1. Image Formation and Optics
3-D Computer Vision CSc 83020
Image Formation and Optics

2. Image Formation & Optics
Image: 2D projection of a 3D scene.
We need to understand Geometric & Radiometric relations between the scene and its image.

3. Topics: Pinhole & Perspective Projection.
Image Formation using Lenses.
Lens Related Issues.
Image Formation in the Eye.
Our Visual World.

4. Pinhole & the Perspective Projection
(x,y)
SCREEN
SCENE
Is there an image being formed on the screen?

5. Pinhole Camera “Camera obscura” – known since antiquity Image plane
Object
Pinhole camera
Image

6. Perspective Camera r r’ Center of Projection r =[x,y,z]T r’=[X,Y,Z]T
From Trucco & Verri r r’
(x,y,z)
(X,Y,Z)
Center of
Projection
r =[x,y,z]T
r’=[X,Y,Z]T
r/f=r’/Z
x=f * X/Z
y=f * Y/Z
z=f
f: effective focal length:
distance of image plane from O.

7. Magnification x/f=X/Z y/f=Y/Z (x+dx)/f=(X+dX)/Z (y+dy)/z=(Y+dY)/Z
From Trucco & Verri
(x,y)
(X,Y,Z) d
Center of
Projection d’
(x+dx,y+dy)
(X+dX,Y+dY,Z)
x/f=X/Z
y/f=Y/Z
(x+dx)/f=(X+dX)/Z
(y+dy)/z=(Y+dY)/Z
dx/f=dX/Z
dy/f=dY/Z
=>

8. Magnification Magnification: |m|=||d’||/||d||=|f/Z| or m=f/Z
From Trucco & Verri
(x,y)
(X,Y,Z) d
Center of
Projection d’
(x+dx,y+dy)
(X+dX,Y+dY,Z)
Magnification: |m|=||d’||/||d||=|f/Z|
or m=f/Z
m is negative when image is inverted…

9. Magnification Area(image)/Area(scene)=?
m can be assumed to be CONSTANT
if range of scene depth (ΔZ) is much smaller
than average scene depth (Z).

10. Implications For Perception*
Same size things get smaller, we hardly notice…
Parallel lines meet at a point…
* A Cartoon Epistemology:

11. Vanishing Points
(from NALWA)

12. Consequences: Parallel lines meet
There exist vanishing points
Marc Pollefeys

13. Vanishing points Different directions correspond
VPL
VPR
VP1
VP2
Different directions correspond
to different vanishing points
VP3
Marc Pollefeys

14. Question ∞ How many vanishing points are there in an image? 1 2 3 6
100 ∞

15. Approximations Linear approximation to perspective equations.
Orthographic: (m=1 => x=X, y=Y).

16. Approximations Linear approximation to perspective equations.
Weak-Perspective: m is CONSTANT.
x=f*X/Z  f*X/Zavg (Zavg average distance of points from camera)
y=f*Y/Z  f*Y/Zavg
Possible when Zavg is much smaller than ΔZ (relative distance of points along the optical axis).

17. Weak-Perspective Cont.
From Trucco & Verri
OBJECT POINTS
Zavg
Zavg: average distance of points along the optical axis.

18. Ioannis Stamos – CSc 83020 Spring 2007
Approximations
Weak
Perspective
Para
Perspective
Ioannis Stamos – CSc Spring 2007

19. Pictorial Comparison
Weak perspective
Perspective 
Marc Pollefeys

20. Problems with Pinholes
Pinhole size (aperture) must be small.
The smaller the size, the less light goes through.
If pinhole is comparable to wavelength λ of light DIFFRACTION effects blur image.
Pinhole diameter d=2*sqrt(f*λ) for sharp images:
If f=50mm and λ=600nm (red light) then d=0.36mm.

21. Pinhole cameras

22. Lenses Used to avoid problems associated with pinholes.
Ideal Lens: Same projection, but gathers more light!
From Trucco & Verri
f: point of convergence of rays that come from infinity.

23. Thin Lens: Projection f z optical axis Image plane
Spherical lense surface: Parallel rays are refracted to single point

24. Thin Lens: Projection f f z optical axis Image plane
Spherical lense surface: Parallel rays are refracted to single point

25. Thin Lens: Properties
Any ray entering a thin lens parallel to the optical axis must go through the focus on other side
Any ray entering through the focus on one side will be parallel to the optical axis on the other side

26. Ioannis Stamos – CSc 83020 Spring 2007
Lenses
Used to avoid problems associated with pinholes.
Ideal Lens: Same projection, but gathers more light!
Ray of light
From Trucco & Verri
Optical Axis
Ioannis Stamos – CSc Spring 2007

27. Lenses Gaussian Lens Formula for thin lenses: 1/Ž + 1/ž = 1/f
Ž=Z+f, ž=z+f
f: focal length of lens: ability to bend light
Ray of light
Optical Axis
From Trucco & Verri
Example: if f=50mm, Ž=300mm, then image distance
ž =60mm.

