Images complex mapping

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Lecture 3

• Images complex mapping
• Asymptotic limits

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Air
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Refraction
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Refraction
Air
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Refraction
Glass
Air
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Refraction
Glass
Air
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Angle of incidence
Glass
Air
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Angle of refraction
Glass
Air
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Two different indices of refraction.
Glass
Air
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Three different indices of refraction,
air, water, and glass.
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Lower n Higher n
Glass
Air
Glass has a higher index of refraction than does
air.
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Snells law
Glass
Air
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Higher velocity Lower velocityof light
of light
Glass
Air
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Lower n Higher n
Glass
Air
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Some refractive indices
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Note that the lights frequency does not change
as it moves from one medium to another, but
wavelength does change, to produce the different
velocities.
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Lens element
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A greater or lesser amount of light energy will
reach the small region of the image shown.
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Depending upon
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Depending upon
- the power per steradian (a measure of solid
angle) reflected off the surface region,
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Depending upon
- the amount of those steradians of reflected
light that make it to the front element of the
compound lens,
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Depending upon
- the duration of the exposure,
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- the amount of energy collected at
this juncture on the image plane will vary. This
variation across the image plane is the image.
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If light from the original very small region
begins
to appear on a more disbursed area of the focal
plane, we say that the camera is out of focus.
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Often, just moving the image plane front or
back is enough to restore focus.
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Unfortunately, different wavelengths of light
within
the visible spectrum are associated with
different n values, for the same material.
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Lens makers challenge of avoiding chromatic
aberrations
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n for various clear glasses.
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With the simple single-element lens, incoming
light from the same source appears on different
locations on the image plane, depending upon
frequency.
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So the path of blue light through the lens
elements would be different for blue light.
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So the path of blue light through the lens
elements would be different for blue light.
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So the path of blue light through the lens
elements would be different for blue light.
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So the path of blue light through the lens
elements would be different for blue light.
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So the path of blue light through the lens
elements would be different for blue light.
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So the path of blue light through the lens
elements would be different for blue light.
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So the path of blue light through the lens
elements would be different for blue light.
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So how do you get blue light from the same point
to focus onto the same small region
on the image plane as the red over a large depth
of field?
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gt By very careful design of the compound lens.
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gt By very careful design of the compound lens.
This and many other challenges make lens design
difficult.
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One consequence of all this complexity is that
mapping from 3D physical space into camera
space is not mathematically simple.
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not mathematically simple.
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Such complexity makes difficult the matter of
visually guided robot control.
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Consider the much simpler pinhole camera model.
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Consider the pinhole camera model.
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Consider the pinhole camera model.
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Consider the pinhole camera model.
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Consider the pinhole camera model.
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Consider the pinhole camera model.
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pinhole, or perspective mapping.
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If chromatic and other abberations are
eliminated, even with real compound lenses the
laser spots will appear in unique camera-space
locations in the two cameras images.
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Laser leaves a small spot on the surface.
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Image of the spot in camera 1.
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Image of the spot in camera 2.
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How can we know well deliver the pen tip there?
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Blue pen tip will be in its place if
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it finds it target in camera 1, and
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it finds it target in camera 2.
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This is true regardless of the complexity of the
real-lens mapping.
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If three or more cameras agree, so much the
better.
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What if we wanted to bear down, leave a mark?
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In the absence of seeing the blue mark through
to c.s. success, we need to return to our lens
mapping. So lets think about a regular map for
a while.
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Just as a flat map accommodates the real
(bulgy, mostly round) world well, over a
sufficiently contained region, so too the simple
pinhole camera model can be relied upon within a
sufficiently small region of physical space.
pinhole, or perspective mapping.
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pinhole, or perspective mapping.
A similar idea can be used for the imperfectly
modeled robot kinematics.
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