Projective Geometry Conversion Euclidean -> Homogenous ... Projective Geometry • Axioms of...
date post
10-Oct-2020Category
Documents
view
5download
0
Embed Size (px)
Transcript of Projective Geometry Conversion Euclidean -> Homogenous ... Projective Geometry • Axioms of...
1
CS252A, Fall 2017
Computer Vision I
Image Formation and Cameras
Computer Vision I CSE 252A Lecture 4
CS252A, Fall 2017
Computer Vision I
Announcements • HW0 is due on today • HW1 will be assigned tomorrow. • Read Chapters 1 & 2 of Forsyth & Ponce • Everyone is off waitlist • Welcome extension students – send email to
Bolun Zhang [email protected] • (Subset of?) Final exam can be use for CSE
MS Comprehensive Exam [Pending]
CS252A, Fall 2017
Computer Vision I
Projective Geometry • Axioms of Projective Plane
1. Every two distinct points define a line 2. Every two distinct lines define a point (intersect
at a point) 3. There exists three points, A,B,C such that C
does not lie on the line defined by A and B. • Different than Euclidean (affine) geometry • Projective plane is “bigger” than affine
plane – includes “line at infinity”
Projective Plane
Affine Plane = + Line at Infinity
CS252A, Fall 2017
Computer Vision I
Conversion Euclidean -> Homogenous -> Euclidean In 2-D • Euclidean -> Homogenous:
(x, y) -> λ(x,y,1)= (λx, λy, λ)
• Homogenous -> Euclidean: (x, y, w) -> (x/w, y/w)
In 3-D • Euclidean -> Homogenous:
(x, y, z) -> λ(x ,y, z, 1) • Homogenous -> Euclidean:
(x, y, z, w) -> (x/w, y/w, z/w)
X
Y (x,y)
(x,y,1)
1 W
CS252A, Fall 2017
Computer Vision I
The equation of projection: Euclidean & Homogenous Coordinates
Cartesian coordinates:
€
(x,y,z)→( f x z , f y
z )
€
U V W
⎛
⎝
⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟
=
1 0 0 0 0 1 0 0 0 0 1 f 0
⎛
⎝
⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟
X Y Z T
⎛
⎝
⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟
Homogenous Coordinates and Camera matrix
CS252A, Fall 2017
Computer Vision I
Affine Camera Model
• Take perspective projection equation, and perform Taylor series expansion about some point (x0,y0,z0).
• Drop terms that are higher order than linear. • Resulting expression is affine camera model
⎥ ⎥ ⎥
⎦
⎤
⎢ ⎢ ⎢
⎣
⎡
0
0
0
z y x
Appropriate in Neighborhood About (x0,y0,z0)
2
CS252A, Fall 2017
Computer Vision I
Simplified Camera Models Perspective Projection
Scaled Orthographic Projection
Affine Camera Model
Orthographic Projection
Approximation: Taylor Series Expansion about (X0, Y0, Z0)
Case of affine where X0= Y0 = 0
Case of scaled orthograhic where f/Z0 = 1
CS252A, Fall 2017
Computer Vision I
Projective Transformation Mapping from Plane to Plane
• 3 x 3 linear transformation of homogenous coordinates
• Points map to points • Lines map to lines
⎥ ⎥ ⎥
⎦
⎤
⎢ ⎢ ⎢
⎣
⎡
⎥ ⎥ ⎥
⎦
⎤
⎢ ⎢ ⎢
⎣
⎡
=
⎥ ⎥ ⎥
⎦
⎤
⎢ ⎢ ⎢
⎣
⎡
3
2
1
333231
232221
211211
3
2
1
x x x
aaa aaa aaa
u u u
X U
CS252A, Fall 2017
Computer Vision I
^
Figure borrowed from Hartley and Zisserman “Multiple View Geometry in computer vision”
Mapping from a Plane to a Plane under Perspective is given by a projective transform H
x’ = Hx H is a 3x3 matrix, x, x’ are 3x1 vectors of homogenous coordinates
î ĵ
Oπ
CS252A, Fall 2017
Computer Vision I
The equation of projection
Cartesian coordinates:
€
(x,y,z)→( f x z , f y
z )
U V W
⎛
⎝
⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ =
1 0 0 0 0 1 0 0 0 0 1 f 0
⎛
⎝
⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟
X Y Z T
⎛
⎝
⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟
Homogenous Coordinates and Camera matrix
CS252A, Fall 2017
Computer Vision I
^
Oπ
x = xî + yĵ +Oπ
3D point coordinates on plane π
As a matrix equation
X = î ĵ Oπ ⎡ ⎣⎢
⎤ ⎦⎥
x y 1
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥
In homogenous coordinates
x ' y ' w '
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ =
1 0 0 0 0 1 0 0 0 0 1 f 0
⎡
⎣
⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥
î 0
ĵ 0
Oπ 1
⎡
⎣
⎢ ⎢
⎤
⎦
⎥ ⎥
x y 1
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ = H
x y 1
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥
And taking perspective projection onto π’
î ĵX = î0
ĵ 0
Oπ 1
⎡
⎣
⎢ ⎢
⎤
⎦
⎥ ⎥
x y 1
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥
x’ = Hx
4x3
CS252A, Fall 2017
Computer Vision I
Application of Homography: OCRs, scan,…
Homography: Can be estimated from four points.
