Homogeneous representation

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Homogeneous representation

• Points
• Vectors
• Transformation
• representation

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Lecture 4 Image formation
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Image Formation

• 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

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Image formation (Chapter 3)
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Representation of images
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Similar triangles ltPFSgt,ltROFgt and ltPSFgtltQOFgt ?
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Pinhole model
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Forward pinhole
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Distant objects are smaller
(Forsyth Ponce)
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Parallel lines meet
Common to draw image plane in front of the focal
point. Moving the image plane merely scales the
image.
(Forsyth Ponce)
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Vanishing points

• 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

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Properties of Projection

• 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).

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Orthographic projection
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Cameras with Lenses
(Forsyth Ponce)
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Assumptions for thin lens equation

• Lens surfaces are spherical
• Incoming light rays make a small angle with the
  optical axis
• The lens thickness is small compared to the radii
  of curvature
• The refractive index is the same for the media on
  both sides of the lens

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Blur circle
Points a t distance are brought into focus
at distance
is imaged at point
A point at distance
from the lens
and so
Thus points at distance will give rise to a
blur circle of diameter
with d the diameter of the lens
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Interaction of light with matter

• Absorption
• Scattering
• Refraction
• Reflection
• Other effects
• Diffraction deviation of straight propagation in
  the presence of obstacles
• Fluorescenceabsorbtion of light of a given
  wavelength by a fluorescent molecule causes
  reemission at another wavelength

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Refraction
n1, n2 indexes of refraction
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Solid Angle
hemisphere
radian
dq
q
Sphere 4p
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Radiometric Terms
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Irradiance and Radiance
Irradiance Definition power per unit area
incident on a surface
W/m2 lux
Radiance Definition power per unit area and
projected solid angle
W/m2sr
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Radiant Intensity
Definition flux per unit solid angle
? Radiant flux W
W/sr cd (candela)
?


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Isotropic Point Source
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Isotropic Point Source
? Radiant flux W
All directions solid angle 4p

• Radiant flux per
• unit solid angle W/sr

r
Radiant intensity

• Note inverse square law fall off.

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Isotropic Point Source
? Radiant flux W
All directions solid angle 4p

• Radiant flux per
• unit solid angle W/sr

r
Radiant intensity
h

• Note cosine dependency.

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Isotropic Point Source
Point source at a finite distance
r
h

• Note inverse square law fall off.

• Note cosine dependency.

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Irradiance from Area Sources
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Hemispherical Source
L
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Reflectance
The surface becomes a light source
Li(x,wi)
qi
Ei
dLrfr dEi
Reflectance ratio of radiance to irradiance
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BRDF
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BRDF
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Reflection Equation
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qi
Perfectly Diffuse Reflection

• Perfectly Diffuse Surface
• Appears equally bright from all viewing
  directions (qr, fr)
• Reflects all incident light, i.e.,

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qi
Common Diffuse Reflection

• Normal Diffuse Surface
• Appears almost equally bright from most viewing
• directions (qr, fr), qr ltlt
  90
• Reflects only a fraction of incident light, i.e.,

Reflectance Albedo
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Perfectly Diffuse Reflection
Distant point light source
Lambertian cosine Law
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Law of Reflection
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Perfectly Specular Reflection
From the definition of BRDF, the surface radiance
is
To satisfy
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Lambertian Examples
Lambertian sphere as the light moves. (Steve
Seitz)
Scene (Oren and Nayar)
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Lambertian Specular Model
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Lambertian specular

• Two parameters how shiny, what kind of shiny.
• Advantages
• easy to manipulate
• very often quite close true
• Disadvantages
• some surfaces are not
• e.g. underside of CDs, feathers of many birds,
  blue spots on many marine crustaceans and fish,
  most rough surfaces, oil films (skin!), wet
  surfaces
• Generally, very little advantage in modelling
  behaviour of light at a surface in more detail --
  it is quite difficult to understand behaviour of
  LS surfaces (but in graphics???)

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LambertianSpecularAmbient

(http//graphics.cs.ucdavis.edu/GraphicsNotes/Shad
ing/Shading.html)