−a12 a11 a230 0 1
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥
x1x2x3
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥
€
u1u2u3
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥
=
a11 a12 a13a21 a22 a230 0 1
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥
x1x2x3
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥
where
€
a11 a12a21 a22
⎡
⎣ ⎢
⎤
⎦ ⎥ =
cosθ sinθ−sinθ cosθ⎡
⎣ ⎢
⎤
⎦ ⎥
CS252A, Fall 2011 Computer Vision I
Figure borrowed from Hartley and Zisserman “Multiple View Geometry in computer vision”
Central Projection
CS252A, Fall 2011 Computer Vision I
Figure borrowed from Hartley and Zisserman “Multiple View Geometry in computer vision”
Planar Homography
CS252A, Fall 2011 Computer Vision I
Figure borrowed from Hartley and Zisserman “Multiple View Geometry in computer vision”
Planar Homography: Pure Rotation
CS252A, Fall 2011 Computer Vision I
Radiometry Read Chapter 4 of Ponce & Forsyth
• Solid Angle • Irradiance • Radiance • BRDF • Lambertian/Phong BRDF
CS252A, Fall 2011 Computer Vision I
, estimate intrinsic and extrinsic camera parameters
• See Text book for how to do it. • Camera Calibration Toolbox for Matlab (Bouguet) http://www.vision.caltech.edu/bouguetj/calib_doc/
Camera Calibration
4
CS252A, Fall 2011 Computer Vision I
Solid Angle
• By analogy with angle (in radians), the solid angle subtended by a region at a point is the area projected on a unit sphere centered at that point
• The solid angle dω subtended by a small patch with area dA at distance r from the origin and at angle θ is:
• Power is energy per unit time
• Radiance: Power traveling at some point in a specified direction, per unit area perpendicular to the direction of travel, per unit solid angle
• Symbol: L(x,θ,φ)
• Units: watts per square meter per steradian : w/(m2sr1)
– CSE 252A
€
L =P
(dAcosα)dω
x
dA
α
dω
(θ, φ)
Power emitted from patch, but radiance in direction different from surface normal
CS252A, Fall 2010 Computer Vision I
Radiance transfer
From definition of radiance
– What is the power received by a small area dA2 at distance r from a small area dA1 emitting radiance L?
€
L =P
(dAcosθ)dω
– From definition of solid angle
• How much light is arriving at a surface?
• Units of irradiance: Watts/m2 • This is a function of incoming angle. • A surface experiencing radiance L
(x,θ,φ) coming in from solid angle dω experiences irradiance:
• Crucial property: Total Irradiance arriving at the surface is given by adding irradiance over all incoming angles Total irradiance is
– CSE 252A
N φ
L(x,θ,φ)
x x
θ
– CSE 252A, Winter 2007 – Image Irradiance !
E ="4
dz'#
$ % &
' ( 2
cos4)*
+ ,
-
. / L
– E: Image irradiance – L: emitted radiance – d : Lens diameter – z’: depth of image plane – α: Angle of patch from optical axis
– δA
– δA’
CS252A, Fall 2011 Computer Vision I
Camera’s sensor
– CSE 252A
• Measured pixel intensity is a function of irradiance integrated over – pixel’s area – over a range of wavelengths – for some period of time
• Ideally, it’s linear to the radiance, but the camera response C(.) may not be linear
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CS252A, Fall 2011 Computer Vision I
Color Cameras
We considered four concepts:
1. Prism (with 3 sensors) 2. Filter mosaic 3. Filter wheel\ 4. Fovean
CS252A, Fall 2011 Computer Vision I
Light at surfaces Many effects when light strikes a
surface -- could be: • transmitted
– Skin, glass • reflected
– mirror • scattered
– milk • travel along the surface and
leave at some other point • absorbed
– sweaty skin
Assume that • surfaces don’t fluoresce
– e.g. scorpions, detergents • surfaces don’t emit light (i.e.
are cool) • all the light leaving a point is
due to that arriving at that point
CS252A, Fall 2011 Computer Vision I
BRDF
^ n (θin,φin)
(θout,φout)
Where ρ is sometimes denoted fr.
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