Perspective algebra: quickanddirty first look

1
Perspective algebra quick-and-dirty first look

• Geometry of similar triangles yields algebra for
  computing world-image transformation.

2
Pin hole lens model

• World surface point reflects light through lens
  center to image plane
• Actual pin hole camera can take sharp pictures
  of static scene with long exposure time
• When a lens replaces the pin hole, light is
  collected over an entire cone enabling much
  faster imaging but suffering some optical effects.

a
b
c
Light rays pass through pin hole to image
A
B
C
3
Actual lens model (more later)

• World surface point reflects light through lens
  center to image plane
• When a lens replaces the pin hole, light is
  collected over an entire cone enabling much
  faster imaging but suffering some optical
  effects.
• real image is inverted

a
b
c
X i
Lens and not pinhole
A
B
C
X c
4
Camera looking down 90 deg
5
General trans in 3D
Camera coord sys
Image point in camera coord sys
3D object point in camera coord sys
6
notes

• Similar triangles mean mapping is from mm in
  world to mm in image
• Or inches in world to inches in image
• Can convert mm to pixels
• nominal resolution is patch of world that
  images to a pixel

7
Perspective vs orthographic projection
Perspective distortion gets worse toward the
periphery of the image. Distance c-a is not equal
to distance a-b in image Orthographic is often
used in design and blueprints. True (scaled)
dimensions can be taken from the image. Distance
c-a distance a-b.
c a b
8
Future work in Chs 12 and 13

• Model 2 cameras to get stereo model
• Model camera and projector for structured light
• Use world coordinate system and relate camera
  model to it via rotation and translation