EEM 561 Machine Vision Week 10 :Image Formation and Cameras
Spring 2015 Instructor: Hatice nar Akakn, Ph.D.
[email protected] Anadolu University
Slide 2
Figures Stephen E. Palmer, 2002 Image formation 3D world2D
image Slide source: A.Torralba Images are projections of the 3-D
world onto a 2-D plane
Slide 3
Image formation Lets design a camera Idea 1: put a piece of
film in front of an object Do we get a reasonable image? Slide
source: Seitz
Slide 4
Pinhole camera Add a barrier to block off most of the rays This
reduces blurring The opening known as the aperture How does this
transform the image? Slide source: Seitz The barrier blocks off
most of the rays It gets inverted!!
Slide 5
Light rays from many different parts of the scene strike the
same point on the paper. Forsyth & Ponce Each point on the
image plane sees light from only one direction, the one that passes
through the pinhole.
Slide 6
Pinhole camera Figure from Forsyth f f = focal length c =
center of the camera c If we treat pinhole as a point, only one ray
from any given point can enter the camera. Pinhole camera is a
simple model to approximate imaging process, perspective
projection
Slide 7
Photograph by Abelardo Morell, 1991 Pinhole camera Slide
source: A.Torralba
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Photograph by Abelardo Morell, 1991 Pinhole camera Slide
source: A.Torralba
Slide 9
Photograph by Abelardo Morell, 1991 Pinhole camera Slide
source: A.Torralba
Slide 10
Photograph by Abelardo Morell, 1991 Pinhole camera Slide
source: A.Torralba
Slide 11
Effect of pinhole size Wandell, Foundations of Vision, Sinauer,
1995
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Shrinking the aperture Why not make the aperture as small as
possible? Less light gets through Diffraction effects... Slide
source:N.Snavely
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Shrinking the aperture Slide source:N.Snavely
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Camera obscura: The pre- camera "Reinerus Gemma-Frisius,
observed an eclipse of the sun at Louvain on January 24, 1544, and
later he used this illustration of the event in his book De Radio
Astronomica et Geometrica, 1545. It is thought to be the first
published illustration of a camera obscura..." Hammond, John H.,
The Camera Obscura, A Chronicle
http://www.acmi.net.au/AIC/CAMERA_OBSCURA.html In Latin, means dark
room
Slide 16
Camera Obscura
Slide 17
Camera obscura Illustration of Camera ObscuraFreestanding
camera obscura at UNC Chapel Hill Photo by Seth Ilys
Slide 18
Camera obscura at home Sketch from
http://www.funsci.com/fun3_en/sky/sky.htm
http://blog.makezine.com/archive/2006/02/how_to_room_
sized_camera_obscu.html
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Accidental pinhole camera See Zomet, A.; Nayar, S.K. CVPR 2006
for a detailed analysis. Outside scene * Aperture Slide source:
A.Torralba
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Measuring distance Object size decreases with distance to the
pinhole There, given a single projection, if we know the size of
the object we can know how far it is. But for objects of unknown
size, the 3D information seems to be lost.
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Adding a lens A lens focuses light onto the film There is a
specific distance at which objects are in focus other points
project to a circle of confusion in the image Changing the shape of
the lens changes this distance circle of confusion Slide
source:N.Snavely
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Cameras with lenses focal point F optical center (Center Of
Projection) A lens focuses parallel rays onto a single focal point
Gather more light, while keeping focus; make pinhole perspective
projection practical Slide source:K.Grauman
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Thin lens equation Any object point satisfying this equation is
in focus Slide source:K.Grauman
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Combining Lenses
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The eye The human eye is a camera Iris - colored annulus with
radial muscles Pupil - the hole (aperture) whose size is controlled
by the iris Whats the film? photoreceptor cells (rods and cones) in
the retina
Slide 26
Perspective projection y y f z camera world Cartesian
coordinates: We have, by similar triangles, that (x, y, z) -> (f
x/z, f y/z, -f) Ignore the third coordinate, and get Slide source:
A.Torralba f: focal length O: camera center
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Points go to points Lines go to lines Planes go to whole image
or half-planes. Polygons go to polygons Degenerate cases line
through focal point to point plane through focal point to line
Geometric properties of projection Slide source: A.Torralba
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Modeling projection Is this a linear transformation?
Homogeneous coordinates to the rescue! homogeneous image
coordinates homogeneous scene coordinates Converting from
homogeneous coordinates nodivision by z is nonlinear Slide by Steve
Seitz
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Perspective Projection Matrix divide by the third coordinate to
convert back to non-homogeneous coordinates Projection is a matrix
multiplication using homogeneous coordinates: Slide by Steve
Seitz
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Perspective Projection -- Ideal Case
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Slide 32
Orthographic projection Given camera at constant distance from
scene World points projected along rays parallel to optical
access
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Projection properties Parallel lines converge at a vanishing
point Each direction in space has its own vanishing point But
parallels parallel to the image plane remain parallel Slide
source:N.Snavely
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Vanishing points and lines Vanishing point Vanishing line
Vanishing point Vertical vanishing point (at infinity) Slide from
Efros, Photo from Criminisi source:J.Hays
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Homogeneous coordinates 2D Points: 2D Lines: d (n x, n y )
Slide source: A.Torralba
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Homogeneous coordinates Intersection between two lines: Slide
source: A.Torralba
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Homogeneous coordinates Line joining two points: Slide source:
A.Torralba
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2D Transformations tx ty = + 1 = 10tx 01ty. = 10tx 01ty 001.
Example: translation Now we can chain transformations Slide source:
A.Torralba
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Recall:Summary of Affine Transformations
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More Realistic Perspective Projection
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Perspective projection (intrinsics) in general, : aspect ratio
(1 unless pixels are not square) : skew (0 unless pixels are shaped
like rhombi/parallelograms) : principal point ((0,0) unless optical
axis doesnt intersect projection plane at origin) (upper triangular
matrix) (converts from 3D rays in camera coordinate system to pixel
coordinates) Slide source:N.Snavely
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 Intrinsic Assumptions
Unit aspect ratio No skew Extrinsic Assumptions No rotation Camera
at (0,0,0)
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Remove assumption: square pixels Intrinsic Assumptions No skew
Extrinsic Assumptions No rotation Camera at (0,0,0)
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Remove assumption: non-skewed pixels Intrinsic
AssumptionsExtrinsic Assumptions No rotation Camera at (0,0,0)
Note: different books use different notation for parameters Slide
Credit: J. Hays
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Oriented and Translated Camera OwOw iwiw kwkw jwjw t R Slide
Credit: J. Hays
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Allow camera translation Intrinsic AssumptionsExtrinsic
Assumptions No rotation Slide Credit: J. Hays
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3D Rotation of Points, : Rotation around the coordinate axes,
counter-clockwise: p pppp y z Slide Credit: Saverese
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Allow camera rotation Slide source:J.Hays
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Degrees of freedom 56 How many known points are needed to
estimate this? Slide source:J.Hays
Slide 52
Camera calibration Use the camera to tell you things about the
world: Relationship between coordinates in the world and
coordinates in the image: geometric camera calibration, see
Szeliski, section 5.2, 5.3 for references (Relationship between
intensities in the world and intensities in the image: photometric
image formation, see Szeliski, sect. 2.2.) Slide source:
A.Torralba
Slide 53
Things to remember Vanishing points and vanishing lines Pinhole
camera model and camera projection matrix Homogeneous coordinates
Vanishing point Vanishing line Vanishing point Vertical vanishing
point (at infinity)