28. Blur Circle (Defocus) Ž Ž’ P ž ž’ IMAGE PLANE APERTURE Blur Circle w/
diameter b P p d
(aperture)
OPTICAL
AXIS ž Ž ž’ Ž’

29. Blur Circle (Defocus) Ž’ P ž’ IMAGE PLANE APERTURE Blur Circle w/
diameter b P p d
OPTICAL
AXIS
3D SCENE ž’ Ž’

30. Blur Circle (Defocus) Ž Ž’ ž ž’ IMAGE PLANE APERTURE Blur Circle w/
diameter b
OPTICAL
AXIS
3D SCENE ž Ž ž’ Ž’

31. Blur Circle (Defocus) Ž Ž’ P ž ž’ IMAGE PLANE APERTURE Blur Circle w/
diameter b P p d
(aperture)
OPTICAL
AXIS ž Ž ž’ Ž’

32. b=? 1/ ž+1/ Ž=1/f ž=Ž*f/(Ž-f) 1/ ž’+1/ Ž’=1/f ž’=Ž’*f/(Ž’-f)
Blur Circle Diameter b= | ž’-ž | d / ž’
(from similar triangles)
Depth of Field
Range of object distances (Ž- Ž’) over which
image is “sufficiently well” focused.
i.e. b is less than resolution of imaging sensor.
Note that b is proportional to d (aperture).

33. Ioannis Stamos – CSc 83020 Spring 2007
Aperture & DOF d=
(From KODAK)
Ioannis Stamos – CSc Spring 2007

34. Ioannis Stamos – CSc 83020 Spring 2007
Blur Circle (Defocus)
IMAGE PLANE
APERTURE
Blur Circle w/
diameter b P p d
(aperture)
OPTICAL
AXIS ž Ž ž’ Ž’
Ioannis Stamos – CSc Spring 2007

35. Ioannis Stamos – CSc 83020 Spring 2007
Focusing
Defocused image can be made focused by:
Moving image plane.
Moving the lens.
Both as a single unit.
Ioannis Stamos – CSc Spring 2007

36. Two Lens System Ž2 Ž1 Magnification: m=x’’/x=(i2/o2)*(i1/o1)
From Shree Nayar’s notes Ž2 Ž1 ž2 ž1
Magnification: m=x’’/x=(i2/o2)*(i1/o1)
Zooming: Varying magnification without moving object or
image plane.
Example: Move LENS2 to change m and then move LENS1
and LENS2 together to re-focus.
ZOOMING=CHANGING EFFECTIVE FOCAL LENGTH

37. Ioannis Stamos – CSc 83020 Spring 2007
Thick Lens
From Horn
Ioannis Stamos – CSc Spring 2007

38. Vignetting
From Horn

39. Vignetting
Effect: Darkens pixels near the image boundary

40. Ioannis Stamos – CSc 83020 Spring 2007
From Shree Nayar’s notes
Ioannis Stamos – CSc Spring 2007

41. Can be corrected! (if parameters are know)
Distortion
magnification/focal length different
for different angles of inclination
pincushion
(tele-photo)
barrel
(wide-angle)
Can be corrected! (if parameters are know)
Marc Pollefeys

42. Chromatic Aberration rays of different wavelengths focused
in different planes
cannot be removed completely
Marc Pollefeys

43. Image Formation in the Eye
Optics in the Eye: Iris, Lens, Retina…
Defects in the Eye’s Lens:
Myopia (Near-Sighted)
Hyperopia (Far-Sighted)
Accomodation (Focusing)

44. Ioannis Stamos – CSc 83020 Spring 2007

45. Ioannis Stamos – CSc 83020 Spring 2007
THE HUMAN EYE!
Ioannis Stamos – CSc Spring 2007

47. Ioannis Stamos – CSc 83020 Spring 2007

49. Ioannis Stamos – CSc 83020 Spring 2007

50. Our Visual World Image Formation: 3D => 2D
Can we recover 3D Scene from 2D Image?
We live in a special world!
Medium (Air): Transparent & Homogeneous.
Objects: Opaque & Reflective.
We need to recover surfaces, not volumes.
Is one image enough?