Geometric Transformations...Distant objects are smaller (Forsyth & Ponce) Parallel lines meet...
Transcript of Geometric Transformations...Distant objects are smaller (Forsyth & Ponce) Parallel lines meet...
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Geometric Transformations
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Let’s design a camera
• Idea 1: put a piece of film in front of an object • Do we get a reasonable image?
Slide by Steve Seitz
Pinhole camera
• Add a barrier to block off most of the rays – This reduces blurring – The opening is known as the aperture
Slide by Steve Seitz
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Camera Obscura
First Idea:Mo-Ti , China (470 BC to 390 BC)
First built: Ibn Al-Haytham or Alhazen, Iraq/Egypt (965 to 1039AD)
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http://www.acmi.net.au/AIC/CAMERA_OBSCURA.html (Russell Naughton)
Camera Obscura
"When images of illuminated objects ... penetrate through a small hole into a very dark room ... you will see [on the opposite wall] these objects in their proper form and color, reduced in size ... in a reversed position, owing to the intersection of the rays".
Da Vinci
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Pinhole cameras
• Abstract camera model - box with a small hole in it
• Pinhole cameras work in practice
(Forsyth & Ponce)
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Distant objects are smaller
(Forsyth & Ponce)
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Parallel lines meet
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Parallel lines meetCommon to draw image plane in front of the focal point. Moving the image plane merely scales the image.
(Forsyth & Ponce)
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Vanishing points
• Each set of parallel lines meets at a different point – The vanishing point for this direction
• Sets of parallel lines on the same plane lead to collinear vanishing points. – The line is called the horizon for that plane
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Properties of Projection
•Points project to points •Lines project to lines •Planes project to the whole image or a half image
•Angles are not preserved •Degenerate cases
• Line through focal point projects to a point.
• Plane through focal point projects to line
• Plane perpendicular to image plane projects to part of the image (with horizon).
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Take out paper and pencil
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http://www.sanford-artedventures.com/create/tech_1pt_perspective.html
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The equation of projection
(Forsyth & Ponce)
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The equation of projection• Cartesian coordinates:
– We have, by similar triangles, that
– Ignore the third coordinate,
and get
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Intrinsic matrix, K
[xy1] =
f 0 px0 f py0 0 1
[XYZ ]
f 0 px0 f py0 0 1
=1 0 px0 1 py0 0 1
f 0 00 f 00 0 1
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Three different coordinate systems
X
x
Ocamera
OimageOworld
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Camera extrinsic (or external) parameters• Transform a point from the world coordinate
to the camera’s coordinate system • Translation and rotation
Xc = R(Xw − Cw)
Xc = RXw − RCw
XcYcZc1
= [R −RCw0 1 ]
XwYwZw1
[R −RCw0 1 ] = [R 0
0 1][I −Cw0 1 ]
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Camera extrinsic (or external) parameters
XcYcZc1
= [R −RCw0 1 ]
XwYwZw1
[R −RCw0 1 ] = [R 0
0 1][I −Cw0 1 ]
[R 00 1] =
r11 r12 r13 0r21 r22 r23 0r31 r32 r33 00 0 0 1
[I −Cw0 1 ] =
1 0 0 −cx0 1 0 −Cy0 0 1 −Cz0 0 0 1
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Camera extrinsic (or external) parameters
XcYcZc1
=
r11 r12 r13 0r21 r22 r23 0r31 r32 r33 00 0 0 1
1 0 0 −Cx0 1 0 −Cy0 0 1 −Cz0 0 0 1
XwYwZw1
XcYcZc1
= R(I3×3 | − Cw)
XwYwZw1
[xy1] = K
XcYcZc1
K =f s px0 mf py0 0 1
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SummaryXwYwzw
XcYczc
[xy]
world coordinates
camera coordinates
pixel coordinates
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Homogeneous coordinates of a 3-D point
Homogeneous coordinates of its 2-D image
Image of a Point
Projection of a 3-D point to an image plane
P
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