Understanding Perspective and Image Formation: A Look into Geometry and Radiometry - Prof., Study notes of Computer Science

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• Vision infers world properties form images.

• How do images depend on these properties?

• Two key elements

– Geometry

– Radiometry

– We consider only simple models of these

• Used to observe eclipses (eg., Bacon, 1214-1294)
• By artists (eg., Vermeer).

http://brightbytes.com/cosite/collection2.html (Jack and Beverly Wilgus)

Jetty at Margate England,

• Abstract camera

model - box with a

small hole in it

• Pinhole cameras

work in practice

(Forsyth & Ponce)

(Forsyth & Ponce)

• 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

• 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).

http://www.sanford-artedventures.com/create/tech_1pt_perspective.html

• Cartesian coordinates:
  • We have, by similar

triangles, that

• Ignore the third coordinate,

and get

z

y f z

x x y z f

z

y y f

z

x x f

( , , ) ( ' , ' , f ')

z

y

f

z

x

x y z f

• s is constant for all points.
• Parallel lines no longer converge, they remain

parallel.

• Weak perspective much simpler math.

– Accurate when object is small and distant.

– Most useful for recognition.

• Pinhole perspective much more accurate for

scenes.

– Used in structure from motion.

• When accuracy really matters, must model

real cameras.