P1
P3
P4
P2
(0,0) (0,1)
(1,1) (1,0)
3
CS252A, Fall 2017
Computer Vision I Figure borrowed from Hartley and Zisserman “Multiple View Geometry in computer vision”
Planar Homography
x =H1X X’ =H2X
x’ = H2X = H2(H1-1 x) = (H2H1-1)x
Application: Two photos of a white board CS252A, Fall 2017
Computer Vision I Figure borrowed from Hartley and Zisserman “Multiple View Geometry in computer vision”
Planar Homography: Pure Rotation
x’ = H2X = H2(H1-1 x) = (H2H1-1)x
Application: Panoramas
CS252A, Fall 2017
Computer Vision I
Application: Panoramas and image stitching
All images are warped to central image CS252A, Fall 2017
Computer Vision I
Lenses
CS252A, Fall 2010 Computer Vision I
Beyond the pinhole Camera Getting more light – Bigger Aperture
CS252A, Fall 2010 Computer Vision I
Pinhole Camera Images with Variable Aperture
1mm
.35 mm
.07 mm
.6 mm
2 mm
.15 mm
4
CS252A, Fall 2010 Computer Vision I
Limits for pinhole cameras
CS252A, Fall 2010 Computer Vision I
The reason for lenses
CS252A, Fall 2010 Computer Vision I
Thin Lens
O
• Rotationally symmetric about optical axis. • Spherical interfaces.
Optical axis
CS252A, Fall 2010 Computer Vision I
Thin Lens: Center
O
• All rays that enter lens along line pointing at O emerge in same direction.
F
CS252A, Fall 2010 Computer Vision I
Thin Lens: Focus
O
Parallel lines pass through the Focal Point, F
F
CS252A, Fall 2010 Computer Vision I
Thin Lens: Image of Point
O
All rays passing through lens and starting at P converge upon P’
F
P
P’
5
CS252A, Fall 2010 Computer Vision I
Thin Lens: Image of Point
O F
P
P’ z’
f
z
1 z ' − 1 z = 1 f
Where z, z’, and f are the z-coordinates of P, P, and F. i.e., z is negative.
j k
i
CS252A, Fall 2010 Computer Vision I
Thin Lens: Image Plane
O F
P
P’
Image Plane
Q’
Q
A price: Whereas the image of P is in focus, the image of Q isn’t.
CS252A, Fall 2010 Computer Vision I
Thin Lens: Aperture
O
P
P’
Image Plane • Smaller Aperture -> Less Blur • Pinhole -> No Blur
CS252A, Fall 2010 Computer Vision I
Field of View
O Field of View
Im age Plane
f
– Telephoto lens: small field of view (large f) – Wide angle lens: large field of view(small f) – Zoom lens: variable f using compound lens
CS252A, Fall 2010 Computer Vision I
Deviations from the lens model Deviations from this ideal are aberrations Two types
2. chromatic
1. geometrical ❑ spherical aberration ❑ astigmatism ❑ distortion ❑ coma
Aberrations are reduced by combining lenses
Compound lenses CS252A, Fall 2010 Computer V
